Planet Musings

October 28, 2025

Scott Aaronson An Experimental Program for AI-Powered Feedback at STOC: Guest Post from David Woodruff

This year for STOC, we decided to run an experiment to explore the use of Large Language Models in the theoretical computer science community, and we’re inviting the entire community to participate.

We—a team from the STOC PC—are offering authors the chance to get automated pre-submission feedback from an advanced, Gemini-based LLM tool that’s been optimized for checking mathematical rigor. The goal is simple: to provide constructive suggestions and, potentially, help find technical mistakes before the paper goes to the PC. Some important points:

  • This is 100% optional and opt-in.
  • The reviews generated WILL NOT be passed on to the PC. They are for your eyes only.
  • Data Privacy is Our #1 Commitment. We commit that your submitted paper will NOT be logged, stored, or used for training.
  • Please do not publicly share these reviews without contacting the organizing team first.

This tool is specifically optimized for checking a paper’s mathematical rigor. It’s a hopefully useful way to check the correctness of your arguments. Note that sometimes it does not possess external, area-specific knowledge (like “folklore” results). This means it may flag sections that rely on unstated assumptions, or it might find simple omissions or typos.

Nevertheless, we hope you’ll find this feedback valuable for improving the paper’s overall clarity and completeness.

If you’re submitting to STOC, we encourage you to opt-in. You’ll get (we hope) useful feedback, and you’ll be providing invaluable data as we assess this tool for future theory conferences.

The deadline to opt-in on the HotCRP submission form is November 1 (5pm EST).

You can read the full “Terms of Participation” (including all privacy and confidentiality details) at the link below.

This experiment is being run by PC members David Woodruff (CMU) and Rajesh Jayaram (Google), as well as Vincent Cohen-Addad (Google) and Jon Schneider (Google).

We’re excited to offer this resource to the community.

Please see the STOC Call for Papers here and specific details on the experiment here.

n-Category Café Applied Category Theory 2026

The next annual conference on applied category theory is in Estonia!

For more details, read on!

The conference particularly encourages participation from underrepresented groups. The organizers are committed to non-discrimination, equity, and inclusion. The code of conduct for the conference is available here.

Deadlines

  • Registration: TBA
  • Abstracts Due: 23 March 2026
  • Full Papers Due: 30 March 2026
  • Author Notification: 11 May 2026
  • Adjoint School: 29 June – 3 July 2026
  • Conference: 6 – 10 July 2026
  • Final versions of papers for proceedings due: TBA

Submissions

ACT2026 accepts submissions in English, in the following three tracks:

  1. Research

  2. Software demonstrations

  3. Teaching and communication

The detailed Call for Papers is available here.

Extended abstracts and conference papers should be prepared with LaTeX. For conference papers please use the EPTCS style files available here. The submission link is here.

Reviewing is single-blind, and we are not making public the reviews, reviewer names, the discussions nor the list of under-review submissions. This is the same as previous instances of ACT.

Program Committee Chairs

  • Geoffrey Cruttwell, Mount Allison University, Sackville
  • Priyaa Varshinee Srinivasan, Tallinn University of Technology, Estonia

Program Committee

  • Alexis Toumi, Planting Space
  • Bryce Clarke, Tallinn University of Technology
  • Barbara König, University of Duisburg-Essen
  • Bojana Femic, Serbian Academy of Sciences and Arts
  • Chris Heunen, The University of Edinburgh
  • Daniel Cicala, Southern Connecticut State University
  • Dusko Pavlovic, University of Hawaii
  • Evan Patterson, Topos Institute
  • Fosco Loregian, Tallinn University of Technology
  • Gabriele Lobbia, Università di Bologna
  • Georgios Bakirtzis, Institut Polytechnique de Paris
  • Jade Master, University of Strathclyde
  • James Fairbanks, University of Florida
  • Jonathan Gallagher, Hummingbird Biosciences
  • Joe Moeller, Caltech
  • Jules Hedges, University of Strathclyde
  • Julie Bergner, University of Virginia
  • Kohei Kishida, University of Illinois, Urbana-Champaign
  • Maria Manuel Clementino, CMUC, Universidade de Coimbra
  • Mario Román, University of Oxford
  • Marti Karvonen, University College London
  • Martina Rovelli, UMass Amherst
  • Masahito Hasegawa, Kyoto University
  • Matteo Capucci, University of Strathclyde
  • Michael Shulman, University of San Diego
  • Nick Gurski, Case Western Reserve University
  • Niels Voorneveld, Cybernetica
  • Paolo Perrone, University of Oxford
  • Peter Selinger, Dalhousie University
  • Paul Wilson, University of Southampton
  • Robin Cockett, University of Calgary
  • Robin Piedeleu, University College London
  • Rory Lucyshyn-Wright, Brandon University
  • Rose Kudzman-Blais, University of Ottawa
  • Ryan Wisnesky, Conexus AI
  • Sam Staton, University of Oxford
  • Shin-Ya Katsumata, Kyoto Sangyo University
  • Simon Willerton, University of Sheffield
  • Spencer Breiner, National Institute of Standards and Technology
  • Tai Danae Bradley, SandboxAQ
  • Titouan Carette, École Polytechnique
  • Tom Leinster, The University of Edinburgh
  • Walter Tholen, York University

Teaching & Communication

  • Selma Dündar-Coecke, University College London, Institute of Education
  • Ted Theodosopoulos, Nueva School

Organizing Committee

  • Pawel Sobocinski, Tallinn University of Technology
  • Priyaa Varshinee Srinivasan, Tallinn University of Technology
  • Sofiya Taskova, Tallinn University of Technology
  • Kristi Ainen, Tallinn University of Technology

Steering Committee

  • John Baez, University of California, Riverside
  • Bob Coecke, University of Oxford
  • Dorette Pronk, Dalhousie University
  • David Spivak, Topos Institute
  • Michael Johnson, Macquarie University
  • Simona Paoli, University of Aberdeen

October 27, 2025

John BaezPhilip Gibbs – Black Holes and White Holes

A white hole is a purely hypothetical time-reversed black hole. What does general relativity say about them? Would they repel you? Could you fall into a white hole—or only fall out? Could the universe be a white hole?

Philip Gibbs answers all these questions and more in my first interview for Edinburgh Explorations—a series put out by the School of Mathematics and the School of Physics and Astronomy at the University of Edinburgh.

I met Philip online on sci.physics in the early 1990s, and together with a bunch of other folks we created the Physics FAQ, which answers a lot of the most fascinating questions about physics. At first I didn’t believe everything he said about white holes, but eventually I realized he was right!

What do I mean by “right”? Be careful: white holes are just solutions of the equations of general relativity, not things we’ve actually seen. But you can work out what general relativity predicts about them: that’s the game here, and that determines what’s “right”. It doesn’t mean white holes actually exist and do these things.

For the physics FAQ, much of it created by Philip Gibbs, go here.

The cover picture, showing the maximally extended Schwarzschild solution containing both a black hole and white hole, was made by Isaak Neutelings.

John PreskillThe sequel

This October, fantasy readers are devouring a sequel: the final installment in Philip Pullman’s trilogy The Book of Dust. The series follows student Lyra Silvertongue as she journeys from Oxford to the far east. Her story features alternate worlds, souls that materialize as talking animals, and a whiff of steampunk. We first met Lyra in the His Dark Materials trilogy, which Pullman began publishing in 1995. So some readers have been awaiting the final Book of Dust volume for 30 years. 

Another sequel debuts this fall. It won’t spur tens of thousands of sales; nor will Michael Sheen narrate an audiobook version of it. Nevertheless, the sequel should provoke as much thought as Pullman’s: the sequel to the Maryland Quantum-Thermodynamics Hub’s first three years.

More deserving of a Carnegie Medal than our hub, but the hub deserves no less enthusiasm!

The Maryland Quantum-Thermodynamics Hub debuted in 2022, courtesy of a grant from the John F. Templeton Foundation. Six theorists, three based in Maryland, have formed the hub’s core. Our mission has included three prongs: research, community building, and outreach. During the preceding decade, quantum thermodynamics had exploded, but mostly outside North America. We aimed to provide a lodestone for the continent’s quantum-thermodynamics researchers and visitors.

Also, we aimed to identify the thermodynamics of how everyday, classical physics emerges from quantum physics. Quantum physics is reversible (doesn’t distinguish the past from the future), is delicate (measuring a quantum system can disturb it), and features counterintuitive phenomena such as entanglement. In contrast, our everyday experiences include irreversibility (time has an arrow), objectivity (if you and I read this article, we should agree about its contents), and no entanglement. How does quantum physics give rise to classical physics at large energy and length scales? Thermodynamics has traditionally described macroscopic, emergent properties. So quantum thermodynamics should inform our understanding of classical reality’s emergence from quantum mechanics.

Our team has approached this opportunity from three perspectives. One perspective centers on quantum Darwinism, a framework for quantifying how interactions spread information about an observed quantum system. Another perspective highlights decoherence, the contamination of a quantum system by its environment. The third perspective features incompatible exchanged quantities, described in an earlier blog post. Or two. Or at least seven

Each perspective led us to discover a tension, or apparent contradiction, that needs resolving. One might complain that we failed to clinch a quantum-thermodynamic theory of the emergence of classical reality. But physicists adore apparent contradictions as publishers love splashing “New York Times bestseller” on their book covers. So we aim to resolve the tensions over the next three years.

Physicists savor paradoxes and their ilk.

I’ll illustrate the tensions with incompatible exchanged quantities, of course. Physicists often imagine a small system, such as a quantum computer, interacting with a large environment, such as the surrounding air and the table on which the quantum computer sits. The system and environment may exchange energy, particles, electric charge, etc. Typically, the small system thermalizes, or reaches a state mostly independent of its initial conditions. For example, after exchanging enough energy with its environment, the system ends up at the environment’s temperature, mostly regardless of the system’s initial temperature. 

For decades, physicists implicitly assumed that the exchanged quantities are compatible: one can measure them simultaneously. But one can’t measure all of a quantum system’s properties simultaneously. Position and momentum form the most famous examples. Incompatibility epitomizes quantum physics, underlying Heisenberg’s uncertainty relation, quantum error correction, and more. So collaborators and I ask how exchanged quantities’ incompatibility alters thermalization, which helps account for time’s arrow. 

Our community has discovered that such incompatibility can hinder certain facets of thermalization—in a sense, stave off certain aspects of certain quantum systems’ experience of time. But incompatible exchanged quantities enhance other features of thermalization. How shall we reconcile the hindrances with the enhancements? Does one of the two effects win out? I hope to report back in three years. For now, I’m rooting for Team Hindrance.

In addition to resolving apparent conflicts, we’re adding a fourth perspective to our quiver—a gravitational one. In our everyday experiences, space-time appears smooth; unlike Lyra’s companion Will in The Subtle Knife, we don’t find windows onto other worlds. But quantum physics, combined with general relativity, suggests that you’d find spikes and dips upon probing space-time over extremely short length scales. How does smooth space-time emerge from its quantum underpinnings? Again, quantum thermodynamics should help us understand.

To address these challenges, we’re expanding the hub’s cast of characters. The initial cast included six theorists. Two more are joining the crew, together with the hub’s first two experimentalists. So is our first creative-writing instructor, who works at the University of Maryland (UMD) Jiménez-Porter Writers’ House.

As the hub has grown, so has the continent’s quantum-thermodynamics community. We aim to continue expanding that community and strengthening its ties to counterparts abroad. As Lyra learned in Pullman’s previous novel, partnering with Welsh miners and Czech book sellers and Smyrnan princesses can further one’s quest. I don’t expect the Maryland Quantum-Thermodynamics Hub to attract Smyrnan princesses, but a girl can dream. The hub is already partnering with the John F. Templeton Foundation, Normal Computing, the Fidelity Center for Applied Technology, the National Quantum Laboratory, Maryland’s Capital of Quantum team, and more. We aim to integrate quantum thermodynamics into North America’s scientific infrastructure, so that the field thrives here even after our new grant terminates. Reach out if you’d like to partner with us.

To unite our community, the hub will host a gathering—a symposium or conference—each year. One conference will feature quantum thermodynamics and quantum-steampunk creative writing. Scientists and authors will present. We hope that both groups will inspire each other, as physicist David Deutsch’s work on the many-worlds formulation of quantum theory inspired Pullman.

That conference will follow a quantum-steampunk creative-writing course to take place at UMD during spring 2026. I’ll co-teach the course with creative-writing instructor Edward Daschle. Students will study quantum thermodynamics, read published science-fiction stories, write quantum-steampunk stories, and critique each other’s writing. Five departments have cross-listed the course: physics, arts and humanities, computer science, chemistry, and mechanical engineering. If you’re a UMD student, you can sign up in a few weeks. Do so early; seats are limited! We welcome graduate students and undergrads, the latter of whom can earn a GSSP general-education credit.1 Through the course, the hub will spread quantum thermodynamics into Pullman’s world—into literature.

Pullman has entitled his latest novel The Rose Field. The final word refers to an object studied by physicists. A field, such as an electric or gravitational field, is a physical influence spread across space. Hence fiction is mirroring physics—and physics can take its cue from literature. As ardently as Lyra pursues the mysterious particle called Dust, the Maryland Quantum-Thermodynamics Hub is pursuing a thermodynamic understanding of the classical world’s emergence from quantum physics. And I think our mission sounds as enthralling as Lyra’s. So keep an eye on the hub for physics, community activities, and stories. The telling of Lyra’s tale may end this month, but the telling of the hub’s doesn’t.

1Just don’t ask me what GSSP stands for.

October 26, 2025

Doug NatelsonScience journalism - dark times

At this point it's old hat to decry the problems facing traditional news media.  Still, it is abundantly clear in our late stage capitalist society that there has been a collective business decision over the last 20+ years that, like local newspapers and television news, real science journalism is not a money maker.   Just a few examples:  Seventeen years ago, CNN cut its entire science, technology and environment reporting team.  In 2022, Popular Science ceased publication.  In 2023, National Geographic laid off their staff writers.  Last week, the Wall Street Journal laid off their science and health reporters.  

I have it on good authority that there is now only one science reporter left at the WSJ.  One, at a time when science and technology are more critically important to our rapidly changing society than ever, and there is enormous tumult in the US and elsewhere about how science is or is not supported and is or is not factoring into policy decisions.  All of this is happening at a time when public trust in science is falling.  (Check out this from Science Friday.)  

To be sure, there are some great publications out there, like Quanta and Nautilus (both founded by nonprofits). There are outstanding public writers of science, like Philip Ball, Helen Czerski, Katie Mack, Ethan Siegel, and many others (apologies for the incompleteness of this list).  There are some excellent freelance journalists.  The internet also means that there are many opportunities for great engagement.  For example, the videos from 3blue1brown are uniformly outstanding.  However, there are no filters, and the temptation to be click-baity or sensationalistic is problematic.  

I have no solutions to offer, except that I encourage you to support good science journalism and reporting when you see it.  It's important.

October 24, 2025

Matt von HippelC. N. Yang, Dead at 103

I don’t usually do obituaries here, but sometimes I have something worth saying.

Chen Ning Yang, a towering figure in particle physics, died last week.

Picture from 1957, when he received his Nobel

I never met him. By the time I started my PhD at Stony Brook, Yang was long-retired, and hadn’t visited the Yang Institute for Theoretical Physics in quite some time.

(Though there was still an office door, tucked behind the institute’s admin staff, that bore his name.)

The Nobel Prize doesn’t always honor the most important theoretical physicists. In order to get a Nobel Prize, you need to discover something that gets confirmed by experiment. Generally, it has to be a very crisp, clear statement about reality. New calculation methods and broader new understandings are on shakier ground, and theorists who propose them tend to be left out, or at best combined together into lists of partial prizes long after the fact.

Yang was lucky. With T. D. Lee, he had made that crisp, clear statement. He claimed that the laws of physics, counter to everyone’s expectations, are not the same when reflected in a mirror. In 1956, Wu confirmed the prediction, and Lee and Yang got the prize the year after.

That’s a huge, fundamental discovery about the natural world. But as a theorist, I don’t think that was Yang’s greatest accomplishment.

Yang contributed to other fields. Practicing theorists have seen his name strewn across concepts, formalisms, and theorems. I didn’t have space to talk about him in my article on integrability for Quanta Magazine, but only just barely: another paragraph or two, and he would have been there.

But his most influential contribution is something even more fundamental. And long-time readers of this blog should already know what it is.

Yang, along with Robert Mills, proposed Yang-Mills Theory.

There isn’t a Nobel prize for Yang-Mills theory. In 1953, when Yang and Mills proposed the theory, it was obviously wrong, a theory that couldn’t explain anything in the natural world, mercilessly mocked by famous bullshit opponent Wolfgang Pauli. Not even an ambitious idea that seemed outlandish (like plate tectonics), it was a theory with such an obvious missing piece that, for someone who prioritized experiment like the Nobel committee does, it seemed pointless to consider.

All it had going for it was that it was a clear generalization, an obvious next step. If there are forces like electromagnetism, with one type of charge going from plus to minus, why not a theory with multiple, interacting types of charge?

Nothing about Yang-Mills theory was impossible, or contradictory. Mathematically, it was fine. It obeyed all the rules of quantum mechanics. It simply didn’t appear to match anything in the real world.

But, as theorists learn, nature doesn’t let a good idea go to waste.

Of the four fundamental forces of nature, as it would happen, half are Yang-Mills theories. Gravity is different, electromagnetism is simpler, and could be understood without Yang and Mills’ insights. But the weak nuclear force, that’s a Yang-Mills theory. It wasn’t obvious in 1953 because it wasn’t clear how the massless, photon-like particles in Yang-Mills theory could have mass, and it wouldn’t become clear until the work of Peter Higgs over a decade later. And the strong nuclear force, that’s also a Yang-Mills theory, missed because of the ability of such a strong force to “confine” charges, hiding them away.

So Yang got a Nobel, not for understanding half of nature’s forces before anyone else had, but from a quirky question of symmetry.

In practice, Yang was known for all of this, and more. He was enormously influential. I’ve heard it claimed that he personally kept China from investing in a new particle collider, the strength of his reputation the most powerful force on that side of the debate, as he argued that a developing country like China should be investing in science with more short-term industrial impact, like condensed matter and atomic physics. I wonder if the debate will shift with his death, and what commitments the next Chinese five-year plan will make.

Ultimately, Yang is an example of what a theorist can be, a mix of solid work, counterintuitive realizations, and the thought-through generalizations that nature always seems to make use of in the end. If you’re not clear on what a theoretical physicist is, or what one can do, let Yang’s story be your guide.

Tommaso DorigoAre We Stochastic Parrots, Too? What LLMs Teach Us About Intelligence And Understanding

Having interacted for a few months with ChatGPT 5 now, both for work-related problems and for private / self-learning tasks, I feel I might share some thoughts here on what these large models can tell us about our own thought processes. 

The sentence above is basically giving away my bottomline from square one, but I suppose I can elaborate a bit more on the concept. LLMs have revolutionized a wide range of information-processing tasks in just three or four years. Looking back, the only comparable breakthrough I can recall is the advent of internet search engines in the early 1990s. But as exciting and awesome this breakthrough is, it inspires me still more to ponder on how this is even possible. Let me unpack this.

read more

October 22, 2025

Terence TaoSmooth numbers and max-entropy

Given a threshold {y>1}, a {y}-smooth number (or {y}-friable number) is a natural number {n} whose prime factors are all at most {y}. We use {\Psi(x,y)} to denote the number of {y}-smooth numbers up to {x}. In studying the asymptotic behavior of {\Psi(x,y)}, it is customary to write {y} as {x^{1/u}} (or {x} as {y^u}) for some {u>0}. For small values of {u}, the behavior is straightforward: for instance if {0 < u < 1}, then all numbers up to {x} are automatically {y}-smooth, so

\displaystyle  \Psi(x,y) \sim x

in this case. If {1 < u < 2}, the only numbers up to {x} that are not {y}-smooth are the multiples of primes {p} between {y} and {x}, so

\displaystyle  \Psi(x,y) \sim x - \sum_{y < p \leq x} (\frac{x}{p} + O(1))

\displaystyle  \sim x - x (\log\log x - \log\log y)

\displaystyle  \sim x (1 - \log u)

where we have employed Mertens’ second theorem. For {2 < u < 3}, there is an additional correction coming from multiples of two primes between {y} and {x}; a straightforward inclusion-exclusion argument (which we omit here) eventually gives

\displaystyle  \Psi(x,y) \sim x (1 - \log u + \int_2^u \frac{\log(t-1)}{t} dt)

in this case.

More generally, for any fixed {u>0}, de Bruijn showed that

\displaystyle  \Psi(x, y) \sim \rho(u) x

where {\rho(u)} is the Dickman function. This function is a piecewise smooth, decreasing function of {u}, defined by the delay differential equation

\displaystyle  u \rho'(u) + \rho(u-1) = 0

with initial condition {\rho(u) = 1} for {0 \leq u \leq 1}.

The asymptotic behavior of {\rho(u)} as {u \rightarrow \infty} is rather complicated. Very roughly speaking, it has inverse factorial behavior; there is a general upper bound {\rho(u) \leq 1/\Gamma(u+1)}, and a crude asymptotic

\displaystyle  \rho(u) = u^{-u+o(u)} = \exp( - u \log u + o(u \log u)).

With a more careful analysis one can refine this to

\displaystyle  \rho(u) = \exp( - u \log u - u \log\log u + u + o(u)); \ \ \ \ \ (1)

and with a very careful application of the Laplace inversion formula one can in fact show that

\displaystyle  \rho(u) \sim \sqrt{\frac{\xi'(u)}{2\pi}} \exp( \gamma - u \xi(u) + \int_0^{\xi(u)} \frac{e^s - 1}{s} ds) \ \ \ \ \ (2)

where {\gamma} is the Euler-Mascheroni constant and {\xi(u)} is defined implicitly by the equation

\displaystyle  e^{\xi(u)} - 1 = u \xi(u). \ \ \ \ \ (3)

One cannot write {\xi(u)} in closed form using elementary functions, but one can express it in terms of the Lambert {W} function as {\xi(u) = -W(-\frac{1}{u} e^{-1/u}) - 1/u}. This is not a particularly enlightening expression, though. A more productive approach is to work with approximations. It is not hard to get the initial approximation {\xi(u) \approx \log u} for large {u}, which can then be re-inserted back into (3) to obtain the more accurate approximation

\displaystyle  \xi(u) = \log u + \log\log u + O(1)

and inserted once again to obtain the refinement

\displaystyle  \xi(u) = \log u + \log\log u + O(\frac{\log\log u}{\log u}).

We can now see that (2) is consistent with previous asymptotics such as (1), after comparing the integral {\int_0^{\xi(u)} \frac{e^s - 1}{s} ds} to

\displaystyle  \int_0^{\xi(u)} \frac{e^s - 1}{\xi(u)} ds = u - 1.

For more details of these results, one can see for instance this survey by Granville.

This asymptotic (2) is quite complicated, and so one does not expect there to be any simple argument that could recover it without extensive computation. However, it turns out that one can use a “maximum entropy” analysis to get a reasonably good heuristic approximation to (2), that at least reveals the role of the mysterious function {\xi(u)}. The purpose of this blog post is to give this heuristic.

Viewing {x = y^u}, the task is to try to count the number of {y}-smooth numbers of magnitude {y^u}. We will propose a probabilistic model to generate {y}-smooth numbers as follows: for each prime {p \leq y}, select the prime {p} with an independent probability {c_p/p} for some coefficient {c_p}, and then multiply all the selected primes together. This will clearly generate a random {y}-smooth number {n}, and by the law of large numbers, the (log-)magnitude of this number should be approximately

\displaystyle  \log n \approx \sum_{p \leq y} \frac{c_p}{p} \log p, \ \ \ \ \ (4)

(where we will be vague about what “{\approx}” means here), so to obtain a number of magnitude about {y^u}, we should impose the constraint

\displaystyle  \sum_{p \leq y} \frac{c_p}{p} \log p = u \log y. \ \ \ \ \ (5)

The indicator {1_{p|n}} of the event that {p} divides this number is a Bernoulli random variable with mean {c_p/p}, so the Shannon entropy of this random variable is

\displaystyle  \mathbf{H}(1_{p|n}) = - \frac{c_p}{p} \log(\frac{c_p}{p}) - (1 - \frac{c_p}{p}) \log(1 - \frac{c_p}{p}).

If {c_p} is not too large, then Taylor expansion gives the approximation

\displaystyle  \mathbf{H}(1_{p|n}) \approx \frac{c_p}{p} \log p - \frac{c_p}{p} \log c_p + \frac{c_p}{p}.

Because of independence, the total entropy of this random variable {n} is

\displaystyle  \mathbf{H}(n) = \sum_{p \leq y} \mathbf{H}(1_{p|n});

inserting the previous approximation as well as (5), we obtain the heuristic approximation

\displaystyle  \mathbf{H}(n) \approx u \log y - \sum_{p \leq y} \frac{c_p}{p} (\log c_p - 1).

The asymptotic equipartition property of entropy, relating entropy to microstates, then suggests that the set of numbers {n} that are typically generated by this random process should have cardinality approximately

\displaystyle  \exp(\mathbf{H}(n)) \approx \exp\left(u \log y - \sum_{p \leq y} \frac{c_p}{p} (\log c_p - 1)\right)

\displaystyle  = \exp\left(- \sum_{p \leq y} \frac{c_p \log c_p - c_p}{p}\right) x.

Using the principle of maximum entropy, one is now led to the approximation

\displaystyle  \rho(u) \approx \exp\left(- \sum_{p \leq y} \frac{c_p \log c_p - c_p}{p}\right). \ \ \ \ \ (6)

where the weights {c_p} are chosen to maximize the right-hand side subject to the constraint (5).

One could solve this constrained optimization problem directly using Lagrange multipliers, but we simplify things a bit by passing to a continuous limit. We take a continuous ansatz {c_p = f(\log p / \log y)}, where {f: [0,1] \rightarrow {\bf R}} is a smooth function. Using Mertens’ theorem, the constraint (5) then heuristically becomes

\displaystyle  \int_0^1 f(t)\ dt = u \ \ \ \ \ (7)

and the expression (6) simplifies to

\displaystyle  \rho(u) \approx \exp( - \int_0^1 \frac{f(t) \log f(t) - f(t)}{t}\ dt). \ \ \ \ \ (8)

So the entropy maximization problem has now been reduced to the problem of minimizing the functional {\int_0^1 \frac{f(t) \log f(t) - f(t)}{t}\ dt} subject to the constraint (7). The astute reader may notice that the integral in (8) might diverge at {t=0}, but we shall ignore this technicality for the sake of the heuristic arguments.

This is a standard calculus of variations problem. The Euler-Lagrange equation for this problem can be easily worked out to be

\displaystyle  \frac{\log f(t)}{t} = \lambda

for some Lagrange multiplier {\lambda}; in other words, the optimal {f} should have an exponential form {f(t)= e^{\lambda t}}. The constraint (7) then becomes

\displaystyle  \frac{e^\lambda - 1}{\lambda} = u

and so the Lagrange multiplier {\lambda} is precisely the mysterious quantity {\xi(u)} appearing in (2)! The formula (8) can now be evaluated as

\displaystyle  \rho(u) \approx \exp\left( - \int_0^1 \frac{e^{\xi(u) t} \xi(u) t - e^{\xi(u) t}}{t}\ dt \right)

\displaystyle  \approx \exp\left( - e^{\xi(u)} + 1 + \int_0^1 \frac{e^{\xi(u) t} - 1}{t}\ dt + \int_0^1 \frac{1}{t}\ dt \right)

\displaystyle  \approx \exp\left( - u \xi(u) + \int_0^{\xi(u)} \frac{e^s - 1}{s}\ ds + C\right)

where {C} is the divergent constant

\displaystyle  C = \int_0^1 \frac{1}{t}\ dt.

This recovers a large fraction of (2)! It is not completely accurate for multiple reasons. One is that the hypothesis of joint independence on the events {p|n} is unrealistic when trying to confine {n} to a single scale {x}; this comes down ultimately to the subtle differences between the Poisson and Poisson-Dirichlet processes, as discussed in this previous blog post, and is also responsible for the otherwise mysterious {e^\gamma} factor in Mertens’ third theorem; it also morally explains the presence of the same {e^\gamma} factor in (2). A related issue is that the law of large numbers (4) is not exact, but admits gaussian fluctuations as per the central limit theorem; morally, this is the main cause of the {\sqrt{\frac{\xi'(u)}{2\pi}}} prefactor in (2).

Nevertheless, this demonstrates that the maximum entropy method can achieve a reasonably good heuristic understanding of smooth numbers. In fact we also gain some insight into the “anatomy of integers” of such numbers: the above analysis suggests that a typical {y}-smooth number {n} will be divisible by a given prime {p \sim y^t} with probability about {e^{\xi(u) t}/p}. Thus, for {t = 1 - O(1/\log u)}, the probability of being divisible by {p} is elevated by a factor of about {\asymp e^{\xi(u)} \asymp u \log u} over the baseline probability {1/p} of an arbitrary (non-smooth) number being divisible by {p}; so (by Mertens’ theorem) a typical {y}-smooth number is actually largely comprised of something like {\asymp u} prime factors all of size about {y^{1-O(1/\log u)}}, with the smaller primes contributing a lower order factor. This is in marked contrast with the anatomy of a typical (non-smooth) number {n}, which typically has {O(1)} prime factors in each hyperdyadic scale {[\exp\exp(k), \exp\exp(k+1)]} in {[1,n]}, as per Mertens’ theorem.

John BaezThe Standard Model (Part 1)

It’s our best theory of elementary particles and forces. It’s absolutely amazing: it took centuries of genius to discover that the world is like this, and it’s absolutely shocking. But nobody believes it’s the last word, so we simply call it The Standard Model.

But what does this theory say? I’ll try to explain part of it in this series of videos. I begin by introducing the cast of characters—the particles—and a bit about their interactions:

If you have questions, please ask—either here or on YouTube! Intelligent questions keep me motivated. Without them, I get bored.

By the way, these videos will contain mistakes. For example, this time I forgot to mention one key particle before saying “So I’ve introduced all the actors in the drama.” When I get better at editing videos, I will correct slips like this. But I will always try to point out errors in a “pinned” comment right below the video. So look down there.

Also: I don’t plan to explain the details of quantum field theory. So even if you watch all my videos, you’ll get just a taste of the Standard Model. But I will get into some of the math, so it will be much more than just chat. It will roughly follow this paper:

• John Baez and John Huerta, The algebra of grand unified theories, Bulletin of the American Mathematical Society 47 (2010), 483–552.

But I may explain more prerequisites, like a bit of quantum theory and group representation theory. That would let more people follow along.

This is part of my Edinburgh Exploration series, which will also include interviews.

October 20, 2025

Scott Aaronson My talk at Columbia University: “Computational Complexity and Explanations in Physics”

Last week, I gave the Patrick Suppes Lecture in the Columbia University Philosophy Department. Patrick Suppes was a distinguished philosopher at Stanford who (among many other things) pioneered remote gifted education through the EPGY program, and who I was privileged to spend some time with back in 2007, when he was in his eighties.

My talk at Columbia was entitled “Computational Complexity and Explanations in Physics.” Here are the PowerPoint slides, and here’s the abstract:

The fact, or conjecture, of certain computational problems being intractable (that is, needing astronomical amounts of time to solve) clearly affects our ability to learn about physics.  But could computational intractability also play a direct role in physical explanations themselves?  I’ll consider this question by examining three possibilities:

(1) If quantum computers really take exponential time to simulate using classical computers, does that militate toward the many-worlds interpretation of quantum mechanics, as David Deutsch famously proposed?

(2) Are certain speculative physical ideas (e.g., time travel to the past or nonlinearities in quantum mechanics) disfavored, over and above any other reasons to disfavor them, because they would lead to “absurd computational superpowers”?

(3) Do certain effective descriptions in physics work only because of the computational intractability of violating those descriptions — as for example with Harlow and Hayden’s resolution of the “firewall paradox” in black hole thermodynamics, or perhaps even the Second Law of Thermodynamics itself?

I’m grateful to David Albert and Lydia Goehr of Columbia’s Philosophy Department, who invited me and organized the talk, as well as string theorist Brian Greene, who came and contributed to the discussion afterward. I also spent a day in Columbia’s CS department, gave a talk about my recent results on quantum oracles, and saw many new friends and old there, including my and my wife’s amazing former student Henry Yuen. Thanks to everyone.


This was my first visit to Columbia University for more than a decade, and certainly my first since the upheavals following the October 7 massacre. Of course I was eager to see the situation for myself, having written about it on this blog. Basically, if you’re a visitor like me, you now need both a QR code and an ID to get into the campus, which is undeniably annoying. On the other hand, once you’re in, everything is pleasant and beautiful. Just from wandering around, I’d have no idea that this campus had recently been Ground Zero for the pro-intifada protests, and then for the reactions against those protests (indeed, the use of the protests as a pretext to try to destroy academia entirely) that rocked the entire country, filling my world and my social media feed.

When I asked friends and colleagues about the situation, I heard a range of perspectives: some were clearly exasperated with the security measures; others, while sharing in the annoyance, suggested the measures seem to be needed, since every time the university has tried to relax them, the “intifada” has returned, with non-university agitators once again disrupting research and teaching. Of course we can all pray that the current ceasefire will hold, for many reasons, the least of which is that perhaps then the obsession of the world’s young and virtuous to destroy the world’s only Jewish state will cool down a bit, and they’ll find another target for their rage. That would also help life at Columbia and other universities return to how it was before.

Before anyone asks: no, Columbia’s Peter Woit never showed up to disrupt my talk with rotten vegetables or a bullhorn—indeed, I didn’t see him at all on his trip, nor did I seek him out. Given that Peter chose to use his platform, one of the world’s best-known science blogs, to call me a mentally ill genocidal fascist week after week, it meant an enormous amount to me to see how many friends and supporters I have right in his own backyard.

All in all, I had a wonderful time at Columbia, and based on what I saw, I won’t hesitate to come back, nor will I hesitate to recommend Jewish or Israeli or pro-Zionist students to study there.

October 18, 2025

Doug NatelsonInteresting preprints: chirality-induced spin selectivity + quantum gravity

This continues to be a very busy time, but I wanted to point out two preprints that caught my eye this week.  Their subjects are completely disparate, but they stand out as essentially reviews written in a much more conversational tone than the usual literature.

The first is this preprint about chirality-induced spin selectivity, a subject that I've mentioned before on this blog.  There is now an extensive body of evidence (of varying quality) that there is a connection between structural chirality of molecules and their interactions with the spin angular momentum of electrons.  This includes monolayers of chiral molecules leading to net spin polarization of photoemitted electrons (here), a lot of electronic transport experiments involving chiral molecules and magnetic electrodes that seem to show spin-dependent transmission that is absent with achiral molecules, and even a chirality dependence of molecular adsorption kinetics on magnetic surfaces (here).  The preprint is a provocative discussion of the topic and possible mechanisms, and the importance of precision in the description of the various phenomena.

On a completely different topic, this preprint is a fun discussion about quantum gravity (!) and how condensed matter ideas of "the vacuum" can lead to insights about how quantum mechanics and gravity might need to play together.  One fun bit early on is a discussion of something I like to point out to my undergrad stat mech students:  A single hydrogen atom in a very very large box will apparently (if the usual stat mech formalism of partition functions is valid) be spontaneously ionized, even when the box and atom are at temperatures faaaaaar below the energy scale for ionization.  This is discussed nicely in this 1966 article in the Journal of Chemical Education.  Anyway, I thought this was an interesting discussion from three condensed matter theorists.

October 17, 2025

Matt von HippelAGI Is an Economic Term, Not a Computer Science Term

Since it resonated with the audience, I’ll recap my main argument against AGI here. ‘General intelligence’ is like phlogiston, or the aether. It’s an outmoded scientific concept that does not refer to anything real. Any explanatory work it did can be done better by a richer scientific frame. 1/3

Shannon Vallor (@shannonvallor.bsky.social) 2025-10-02T22:09:06.610Z

I ran into this Bluesky post, and while a lot of the argument resonated with me, I think the author is missing something important.

Shannon Vallor is a philosopher of technology at the University of Edinburgh. She spoke recently at a meeting honoring the 75th anniversary of the Turing Test. The core of her argument, recapped in the Bluesky post, is that artificial general intelligence, or AGI, represents an outdated scientific concept, like phlogiston. While some researchers in the past thought of humans as having a kind of “general” intelligence that a machine would need to replicate, scientists today break down intelligence into a range of capabilities that can be present in different ways. From that perspective, searching for artificial general intelligence doesn’t make much sense: instead, researchers should focus on the particular capabilities they’re interested in.

I have a lot of sympathy for Vallor’s argument, though perhaps from a different direction than what she had in mind. I don’t know enough about intelligence in a biological context to comment there. But from a computer science perspective, intelligence obviously is composed of distinct capabilities. Something that computes, like a human or a machine, can have different amounts of memory, different processing speeds, different input and output rates. In terms of ability to execute algorithms, it can be a Turing machine, or something less than a Turing machine. In terms of the actual algorithms it runs, they can have different scaling for large inputs, and different overhead for small inputs. In terms of learning, one can have better data, or priors that are closer to the ground truth.

These days, all of these Turing machine algorithm capabilities are in some sense obviously not what the people interested in AGI are after. We already have them in currently-existing computers, after all. Instead, people who pursue AGI, and AI researchers more generally, are interested in heuristics. Humans do certain things without reliable algorithms, instead we do them faster, but unreliably. And while some human heuristics seem pretty general, it’s widely understood that in the heuristics world there is no free lunch. No heuristic is good for everything, and no heuristic is bad for everything.

So is “general intelligence” a mirage, like phlogiston?

If you think about it as a scientific goal, sure. But as a product, not so much.

Consider a word processor.

Obviously, from a scientific perspective, there are lots of capabilities that involve processing words. Some were things machines could do well before the advent of modern computers: consider typewriters, for instance. Others still are out of reach, after all, we do still pay people to write. (I myself am such person!)

But at the same time, if I say that a computer program is a word processor, you have a pretty good idea of what that means. There was a time when processing words involved an enormous amount of labor, work done by a large number of specialized people (mostly women). Look at a workplace documentary from the 1960’s, and compare it to a workplace today, and you’ll see that word processor technology has radically changed what tasks people do.

AGI may not make sense as a scientific goal, but it’s perfectly coherent in these terms.

Right now, a lot of tasks are done by what one could broadly call human intelligence. Some of these tasks have already fallen to technology, others will fall one by one. But it’s not unreasonable to think of a package deal, a technology that covers enough of such tasks that human intelligence stops being economically viable. That’s not because there will be some scientific general intelligence that the technology would then have, but because a decent number of intellectual tasks do seem to come bundled together. And you don’t need to cover 100% of human capabilities to radically change workplaces, any more than you needed to cover 100% of the work of a 1960’s secretary with a word processor for modern secretarial work to have a dramatically different scope and role.

It’s worth keeping in mind what is and isn’t scientifically coherent, to be aware that you can’t just extrapolate the idea of general intelligence to any future machine. (For one, it constrains what “superintelligence” could look like.) But that doesn’t mean we should be complacent, and assume that AGI is impossible in principle. AGI, like a word processor, would be a machine that covers a set of tasks well enough that people use it instead of hiring people to do the work by hand. It’s just a broader set of tasks.

October 16, 2025

John BaezThe Kepler Problem (Part 12)

It’s been a while. Let me finally wrap up this this series! I’ll show you how we can get the periodic table of elements from a quantum field theory of massless spin-1/2 particles.

We’ve been studying the hydrogen atom using Schrödinger’s equation, but still taking the electron’s spin into account. This is more realistic than the basic Schrödinger equation that ignores spin, but less realistic than a full-fledged relativistic treatment using the Dirac equation. It’s a pretty good approximation. We saw that bound states of this hydrogen atom can be reinterpreted as states of a massless left-handed spin-1/2 particle in the Einstein universe—a universe where space is the 3-sphere S^3.

Then I ‘second quantized’ both theories. If we do this to the massless spin-1/2 particle, we get a free quantum field theory on the Einstein universe! This describes arbitrary collections of noninteracting massless spin-1/2 particles. If we second quantize the hydrogen atom, we get a theory of multi-electron atoms—but a naive theory where the electrons don’t interact. In this naive theory we don’t get the periodic table of elements. But today I want to tweak the Hamiltonian so we do get the observed periodic table, at least roughly.

Let’s do it!

In Part 7 we saw how the Hilbert space of bound states of the hydrogen atom, taking the electron’s spin into account, is

\mathcal{H} = L^2(S^3) \otimes \mathbb{C}^2

In atomic physics, the eigenspaces of the Hamiltonian H on \mathcal{H} are called shells, while the joint eigenspaces of the operators H and the angular momentum squared, L^2, are called subshells. We denote the shells as

\displaystyle{ \mathcal{H}_n = \{ \psi \in \mathcal{H} \; H\psi =  -\frac{1}{2n^2} \psi \} }

and the subshells as

\mathcal{H}_{n,\ell} = \{ \psi \in \mathcal{H}_n : \;  L^2 \psi = \ell(\ell + 1) \psi \}

The shells are direct sums of subshells as follows:

\mathcal{H}_n = \bigoplus_{\ell = 0}^{n-1} \mathcal{H}_{n,\ell}

and the direct sum of all the shells is the whole Hilbert space of bound states of the hydrogen atom:

\mathcal{H} = \bigoplus_{n = 1}^\infty  \mathcal{H}_n

The dimension of the subshell \mathcal{H}_{n,\ell} is 2(2\ell + 1), so the dimension of the shell \mathcal{H}_n is

2(1 + 3 + 5 + \cdots + (2n - 1)) = 2n^2

In Part 11 we second quantized the hydrogen atom and got the fermionic Fock space \mathbf{\Lambda} \mathcal{H} which describes collections of electrons bound to the nucleus. This suggests that we could understand the periodic table this way!

The Aufbau principle is a famous approximate way to describe the ground state of an N-electron atom as a state \phi in the N-particle subspace of the Fock space \mathbf{\Lambda} \mathcal{H}. To do this, we choose a Hamiltonian H_{\text{Fock}} on \mathbf{\Lambda} \mathcal{H} and decree that \phi must minimize the expected energy \langle \phi , H_{\text{Fock}}\, \phi \rangle among all unit vectors in the N-particle subspace. However, we choose the Hamiltonian H_{\text{Fock}} in a very simplistic way. We ignore the details of electron-electron interactions! Instead, we simply assign an energy E_{n,\ell} to each subshell, let H_{\text{single}} be the unique Hamiltonian on \mathcal{H} such that

\psi \in \mathcal{H}_{n,\ell} \implies   H_{\text{single}} \psi = E_{n,\ell} \,\psi

and then we let

H_{\text{Fock}} = d\mathbf{\Lambda}(H_{\text{single}})

Thus, H_{\text{Fock}} has a basis of eigenvectors that are wedge products of single-particle states lying in various subshells. Explicitly, if

\phi = \psi_1 \wedge \cdots \wedge \psi_N where \psi_i \in \mathcal{H}_{n_i, \ell_i}

then we have

H_{\text{Fock}} \, \phi = (E_{n_1,\ell_1} + \cdots + E_{n_N, \ell_N}) \phi

Thus, we can minimize \langle \phi , H_{\text{Fock}}\, \phi \rangle among unit vectors in the N-particle subspace by choosing N distinct basis vectors

\psi_i = |n_i, \ell_i, m_i, s_i \rangle

in a way that minimizes the total energy E_{n_1,\ell_1} + \cdots + E_{n_N, \ell_N}.

If we follow this recipe taking H_{\text{single}} to be the hydrogen atom Hamiltonian H, we get results that do not closely match the observed periodic table of elements. With this choice we get energies

\displaystyle{ E_{n,\ell} = -\frac{1}{2n^2} }

that depend only on the shell, not the subshell. Thus, this choice makes no prediction about the order in which subshells are filled! That’s no good.

For the recipe to give results that more closely match the periodic table, we need to choose the energies E_{n,\ell} in a more clever way. In 1936, Madelung argued for these rules:

• subshells are filled in order of increasing value of n + \ell;

• for subshells with the same value of n + \ell, subshells are filled in order of decreasing \ell (or equivalently, increasing n).

In reality this nice pattern is broken by quite a few elements, but here we only consider a simple model in which the Madelung rules hold. The pattern of subshell filling then looks like this:

The above chart uses some old but still popular notation from spectroscopy:

\ell = 0: s
\ell = 1: p
\ell = 2: d
\ell = 3: f

For example, the subshell \mathcal{H}_{3,2} is denoted 3d while \mathcal{H}_{5,3} is denoted 5f.

In 1945, a chemist named Wiswesser noted that the Madelung rules follow from the recipe we outlined if we choose

E_{n,\ell} = n + \ell - \frac{\ell}{\ell + 1}

There are many other functions of n and \ell that achieve the same effect. For example, we can also obtain the Madelung rules if we take

E_{n,\ell} = 2n + (2\ell + 1) + (2\ell + 1)^{-1}

and this formula is more convenient for us.

The Madelung rules do not always hold! The first exception is element 24, chromium. The Madelung rules predict that chromium has 2 electrons in the 4s subshell and 4 electrons in the 3d subshell, while in fact it has 1 in the 4s and 5 in the 3d. There are eleven exceptions to the Madelung rules in the so-called d-block elements (shown in purple in the periodic table below), and nine exceptions in the f-block elements (shown in pink). Nonetheless the general structure of the periodic table is in reasonably good accord with the Madelung rules for all the elements studied chemically so far, though relativistic effects may end this for very heavy elements.

Thus it is of some interest, if only as a curiosity, to define a Hamiltonian on the Hilbert space \mathcal{H} that takes the eigenvalue E_{n,\ell} in the subshell \mathcal{H}_{n,\ell}. These energies are not at all close to the actual energies of the various multi-electron atoms, and any monotone function of E_{n,\ell} would also give the Madelung rules—but this particular Hamiltonian is fairly simple.

Recall from Part 7:

\begin{array}{cclll}      A^2 |n , \ell, m \rangle  &=& \frac{1}{4}(n^2 - 1) |n , \ell, m, s \rangle \\ [3pt]       L^2 |n , \ell, m \rangle &=& \ell(\ell + 1) |n , \ell, m, s \rangle    \end{array}

The Duflo isomorphism, as discussed in Part 6, makes it natural to define operators

\tilde{A}^2 = A^2 + \frac{1}{4}, \qquad \tilde{L}^2 = L^2 + \frac{1}{4}

If we then define \tilde{A} and \tilde{L} to be the square roots of these operators, we have

\begin{array}{cclcl}      \tilde{A} |n , \ell, m \rangle  &=& \frac{1}{2} n |n , \ell, m \rangle    \\ [3pt]       \tilde{L} |n , \ell, m \rangle &=& (\ell + \frac{1}{2}) |n , \ell, m \rangle   \end{array}

and thus

(2\tilde{A} + 2\tilde{L} + (2\tilde{L})^{-1})|n , \ell, m, s \rangle =   E_{n,\ell} |n , \ell, m,  s\rangle

This suggests taking our single-particle Hamiltonian to be

H_{\text{single}} = 2\tilde{A} + 2\tilde{L} + (2\tilde{L})^{-1}

If we then define a Hamiltonian on the Fock space \mathbf{\Lambda} \mathcal{H} by

H_{\text{Fock}} = d\mathbf{\Lambda}(H_{\text{single}})

and create an orthonormal basis \psi_i of eigenvectors of H_{\text{Fock}}, listed in order of increasing eigenvalue, these eigenvectors correspond to the elements with subshells filled as predicted by the Madelung rules. The one exception is the state \psi_0 with no electrons, sometimes called ‘element zero’ and identified with the neutron. For example:

\begin{array}{ccll}   \psi_1 &=& |1,0,0,\frac{1}{2} \rangle & \text{hydrogen} \\ \\  \psi_2 &=& |1,0,0,\frac{1}{2} \rangle \wedge |1,0,0,-\frac{1}{2}\rangle & \text{helium} \\ \\  \psi_3 &=& |1,0,0,\frac{1}{2} \rangle \wedge |1,0,0,-\frac{1}{2}\rangle   \wedge |2,0,0,\frac{1}{2}\rangle & \text{lithium}   \end{array}

and so on.

Here the assignments of magnetic quantum numbers m and spins s are not determined by the rules we have laid out. These are governed, at least approximately, by Hund’s rules:

• every m state in a subshell is singly occupied before any is doubly occupied;

• all of the electrons in singly occupied orbitals have the same spin.

We could go further and attempt to choose a simple Hamiltonian for which the principle of energy mimization also gives Hund’s rules. However, we prefer to stop here, leaving you with the challenge of finding a better-behaved quantum field theory on the Einstein universe whose Hamiltonian gives the Madelung rules, or perhaps better understanding the Hamiltonian we have given here.

Here is the periodic table we get from our approach:

and here are the energies for subshells

E_{n,\ell} = 2n + (2\ell + 1) + (2\ell + 1)^{-1}

that we’re using in our approach:


For more, read my paper:

Second quantization for the Kepler problem.

or these blog articles, which are more expository and fun:

Part 1: a quick overview of Kepler’s work on atoms and the solar system, and more modern developments.

Part 2: why the eccentricity vector is conserved for a particle in an inverse square force, and what it means.

Part 3: why the momentum of a particle in an inverse square force moves around in a circle.

Part 4: why the 4d rotation group \text{SO}(4) acts on bound states of a particle in an attractive inverse square force.

Part 5: quantizing the bound states of a particle in an attractive inverse square force, and getting the Hilbert space L^2(S^3) for bound states of a hydrogen atom, neglecting the electron’s spin.

Part 6: how the Duflo isomorphism explains quantum corrections to the hydrogen atom Hamiltonian.

Part 7: why the Hilbert space of bound states for a hydrogen atom including the electron’s spin is L^2(S^3) \otimes \mathbb{C}^2.

Part 8: why L^2(S^3) \otimes \mathbb{C}^2 is also the Hilbert space for a massless spin-1/2 particle in the Einstein universe.

Part 9: a quaternionic description of the hydrogen atom’s bound states (a digression not needed for later parts).

Part 10: changing the complex structure on L^2(S^3) \otimes \mathbb{C}^2 to eliminate negative-energy states of the massless spin-1/2 particle, as often done.

Part 11: second quantizing the massless spin-1/2 particle and getting a quantum field theory on the Einstein universe, or alternatively a theory of collections of electrons orbiting a nucleus.

Part 12: obtaining the periodic table of elements from a quantum field theory on the Einstein universe.

October 15, 2025

Clifford JohnsonNobel Prize in Physics 2025: Who/What/Why

I started a tradition a little while back where every year we have a special departmental colloquium entitled "The Nobel Prize in Physics: Who/What/Why". This year my job in finding speakers was made easier by having 2/3 of this years newly-minted Nobel Prize winners in physics in the Department! (Michel Devoret and John Martinis.) So our room was a bit more well-attended than normal...(hundreds and hundreds rather than dozens and dozens). Here is a recording of the event, which I was delighted to host, and there's a celebration afterwards too. (Pls share widely!)
[...] Click to continue reading this post

The post Nobel Prize in Physics 2025: Who/What/Why appeared first on Asymptotia.

Jordan EllenbergInternational Conference on Ancient Magic

I have nothing to say about this, I just think it’s cool that Ohio State is hosting an International Conference on Ancient Magic this weekend. Open to the public! Go to this if you’re in Columbus and feeling eldritch!

October 14, 2025

Scott Aaronson Sad and happy day

Today, of course, is the second anniversary of the genocidal Oct. 7 invasion of Israel—the deadliest day for Jews since the Holocaust, and the event that launched the current wars that have been reshaping the Middle East for better and/or worse. Regardless of whether their primary concern is for Israelis, Palestinians, or both, I’d hope all readers of this blog could at least join me in wishing this barbaric invasion had never happened, and in condemning the celebrations of it taking place around the world.


Now for the happy part: today is also the day when the Nobel Prize in Physics is announced. I was delighted to wake up to the news that this year, the prize goes to John Clarke of Berkeley, John Martinis of UC Santa Barbara, and Michel Devoret of UC Santa Barbara (formerly Yale), for their experiments in the 1980s that demonstrated the reality of macroscopic quantum tunneling in superconducting circuits. Among other things, this work laid the foundation for the current effort by Google, IBM, and many others to build quantum computers with superconducting qubits. To clarify, though, today’s prize is not for quantum computing per se, but for the earlier work.

While I don’t know John Clarke, and know Michel Devoret only a little, I’ve been proud to count John Martinis as a good friend for the past decade—indeed, his name has often appeared on this blog. When Google hired John in 2014 to build the first programmable quantum computer capable of demonstrating quantum supremacy, it was clear that we’d need to talk about the theory, so we did. Through many email exchanges, calls, and visits to Google’s Santa Barbara Lab, I came to admire John for his iconoclasm, his bluntness, and his determination to make sampling-based quantum supremacy happen. After Google’s success in 2019, I sometimes wondered whether John might eventually be part of a Nobel Prize in Physics for his experimental work in quantum computing. That may have become less likely today, now that he’s won the Nobel Prize in Physics for his work before quantum computing, but I’m guessing he doesn’t mind! Anyway, huge congratulations to all three of the winners.

October 12, 2025

Jordan EllenbergA favorite tactic of the self-identified iconoclast

A good line by Lauren Oyler, in a Harper’s article about the Goop Cruise:

“A favorite tactic of the self-identified iconoclast is to argue that things that have already happened need, still, to happen.”

I like the placement of “still” and the commas a lot. Is “self-identified” one notch too sneery? Would “would-be” be better? Or even no adjective at all? Maybe the second part of the sentence does enough to establish that the title of iconoclast has not been firmly earned by its claimer.

October 11, 2025

John BaezCrown Ethers

Students who memorize organic chemistry consider it torture. But to explore it is a delight. It’s like the difference between being force-fed and grazing at a buffet.

For example, I recently learned about ‘crown ethers‘: beautiful crown-shaped molecules of carbon, hydrogen and oxygen. Above is a ball-and-stick model of a crown ether called 15-crown-5.

Why is it called that? I guess because it has 5 oxygens (in red) and 10 carbons (in black) for a total of 15 heavier atoms. Then there are a bunch of hydrogens poking out. It has perfect 5-fold symmetry.

Yasui Yoshio won the Nobel Prize in 1987 for discovering crown ethers around 1967. He’s usually known as Charles J. Pedersen. He was born in Korea but later moved to Japan and then the US.

One use of crown ethers is to hold metal ions. A sodium atom with one electron missing can snugly fit inside 15-crown-5. So can a transition metal with two electrons missing, like cobalt (Co²⁺), nickel (Ni²⁺), copper (Cu²⁺), or zinc (Zn²⁺).

Other metal ions fit better in larger crown ethers. For example, here is 18-crown-6 holding a potassium ion. Beautiful 6-fold symmetry!

The 6 bonds here come from 6 oxygens connecting to the central potassium atom in purple. I don’t know, but I bet each oxygen donates 1/6 of an electron to the potassium atom, which is missing one.

1/6 of an electron?!? 🤯

You can’t actually chop an electron into parts, but you can have a single electron whose wavefunction is smeared out around all 6 oxygen atoms and the central potassium.

So, this ball-and-stick model is pretty misleading! The molecule doesn’t really have sticks in it. Let me show you another picture of it:

This more chubby depiction is called a space-filling model. This makes it clear how the potassium fits snugly inside the crown.

Another thing you can put in 18-crown-6 is hydronium, H3O+:

Hydronium is the ion formed by a water molecule and an extra proton. We see here that its 3-fold symmetry fits nicely into the 6-fold symmetry of 18-crown-6.

But when 18-crown-6 does not have an ion inside it, it folds into a shape that lacks 6-fold symmetry:

Here are some more crown ethers:

They are:

1) 12-crown-4,
2) 15-crown-5,
3) 18-crown-6,
4) dibenzo-18-crown-6,
6) an aza-crown ether, meaning one that contains nitrogen.

For a final burst of fun, here’s an aza-crown ether holding a nickel atom in the center, and two chlorines (in green):

It’s called transNi(cyclam)Cl2.

The diversity of highly symmetrical structures in chemistry is truly wondrous! And needless to say, I didn’t draw any of the pictures here. I got them from Wikicommons, and you can find out where by clicking on them. Most are in the public domain, thanks to the generosity of their creators.

Doug NatelsonACS National Nanotechnology Day webinar, Thursday Oct 9

Time for a rare bit of explicit self-promotion on this blog.  This coming Thursday, October 9, as part of the American Chemical Society's activities for National Nanotechnology Day (Why October 9?  In US convention, Oct 9 = 10/9, and 10-9 m = 1 nm.  Look, it wasn't my idea....), I'm speaking in a free webinar titled "Illuminating the Nano Frontier", with Prof. Dongling Ma of INRS in Quebec.  The event is 11am-12:30pm EDT, and there will also be a recording for people who are unable to watch it live.  Should be a fun event. 

UpdateHere is the link to the webinar recording.  It's free and open-access.

October 10, 2025

Matt von HippelCongratulations to John Clarke, Michel Devoret, and John Martinis!

The 2025 Physics Nobel Prize was announced this week, awarded to John Clarke, Michel Devoret, and John Martinis for building an electrical circuit that exhibited quantum effects like tunneling and energy quantization on a macroscopic scale.

Press coverage of this prize tends to focus on two aspects: the idea that these three “scaled up” quantum effects to medium-sized objects (the technical account quotes a description that calls it “big enough to get one’s grubby fingers on”), and that the work paved the way for some of the fundamental technologies people are exploring for quantum computing.

That’s a fine enough story, but it leaves out what made these folks’ work unique, why it differs from other Nobel laureates working with other quantum systems. It’s a bit more technical of a story, but I don’t think it’s that technical. I’ll try to tell it here.

To start, have you heard of Bose-Einstein Condensates?

Bose-Einstein Condensates are macroscopic quantum states that have already won Nobel prizes. First theorized based on ideas developed by Einstein and Bose (the namesake of bosons), they involve a large number of particles moving together, each in the same state. While the first gas that obeyed Einstein’s equations for a Bose-Einstein Condensate was created in the 1990’s, after Clarke, Devoret, and Martinis’s work, other things based on essentially the same principles were created much earlier. A laser works on the same principles as a Bose-Einstein condensate, as do phenomena like superconductivity and superfluidity.

This means that lasers, superfluids, and superconductors had been showing off quantum mechanics on grubby finger scales well before Clarke, Devoret, and Martinis’s work. But the science rewarded by this year’s Nobel turns out to be something quite different.

Because the different photons in laser light are independently in identical quantum states, lasers are surprisingly robust. You can disrupt the state of one photon, and it won’t interfere with the other states. You’ll have weakened the laser’s consistency a little bit, but the disruption won’t spread much, if at all.

That’s very different from the way quantum systems usually work. Schrodinger’s cat is the classic example. You have a box with a radioactive atom, and if that atom decays, it releases poison, killing the cat. You don’t know if the atom has decayed or not, and you don’t know if the cat is alive or not. We say the atom’s state is a superposition of decayed and not decayed, and the cat’s state is a superposition of alive and dead.

But unlike photons in a laser, the atom and the cat in Schrodinger’s cat are not independent: if the atom has decayed, the cat is dead, if the atom has not, the cat is alive. We say the states of atom and cat are entangled.

That makes these so-called “Schrodinger’s cat” states much more delicate. The state of the cat depends on the state of the atom, and those dependencies quickly “leak” to the outside world. If you haven’t sealed the box well, the smell of the room is now also entangled with the cat…which, if you have a sense of smell, means that you are entangled with the cat. That’s the same as saying that you have measured the cat, so you can’t treat it as quantum any more.

What Clarke, Devoret, and Martinis did was to build a circuit that could exhibit, not a state like a laser, but a “cat state”: delicately entangled, at risk of total collapse if measured.

That’s why they deserved a Nobel, even in a world where there are many other Nobels for different types of quantum states. Lasers, superconductors, even Bose-Einstein condensates were in a sense “easy mode”, robust quantum states that didn’t need all that much protection. This year’s physics laureates, in contrast, showed it was possible to make circuits that could make use of quantum mechanics’ most delicate properties.

That’s also why their circuits, in particular, are being heralded as a predecessor for modern attempts at quantum computers. Quantum computers do tricks with entanglement, they need “cat states”, not Bose-Einstein Condensates. And Clarke, Devoret, and Martinis’s work in the 1980’s was the first clear proof that this was a feasible thing to do.

October 09, 2025

Doug NatelsonPostdoctoral opportunity in materials

The Rice Advanced Materials Research Institute is having its 2025-2026 competition for prestigious postdoctoral fellowships - see here:  https://rami.rice.edu/rami-postdoctoral-fellowship-program  .

If you are interested and meet the criteria, I'd be happy to talk.  I have some ideas that lean into the materials for electronics direction, and other possibilities are welcome.  

Jordan EllenbergMacArthur Fellows: Wisconsin and Williams

The 2025 MacArthur Fellows have been announced, and I’m very pleased that two of the 22 awardees are my UW-Madision colleagues: congratulations to Ángel F. Adames Corraliza from atmospheric and oceanic sciences, and Sébastien Phillippe from nuclear engineering and engineering physics. The only mathematician to win the award this year was Lauren Williams, who true devotees of this blog will remember as the person who gave the best talk I’ve ever seen about cluster algebras. Congratulations to all the winners!

October 08, 2025

Doug Natelson2025 Physics Nobel: Macroscopic quantum tunneling

As announced this morning, the 2025 Nobel Prize in Physics has been awarded to John Clarke, Michel Devoret, and John Martinis, for a series of ground-breaking experiments in the 1980s that demonstrated macroscopic quantum tunneling. 

For non-experts: "Tunneling" was originally coined to describe the physical motion of a quantum object, which can pass through a "classically forbidden" region.  I've written about this here, and here is an evocative picture. Suppose there is a particle with a certain amount of total energy in the left region.  Classically, the particle is trapped, because going too far to the left (gray region) or too far to the right (gray region) is forbidden:  Putting the particle inside the shaded regions is "classically forbidden" by conservation of energy.  The particle bounces back and forth in the left well.  If the particle is a quantum object, though, it is described by a wave function, and that wave function has some non-zero amplitude on the far side of barrier in the middle.  The particle can "tunnel" through the barrier, with a probability that decreases exponentially with the height of the barrier and its width.

Fig. 2 from here

Clarke, Devoret, and Martinis were working not with a single particle, but with electrons in a superconductor (many many electrons in a coherent quantum state).  The particular system they chose was a Josephson junction made from an oxide-coated Nb film contacted by a PbIn electrode with a dc current flowing through it.  Instead of an x coordinate of a particle, the relevant coordinate in this system is the phase difference \(\delta\) of the superconducting wave function across the junction.  There is an effective potential energy for this system called a "washboard" potential, \(U(\delta)\), as in this figure.  At the particular DC current, which tilts \(U(\delta)\), the system can transition from one state (\(\delta\) bopping around a constant value, no voltage across the junction) to a state where \(\delta\) is continuously ramping (corresponding to a nonzero voltage across the junction).  The system can get thermally kicked from the zero voltage state to the nonzero voltage state (thermal energy doinks it over the barrier), but the really interesting thing is that the system can quantum mechanically tunnel "through" the barrier as well.

This idea, that a macroscopic (in the sense of comprising many many electrons) system could tunnel out of a metastable state like this, had been investigated by Amir Caldeira and Tony Leggett in this important paper, where they worried about the role of dissipation in the environment.  People tried hard to demonstrate this, but issues with thermal radiation and other noise in the experiments were extremely challenging.  With great care in experimental setup, the three laureates put together a remarkable series of papers (here, here, here) that showed all the hallmarks, including resonantly enhancing tunneling with tuned microwaves (designed to kick the system between the levels shown in panel (d) of the figure above).  

This was an impressive demonstration of controllable, macroscopic quantum tunneling, and it also laid the foundation for the devices now used by the whole superconducting quantum computing community.  


Tommaso DorigoInterna

With this post I would like to present a short update of my personal life to the few readers who are interested in that topic. You know, when I started writing online (over 20 years ago!), blogs used to contain a much more personal, sometimes introspective, description of the owner's private life and actions. Since long, they have been substituted by much more agile, quick-to-consume videos. But the old-fashioned bloggers who stuck with that medium continue to have a life - albeit certainly a less glamorous one than that of today's influencers; so some reporting of personal affairs is not out of place here. 

read more

October 07, 2025

n-Category Café A Complex Qutrit Inside an Octonionic One

Dubois-Violette and Todorov noticed that the Standard Model gauge group is the intersection of two maximal subgroups of F 4\mathrm{F}_4. I’m trying to understand these subgroups better.

Very roughly speaking, F 4\mathrm{F}_4 is the symmetry group of an octonionic qutrit. Of the two subgroups I’m talking about, one preserves a chosen octonionic qubit, while the other preserves a chosen complex qutrit.

A precise statement is here:

Over on Mathstodon I’m working with Paul Schwahn to improve this statement. He made a lot of progress on characterizing the first subgroup. F 4\mathrm{F}_4 is really the group of automorphisms of the Jordan algebra of 3×33 \times 3 self-adjoint octonion matrices, 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). He showed the first subgroup, the one I said “preserves a chosen octonionic qubit”, is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to 𝔥 2(𝕆)\mathfrak{h}_2(\mathbb{O}).

Now we want to show the second subgroup, the one I said “preserves a chosen complex qutrit”, is really the subgroup that preserves any chosen Jordan subalgebra isomorphic to 𝔥 3()\mathfrak{h}_3(\mathbb{C}).

I want to sketch out a proof strategy. So, I’ll often say “I hope” for a step that needs to be filled in.

Choose an inclusion of algebras 𝕆\mathbb{C} \hookrightarrow \mathbb{O}. All such choices are related by an automorphism of the octonions, so it won’t matter which one we choose.

There is then an obvious copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) sitting inside 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). I’ll call this the standard copy. To prove the desired result, it’s enough to show:

  1. The subgroup of F 4\mathrm{F}_4 preserving the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) is a maximal subgroup of F 4\mathrm{F}_4, namely (SU(3)×SU(3))/ 3(\text{SU}(3) \times \text{SU}(3))/\mathbb{Z}_3.

  2. All Jordan subalgebras of 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) isomorphic to 𝔥 3()\mathfrak{h}_3(\mathbb{C}) are related to the standard copy by an F 4\mathrm{F}_4 transformation.

Part 1. should be the easier one to show, but I don’t even know if this one is true! (SU(3)×SU(3))/ 3(\text{SU}(3) \times \text{SU}(3))/\mathbb{Z}_3 is a maximal subgroup of F 4\mathrm{F}_4, and Yokota shows it preserves the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). But he shows it also preserves more, seemingly: it preserves a complex structure on the orthogonal complement of that standard copy. Is this really ‘more’ or does it hold automatically for any element of F 4\mathrm{F}_4 that preserves the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C})? I don’t know.

But I want to focus on part 2). Here’s what we’re trying to show: any Jordan subalgebra of 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) isomorphic to 𝔥 3()\mathfrak{h}_3(\mathbb{C}) can be obtained from the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) by applying some element of F 4\mathrm{F}_4.

So, pick a Jordan subalgebra A of 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) isomorphic to 𝔥 3()\mathfrak{h}_3(\mathbb{C}). Pick an isomorphism A ≅ 𝔥 3()\mathfrak{h}_3(\mathbb{C}).

Consider the idempotents

e 1 = diag(1,0,0) e 2 = diag(0,1,0) e 3 = diag(0,0,1) \begin{array}{ccl} e_1 &=& diag(1,0,0) \\ e_2 &=& diag(0,1,0) \\ e_3 &=& diag(0,0,1) \end{array}

in 𝔥 3()\mathfrak{h}_3(\mathbb{C}). Using our isomorphism A𝔥 3()A \cong \mathfrak{h}_3(\mathbb{C}) they give idempotents in AA, which I’ll call f 1,f 2,f 3f_1, f_2, f_3. Since A𝔥 3(𝕆)A \subset \mathfrak{h}_3(\mathbb{O}) these are also idempotents in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}).

Hope 1: I hope there is an element gg of F 4\mathrm{F}_4 mapping f 1,f 2,f 3𝔥 3(𝕆)f_1, f_2, f_3 \mathfrak{h}_3(\mathbb{O}) to e 1,e 2,e 3𝔥 3()e_1, e_2, e_3 \in \mathfrak{h}_3(\mathbb{C})𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}).

Hope 2: Then I hope there is an element hh of F 4\mathrm{F}_4 that fixes e 1,e 2,e 3e_1, e_2, e_3 and maps the subalgebra gAg A to the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}).

If so, we’re done: hgh g maps AA to the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}).

Hope 1 seems to be known. The idempotents e 1,e 2,e 3e_1, e_2, e_3 form a so-called ‘Jordan frame’ for 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}), and so do f 1,f 2,f 3f_1, f_2, f_3. Faraut and Korányi say that “in the irreducible case, the group KK acts transitively on the set of all Jordan frames”, and I think that implies Hope 1.

As for Hope 2, I know the subgroup of F 4\mathrm{F}_4 that fixes e 1,e 2,e 3e_1, e_2, e_3 contains Spin(8)\text{Spin}(8). I bet it’s exactly Spin(8)\text{Spin}(8). But to prove Hope 2 it may be enough to use Spin(8)\text{Spin}(8).

Let me say a bit more about how we might realize Hope 2. It suffices to consider a Jordan subalgebra BB of 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) that is isomorphic to 𝔥 3()\mathfrak{h}_3(\mathbb{C}) and contains

e 1 = diag(1,0,0) e 2 = diag(0,1,0) e 3 = diag(0,0,1) \begin{array}{ccl} e_1 &=& diag(1,0,0) \\ e_2 &=& diag(0,1,0) \\ e_3 &=& diag(0,0,1) \end{array}

and prove that there is an element hh of F 4\mathrm{F}_4 that fixes e 1,e 2,e 3e_1, e_2, e_3 and maps the subalgebra BB to the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}) in 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}). (In case you’re wondering, this BB is what I was calling gAg A.)

Hope 3: I hope that we can show BB consists of matrices

(a 1 z * y * z a 2 x y x * a 3) \left( \begin{array}{ccc} a_1 & z^\ast & y^\ast \\ z & a_2 & x \\ y & x^\ast & a_3 \end{array} \right)

where a 1,a 2,a 3a_1, a_2, a_3 are arbitrary real numbers and x,y,zx, y, z range over 2-dimensional subspaces V 1,V 2,V 3V_1, V_2, V_3 of 𝕆\mathbb{O}. This would already make it look fairly similar to the standard copy of 𝔥 3()\mathfrak{h}_3(\mathbb{C}), where the subspaces V 1,V 2,V 3V_1, V_2, V_3 are all our chosen copy of \mathbb{C} in 𝕆\mathbb{O}.

If Hope 3 is true, the subspaces V 1,V 2,V 3V_1, V_2, V_3 don’t need to be the same, but I believe they do need to obey V 1V 2V 3V_1 V_2 \subseteq V_3 and cyclic permutations thereof, simply because BB is closed under the Jordan product.

So, we naturally want to know if such a triple of 2d subspaces of 𝕆\mathbb{O} must be related to the ‘standard’ one (where they are all \mathbb{C}) by an element of Spin(8)\text{Spin}(8), where Spin(8)\text{Spin}(8) acts on the three copies of 𝕆\mathbb{O} by the vector, left-handed spinor, and right-handed spinor representations, respectively — since this is how Spin(8)\text{Spin}(8) naturally acts on 𝔥 3(𝕆)\mathfrak{h}_3(\mathbb{O}) while fixing all the diagonal matrices.

This is a nice algebra question for those who have thought about triality, and more general ‘trialities’.

So, that’s where I am now: a bunch of hopes which might add up to a clarification of what I mean by “the subgroup of symmetries of an octonionic qutrit that preserve a complex qutrit”.

October 05, 2025

Jordan EllenbergThis condition of ill-training

This condition of ill-training is intensified considerably in an institution like the state university, because of the large number of technical students in attendance, many of whom are more interested in acquiring information than getting a real education, and who look upon time as wasted unless it is put in in the acquiring of cold facts which may later be put to use in the earning of money.

That’s Thomas Arkle Clark, Dean of Men at UIUC, writing in 1921 in his book Discipline and the Derelict. He also writes that 70% of students in his anonymous survey admitted to cheating (“cribbing,” as it was then called.) He is pretty high on the student-athlete, who he says subscribes to ideals that were less-well known in his own time as an Illinois undergrad, back in the ’80s:

The athlete was not always so worthy of emulation as he is at present. I do not have to go back farther than my own college days nor even so far as that to recall instances of men who found their way into colleges for the sole purpose of developing or exhibiting their physical powers, of making an athletic team, and without any intention of adding to their intellectual strength.

John BaezThe Kepler Problem (Part 7)

In Part 5 I explained a cool way to treat bound states of the hydrogen atom as wavefunctions on a sphere in 4-dimensional space. But so far I’ve been neglecting the electron’s spin. Now let’s throw that in too!

This will wind up leading us in some surprising directions. So far I’ve just been reviewing known ideas, but now we’re getting into my new paper:

Second quantization for the Kepler problem.

It starts out being quite routine: to include spin, we just tensor our previous Hilbert space L^2(S^3) with a copy of \mathbb{C}^2 describing the electron’s spin. The resulting space

L^2(S^3) \otimes \mathbb{C}^2

is the Hilbert space of bound states of a spinor-valued version of the Schrödinger equation for the hydrogen atom.

Beware: this is a simplification of a more careful treatment of hydrogen using the Dirac equation: it neglects all spin-dependent terms in Hamiltonian, like spin-orbit interactions. These spin-dependent terms give corrections that go to zero in the limit where the speed of light approaches infinity. So what we’re doing now is giving a nonrelativistic treatment of the hydrogen atom, but taking into account the fact that the electron is a spin-½ particle.

Things get fun now. The Hilbert space L^2(S^3) \otimes \mathbb{C}^2 becomes a unitary representation of \text{SU}(2) in three important ways. The first two come from the actions of \text{SU}(2) on L^2(S^3) by left and right translation, which I explained in Part 5. The third comes from the natural action of \text{SU}(2) on \mathbb{C}^2. All three of these actions of \text{SU}(2) on L^2(S^3) \otimes \mathbb{C}^2 commute with each other. We thus get a unitary representation of \text{SU}(2) \times \text{SU}(2) \times \text{SU}(2) on L^2(S^3) \otimes \mathbb{C}^2.

It is useful to spell this out at the Lie algebra level. In Part 5, I introduced self-adjoint operators A_j and B_j on L^2(S^3): the self-adjoint generators of the left and right translation actions of \text{SU}(2), respectively. Now we’ll tensor these operators with the identity on \mathbb{C}^2 and get operators on L^2(S^3) \otimes \mathbb{C}^2, which by abuse of notation we’ll denote with the same names: A_j and B_j. But we’ll also introduce spin angular momentum operators

S_j = 1 \otimes \frac{1}{2} \sigma_j

on L^2(S^3) \otimes \mathbb{C}^2. These operators obey the following commutation relations:

\begin{array}{cclcccl}    [A_j, A_k] &=&  i\epsilon_{jk\ell} A_\ell  &\quad &  [A_j, B_k] &=& 0 \\ [2pt]    [B_j, B_k] &=&  i\epsilon_{jk\ell} B_\ell &&  [A_j, S_k] &=& 0  \\ [2pt]   [S_j, S_k] &=&  i\epsilon_{jk\ell} S_\ell && [B_j, S_k] &=& 0  \end{array}

Once we have 3 commuting actions of \text{SU}(2) on a Hilbert space we can get more by mixing and matching them. I won’t go overboard and describe all 23 = 8 of them, but I’ll mention some that we need for physics. First we can define orbital angular momentum operators

L_j = A_j + B_j

These obey

\begin{array}{ccl}     [L_j, L_k] &=&  i\epsilon_{jk\ell} L_\ell \\  [2pt]    [S_j, S_k] &=& i \epsilon_{jk\ell} S_\ell \\  [2pt]    [L_j, S_k] &=&  0  \end{array}

Physically speaking, the L_j generate an action of \text{SU}(2) that rotates the position of the electron in space while not changing its spin state, just as the S_j rotate the electron’s spin state while not changing its position.

Adding the spin and orbital angular momentum, we get total angular momentum operators

J_j = L_j + S_j

which obey

[J_j, J_k] = i \epsilon_{jk\ell} J_\ell

These generate an action of \text{SU}(2) that rotates the electron’s wavefunction along with its spin state!

Finally, we define a Hamiltonian for our new hydrogen atom with spin:

\displaystyle{   H \; = \; - \frac{1}{8(A^2 + \frac{1}{4})} \; = \; - \frac{1}{8(B^2 + \frac{1}{4})} }

This is just the Hamiltonian H_0 for the simplified hydrogen atom neglecting spin that we studied in Part 5, tensored with the identity operator on \mathbb{C}^2. Thus it has the same spectrum, but the multiplicity of each eigenvalue has doubled. This Hamiltonian H commutes with all the operators A_j, B_j, S_j, and thus also L_j and J_j.

Now we can reuse our work from Part 5 and decompose our new Hilbert space into eigenspaces of the Hamiltonian H, labeled by n = 1, 2, 3, \dots, and the orbital angular momentum operator J^2, labeled by \ell = 0 , \dots, n-1. We get this:

\displaystyle{  L^2(S^3) \otimes \mathbb{C}^2 \cong      \bigoplus_{n = 1}^\infty \bigoplus_{\ell = 0}^{n-1} V_\ell \otimes \mathbb{C}^2 }

where V_\ell is the spin-\ell representation of the \text{SU}(2) that rotates the electron’s position but not its spin.

In Part 5 we saw a basis |n, \ell, m \rangle of L^2(S^3). If we tensor that with the standard basis of \mathbb{C}^2, we get an orthonormal basis |n , \ell, m, s \rangle of L^2(S^3) \otimes \mathbb{C}^2 where:

• the principal quantum number n ranges over positive integers;

• the azimuthal quantum number \ell ranges from 0 to n-1 in integer steps;

• the magnetic quantum number m ranges from -\ell to \ell in integer steps;

• the spin quantum number s is +\frac{1}{2} or -\frac{1}{2}.

The calculations we did in Part 5 now imply that

\begin{array}{ccl}     A^2 |n, \ell, m, s \rangle &=& B^2 |n, \ell, m, s \rangle \; =  \;    \frac{1}{4}( n^2 - 1) |n, \ell, m, s\rangle  \\  [8pt]  H |n, \ell, m, s \rangle &=& \displaystyle{ - \frac{1}{2n^2}\,  |n, \ell, m, s \rangle } \\ [12pt]      L^2 |n, \ell, m, s\rangle &=& \ell(\ell + 1) |n , \ell, m, s \rangle  \\ [3pt]      L_3 |n , \ell, m, s \rangle &=& m |n , \ell, m, s \rangle  \\ [3pt]      S^2 |n , \ell, m, s \rangle &=& \frac{3}{4} |n , \ell, m, s \rangle  \\  [3pt]      S_3 |n , \ell, m, s \rangle &=& s |n , \ell, m, s \rangle    \end{array}

Combining this with the textbook treatment of the hydrogen atom, it follows that L^2(S^3) \otimes \mathbb{C}^2 is indeed unitarily equivalent to the subspace of L^2(\mathbb{R}^3) \otimes \mathbb{C}^2 consisting of bound states of the spinor-valued Schrödinger equation

i \frac{\partial \psi}{\partial t} = -\frac{1}{2} \nabla^2 \psi - \frac{1}{r} \psi

with the operators H, L_j and S_j having their usual definitions:

\begin{array}{ccl}      H &=&  -\frac{1}{2} \nabla^2 - \frac{1}{r}   \\ [12pt]  L_j &=&  -i\epsilon_{jk\ell} x_k \frac{\partial}{\partial x_\ell}  \\ [10pt]  S_j &=& \frac{1}{2} \sigma_j     \end{array}

In short, the Hamiltonian H on L^2(S^3) \otimes \mathbb{C}^2 is unitarily equivalent to the Hamiltonian on bound states of the hydrogen atom defined in the usual way! We’ve turned hydrogen into a festival of commuting \text{SU}(2) actions.

Next we’ll do something a bit wild, and new.


For more, read my paper:

Second quantization for the Kepler problem.

or these blog articles, which are more expository and fun:

Part 1: a quick overview of Kepler’s work on atoms and the solar system, and more modern developments.

Part 2: why the eccentricity vector is conserved for a particle in an inverse square force, and what it means.

Part 3: why the momentum of a particle in an inverse square force moves around in a circle.

Part 4: why the 4d rotation group \text{SO}(4) acts on bound states of a particle in an attractive inverse square force.

Part 5: quantizing the bound states of a particle in an attractive inverse square force, and getting the Hilbert space L^2(S^3) for bound states of a hydrogen atom, neglecting the electron’s spin.

Part 6: how the Duflo isomorphism explains quantum corrections to the hydrogen atom Hamiltonian.

Part 7: why the Hilbert space of bound states for a hydrogen atom including the electron’s spin is L^2(S^3) \otimes \mathbb{C}^2.

Part 8: why L^2(S^3) \otimes \mathbb{C}^2 is also the Hilbert space for a massless spin-1/2 particle in the Einstein universe.

Part 9: a quaternionic description of the hydrogen atom’s bound states (a digression not needed for later parts).

Part 10: changing the complex structure on L^2(S^3) \otimes \mathbb{C}^2 to eliminate negative-energy states of the massless spin-1/2 particle, as often done.

Part 11: second quantizing the massless spin-1/2 particle and getting a quantum field theory on the Einstein universe, or alternatively a theory of collections of electrons orbiting a nucleus.

Part 12: obtaining the periodic table of elements from a quantum field theory on the Einstein universe.

October 04, 2025

Scott Aaronson Staying sane on a zombie planet

Above is a typical sample of what’s been filling my inbox, all day every day. The emailers first ask me for reasoned dialogue—then, if I respond, they hit me with this stuff. I’m sharing because I think it’s a usefully accurate depiction of what several billion people, most academics in humanities fields, most who call themselves “on the right side of history,” and essentially all those attacking me genuinely believe about the world right now. Because of their anti-Nazism.

Hardly for the first time in my life, this weekend I got floridly denounced every five minutes—on SneerClub, on the blog of Peter Woit, and in my own inbox. The charge this time was that I’m a genocidal Zionist who wants to kill all Palestinian children purely because of his mental illness and raging persecution complex.

Yes, that’s right, I’m the genocidal one—me, whose lifelong dream is that, just like Germany and Japan rose from their necessary devastation in WWII to become pillars of our global civilization, so too the children in Gaza, the West Bank, Syria, Lebanon, and Iran will one day grow up in free and prosperous societies at peace with the West and with Israel. Meanwhile, those who demand an actual genocide of the Jews, another one—those who pray to Allah for it, who attempt it over and over, who preach it to schoolchildren, who celebrate their progress toward it in the streets—they’re all as innocent as lambs.

Yesterday, in The Free Press, came the report of a British writer who traveled to southern Lebanon, and met an otherwise ordinary young man there … who turned out to be excited for Muslims and Christians to join forces to slaughter all the Yahood, and who fully expected that the writer would share his admiration for Hitler, the greatest Yahood-killer ever.

This is what the global far left has now allied itself with. This is what I’m right now being condemned for standing against, with commenter after commenter urging me to seek therapy.

To me, this raises a broader question: how exactly do you keep your sanity, when you live on a planet filled with brain-eaten zombies?

I’m still struggling with that question, but the best I’ve come up with is what I think of as the Weinberg Principle, after my much-missed friend and colleague here at UT Austin. Namely, I believe that it’s better to have one Steven Weinberg on your side while the rest of humanity is against you, than the opposite. Many other individuals (including much less famous ones) would also work here in place of Steve, but I’ll go with him because I think most of my readers would agree to three statements:

  1. Steve’s mind was more in sync with the way the universe really works, than nearly anyone else’s in history. He was to being free from illusions what Usain Bolt is to running or Magnus Carlsen is to chess.
  2. Steve’s toenail clippings constituted a greater contribution to particle physics than would the life’s work of a hundred billion Peter Woits.
  3. Steve’s commitment to Israel’s armed self-defense, and to Zionism more generally, made mine look weak and vacillating in comparison. No one need wonder what he would’ve said about Israel’s current war of survival against the Iranian-led terror axis.

Maybe it’s possible to wake the zombies up. Yoram Arnon, for example, wrote the following eloquent answer on Quora, in response to the question “Why are so many against freeing Palestine?”:

When Westerners think about freedom they think about freedom of speech, freedom of expression, freedom of movement, freedom of religion, freedom to form political parties, etc.

When Palestinians say “Free Palestine” they mean freedom from Jews, and from Israel’s existence. They’re advocating for the abolition of Israel, replacing it with an Arab country.

Israel is the only country in the Middle East that is free, in the Western sense of the word. If Israel were to disappear, Palestinians would fall under an autocratic regime, just like every other Arab country, with none of the above freedoms. And, of course, Israelis would suffer a terrible fate at their hands.

Pro Palestinians are either unable to see this, or want exactly that, but thankfully many in the West do see this – the same “many” that are against “freeing Palestine”.

Palestinians need to accept Israel’s right to exist, and choose to coexist peacefully alongside it, for them to have the peace and freedom the West wants for them.

Maybe reading words like these—or the words of Coleman Hughes, or Douglas Murray, or Hussein Aboubakr Mansour, or Yassine Meskhout, or John Aziz, or Haviv Rettig Gur, or Sam Harris, or the quantum computing pioneer David Deutsch—can boot a few of the zombies’ brains back up. But even then, I fear that these reboots will be isolated successes. For every one who comes back online, a thousand will still shamble along in lockstep, chanting “brainsssssss! genocide! intifada!”

I’m acutely aware of how sheer numbers can create the illusion of argumentative strength. I know many people who were sympathetic to Israel immediately after October 7, but then gradually read the room, saw which side their bread was buttered on, etc. etc. and became increasingly hostile. My reaction, of course, has been exactly the opposite. The bigger the zombie army I see marching against me, the less inclined I feel to become a zombie myself—and the clearer to me becomes the original case for the Zionist project.

So to the pro-Zionist students—Jewish of course, but also Christian, Muslim, Hindu, atheist, and everyone else—who feel isolated and scared to speak right up now, and who also often email me, here’s what I say. Yes, the zombies vastly outnumber us, but on the other hand, they’re zombies. Some of the zombies know longer words than others, but so far, not one has turned out to have a worldview terribly different from that of the image at the top of this post.


I’ll keep the comments closed, for much the same reasons I did in my last post.  Namely, while there are many people of all opinions and backgrounds with whom one can productively discuss these things, there are many more with whom one can’t. Furthermore, experience has shown that the latter can disguise themselves as the former for days on end, and thereby execute a denial-of-service attack on any worthwhile and open public discussion.

Addendum: The troll who sent the antisemitic image now says that he regrets and apologizes for it, and that he’s going to read books on Jewish history to understand his error. I’ll believe that when he actually sends me detailed book reports or other evidence, but just wanted to update.

Jordan EllenbergTaylor Swift and Stephanie Burt, or: Life of a Harvard English Professor

It’s Taylor Swift album day! We listened to it last night at 11pm central when it went live. Snap reaction from AB, for whom 1989 is apex Swift and the last two albums have been too moody and murky, is — this is a winner. She’s happy to have Max Martin (per AB: “that Norwegian guy”) back.

Friend of the blog Stephanie Burt is probably the world’s foremost academic expert on Taylor Swift. She teaches a class on Swift (which is really a class on how songs work, how poems work, how reputations work, how fandoms work, and Swift) at Harvard. And in the kind of publicity no publisher can plan for, her big Swift book, Taylor’s Version, comes out on Monday, in the middle of a global Taylor Swift media blitz. So thoughtful of Tay to drop the album just in time for Stephanie’s pub date!

I, as a trusted friend of Stephanie, already have a copy. I finished reading it yesterday, just in time for the Life of a Showgirl release. It is good, people, really good. If you are interested in how songs work, how poems work, how reputations work, how fandoms work, or Taylor Swift, I implore you to buy a copy at Bookshop, Amazon, or your local store.

(This blog’s favorite TS songs: “Shake it Off,” “Getaway Car,” “Invisible String,” “We Are Never Ever Getting Back Together,” “Welcome to New York.” AB asked me when I first became aware of Taylor Swift. She’s one of the rare acts for whom I can tell you exactly when. Driving east on Mineral Point Rd., “WANEGBT” came on the radio, and it was a “WHAT IS THIS” moment for me — the two other times I remember this happening were the first time I heard Green Day (“Longview”) and the first time I heard New Pornographers (“The Slow Descent into Alcoholism.”) The time David Carlton explained the functor of points to me in the Beacon St. Star Market in Somerville, MA was actually a very similar experience.)

October 03, 2025

Matt von HippelWhen Your Theory Is Already Dead

Occasionally, people try to give “even-handed” accounts of crackpot physics, like people who claim to have invented anti-gravity devices. These accounts don’t go so far as to say that the crackpots are right, and will freely point out plausible doubts about the experiments. But at the end of the day, they’ll conclude that we still don’t really know the answer, and perhaps the next experiment will go differently. More tests are needed.

For someone used to engineering, or to sciences without much theory behind them, this might sound pretty reasonable. Sure, any one test can be critiqued. But you can’t prove a negative: you can’t rule out a future test that might finally see the effect.

That’s all well and good…if you have no idea what you’re doing. But these people, just like anyone else who grapples with physics, aren’t just proposing experiments. They’re proposing theories: models of the world.

And once you’ve got a theory, you don’t just have to care about future experiments. You have to care about past experiments too. Some theories…are already dead.

The "You're already dead" scene from the anime North Star
Warning: this is a link to TVTropes, enter only if you have lots of time on your hands

To get a little more specific, let’s talk about antigravity proposals that use scalar fields.

Scalar fields seem to have some sort of mysticism attached to them in the antigravity crackpot community, but for physicists they’re just the simplest possible type of field, the most obvious thing anyone would have proposed once they were comfortable enough with the idea of fields in the first place. We know of one, the Higgs field, which gives rise to the Higgs boson.

We also know that if there are any more, they’re pretty subtle…and as a result, pretty useless.

We know this because of a wide variety of what are called “fifth-force experiments“, tests and astronomical observations looking for an undiscovered force that, like gravity, reaches out to long distances. Many of these experiments are quite general, the sort of thing that would pick up a wide variety of scalar fields. And so far, none of them have seen anything.

That “so far” doesn’t mean “wait and see”, though. Each time physicists run a fifth-force experiment, they establish a limit. They say, “a fifth force cannot be like this“. It can’t be this strong, it can’t operate on these scales, it can’t obey this model. Each experiment doesn’t just say “no fifth force yet”, it says “no fifth force of this kind, at all”.

When you write down a theory, if you’re not careful, you might find it has already been ruled out by one of these experiments. This happens to physicists all the time. Physicists want to use scalar fields to understand the expansion of the universe, they use them to think about dark matter. And frequently, a model one physicist proposed will be ruled out, not by new experiments, but by someone doing the math and realizing that the model is already contradicted by a pre-existing fifth-force experiment.

So can you prove a negative? Sort of.

If you never commit to a model, if you never propose an explanation, then you can never be disproven, you can always wait for the experiment of your dreams to come true. But if you have any model, any idea, any explanation at all, then your explanation will have implications. Those implications may kill your theory in a future experiment. Or, they may have already killed it.

October 01, 2025

Robert HellingHolosplit

 Recently I had to update Mathematica on my laptop and after having solved the challenges of the license manager that keeps looking different every time I have to use it, I learned that Mathematica 14 can now officially work with finite fields.

This reminded me that for a while I wanted to revive an old project that had vanished together with the hard drive of some old computer: Holosplit. So, over the last two days and with the help of said version of Mathematica I did a complete rewrite which you can now find on Github.

It consists of two C programs "holosplit" and "holojoin". To the first you give a positive integer \(N\) and a file and it spits out a new file ("fragment") that is roughly \(1/N\) of the size. Every time you do that you obtain a new random fragment.

The later you give any collection of \(N\) of these fragments and it reproduces the original file. So you can for example distribute a file over 10 people such that when any 3 of them work together, they can recover the original. 

How does it work? I uses the finite field \(F\) of \(2^3=256\) elements (in the Github repository, there is also a header file that implements arithmetic in \(F\) and matrix operations like product and inverse over it). Each time, it is invoked, it picks a random vector \(v\in F^N\) and writes it to the output. Then it reads \(N\) bytes from the file at a time which it also interprets as a vector \(d\in F^N\). It then outputs the byte that corresponds to the scalar product \(v\cdot d\).

To reassemble the file, holojoin takes the \(N\) files with its random vectors \(v_1,\ldots,v_N\) and interprets those as the rows of a \(N\times N\) matrix \(A\). With probability

$$\frac{\prod_{k=1}^N \left(256^N-k\right)}{(256)^{N^2}}$$

which exponentially in \(N\) approaches 1 this matrix is invertible (homework: why?). So we can read one byte from each file, assemble those into yet another vector \(e\in F^N\) and recover

$$d=A^{-1}e.$$

Besides the mathematics, it also poses philosophical/legal questions: Consider for example the original file is copyrighted, for example an mp3 or a video. The fragments are clearly derived works. But individually, they do not contain the original work, without sufficiently many other fragments they are useless (although not in a cryptographic sense). So by publishing one fragment, I do not provide access to the original work. What if others publish other fragments? Then my fragment could be the last remaining one that was missing. If there are more, any individual fragment is redundant so publishing it strictly speaking does not provide new information. 

September 29, 2025

Scott Aaronson The QMA Singularity

Update (Sep. 29): Since this post has now gone semi-viral on X, Hacker News, etc., with people arguing about how trivial or nontrivial was GPT5’s “discovery,” it seems worthwhile to say something that was implicit in the post.

Namely, GPT5-Thinking’s suggestion of a function to use “should have” been obvious to us. It would have been obvious to us had we known more, or had we spent more time studying the literature or asking experts.

The point is, anyone engaged in mathematical research knows that an AI that can “merely” fill in the insights that “should’ve been” obvious to you is a really huge freaking deal! It speeds up the actual discovery process, as opposed to the process of writing LaTeX or preparing the bibliography or whatever. This post gave one tiny example of what I’m sure will soon be thousands.

I should also add that, since this post went up, a commenter named Phillip Harris proposed a better function to use than GPT-5’s: det(I-E) rather than Tr[(I-E)-1]. While we’re still checking details, not only do we think this works, we think it simplifies our argument and solves one of our open problems. So it seems human supremacy has been restored, at least for now!


A couple days ago, Freek Witteveen of CWI and I posted a paper to the arXiv called “Limits to black-box amplification in QMA.” Let me share the abstract:

We study the limitations of black-box amplification in the quantum complexity class QMA. Amplification is known to boost any inverse-polynomial gap between completeness and soundness to exponentially small error, and a recent result (Jeffery and Witteveen, 2025) shows that completeness can in fact be amplified to be doubly exponentially close to 1. We prove that this is optimal for black-box procedures: we provide a quantum oracle relative to which no QMA verification procedure using polynomial resources can achieve completeness closer to 1 than doubly exponential, or a soundness which is super-exponentially small. This is proven by using techniques from complex approximation theory, to make the oracle separation from (Aaronson, 2008), between QMA and QMA with perfect completeness, quantitative.

You can also check out my PowerPoint slides here.

To explain the context: QMA, or Quantum Merlin Arthur, is the canonical quantum version of NP. It’s the class of all decision problems for which, if the answer is “yes,” then Merlin can send Arthur a quantum witness state that causes him to accept with probability at least 2/3 (after a polynomial-time quantum computation), while if the answer is “no,” then regardless of what witness Merlin sends, Arthur accepts with probability at most 1/3. Here, as usual in complexity theory, the constants 2/3 and 1/3 are just conventions, which can be replaced (for example) by 1-2-n and 2-n using amplification.

A longstanding open problem about QMA—not the biggest problem, but arguably the most annoying—has been whether the 2/3 can be replaced by 1, as it can be for classical MA for example. In other words, does QMA = QMA1, where QMA1 is the subclass of QMA that admits protocols with “perfect completeness”? In 2008, I used real analysis to show that there’s a quantum oracle relative to which QMA ≠ QMA1, which means that any proof of QMA = QMA1 would need to use “quantumly nonrelativizing techniques” (not at all an insuperable barrier, but at least we learned something about why the problem is nontrivial).

Then came a bombshell: in June, Freek Witteveen and longtime friend-of-the-blog Stacey Jeffery released a paper showing that any QMA protocol can be amplified, in a black-box manner, to have completeness error that’s doubly exponentially small, 1/exp(exp(n)). They did this via a method I never would’ve thought of, wherein a probability of acceptance is encoded via the amplitudes of a quantum state that decrease in a geometric series. QMA, it turned out, was an old friend that still had surprises up its sleeve after a quarter-century.

In August, we had Freek speak about this breakthrough by Zoom in our quantum group meeting at UT Austin. Later that day, I asked Freek whether their new protocol was the best you could hope to do with black-box techniques, or whether for example one could amplify the completeness error to be triply exponentially small, 1/exp(exp(exp(n))). About a week later, Freek and I had a full proof written down that, using black-box techniques, doubly-exponentially small completeness error is the best you can do. In other words: we showed that, when one makes my 2008 QMA ≠ QMA1 quantum oracle separation quantitative, one gets a lower bound that precisely matches Freek and Stacey’s protocol.

All this will, I hope, interest and excite aficianados of quantum complexity classes, while others might have very little reason to care.

But here’s a reason why other people might care. This is the first paper I’ve ever put out for which a key technical step in the proof of the main result came from AI—specifically, from GPT5-Thinking. Here was the situation: we had an N×N Hermitian matrix E(θ) (where, say, N=2n), each of whose entries was a poly(n)-degree trigonometric polynomial in a real parameter θ. We needed to study the largest eigenvalue of E(θ), as θ varied from 0 to 1, to show that this λmax(E(θ)) couldn’t start out close to 0 but then spend a long time “hanging out” ridiculously close to 1, like 1/exp(exp(exp(n))) close for example.

Given a week or two to try out ideas and search the literature, I’m pretty sure that Freek and I could’ve solved this problem ourselves. Instead, though, I simply asked GPT5-Thinking. After five minutes, it gave me something confident, plausible-looking, and (I could tell) wrong. But rather than laughing at the silly AI like a skeptic might do, I told GPT5 how I knew it was wrong. It thought some more, apologized, and tried again, and gave me something better. So it went for a few iterations, much like interacting with a grad student or colleague. Within a half hour, it had suggested to look at the function

$$ Tr[(I-E(\theta))^{-1}] = \sum_{i=1}^N \frac{1}{1-\lambda_i(\theta)}. $$

It pointed out, correctly, that this was a rational function in θ of controllable degree, that happened to encode the relevant information about how close the largest eigenvalue λmax(E(θ)) is to 1. And this … worked, as we could easily check ourselves with no AI assistance. And I mean, maybe GPT5 had seen this or a similar construction somewhere in its training data. But there’s not the slightest doubt that, if a student had given it to me, I would’ve called it clever. Obvious with hindsight, but many such ideas are.

I had tried similar problems a year ago, with the then-new GPT reasoning models, but I didn’t get results that were nearly as good. Now, in September 2025, I’m here to tell you that AI has finally come for what my experience tells me is the most quintessentially human of all human intellectual activities: namely, proving oracle separations between quantum complexity classes. Right now, it almost certainly can’t write the whole research paper (at least if you want it to be correct and good), but it can help you get unstuck if you otherwise know what you’re doing, which you might call a sweet spot. Who knows how long this state of affairs will last? I guess I should be grateful that I have tenure.

September 26, 2025

Matt von HippelRequests for an Ethnography of Cheating

What is AI doing to higher education? And what, if anything, should be done about it?

Chad Orzel at Counting Atoms had a post on this recently, tying the question to a broader point. There is a fundamental tension in universities, between actual teaching and learning and credentials. A student who just wants the piece of paper at the end has no reason not to cheat if they can get away with it, so the easier it becomes to get away with cheating (say, by using AI), the less meaningful the credential gets. Meanwhile, professors who want students to actually learn something are reduced to trying to “trick” these goal-oriented students into accidentally doing something that makes them fall in love with a subject, while being required to police the credential side of things.

Social science, as Orzel admits and emphasizes, is hard. Any broad-strokes picture like this breaks down into details, and while Orzel talks through some of those details he and I are of course not social scientists.

Because of that, I’m not going to propose my own “theory” here. Instead, think of this post as a request.

I want to read an ethnography of cheating. Like other ethnographies, it should involve someone spending time in the culture in question (here, cheating students), talking to the people involved, and getting a feeling for what they believe and value. Ideally, it would be augmented with an attempt at quantitative data, like surveys, that estimate how representative the picture is.

I suspect that cheating students aren’t just trying to get a credential. Part of why is that I remember teaching pre-meds. In the US, students don’t directly study medicine as a Bachelor’s degree. Instead, they study other subjects as pre-medical students (“pre-meds”), and then apply to Medical School, which grants a degree on the same level as a PhD. As part of their application, they include a standardized test called the MCAT, which checks that they have the basic level of math and science that the medical schools expect.

A pre-med in a physics class, then, has good reason to want to learn: the better they know their physics, the better they will do on the MCAT. If cheating was mostly about just trying to get a credential, pre-meds wouldn’t cheat.

I’m pretty sure they do cheat, though. I didn’t catch any cheaters back when I taught, but there were a lot of students who tried to push the rules, pre-meds and not.

Instead, I think there are a few other motivations involved. And in an ethnography of cheating, I’d love to see some attempt to estimate how prevalent they are:

  1. Temptation: Maybe students know that they shouldn’t cheat, in the same way they know they should go to the gym. They want to understand the material and learn in the same way people who exercise have physical goals. But the mind, and flesh, are weak. You have a rough week, you feel like you can’t handle the work right now. So you compensate. Some of the motivation here is still due to credentials: a student who shrugs and accepts that their breakup will result in failing a course is a student who might have to pay for an extra year of ultra-expensive US university education to get that credential. But I suspect there is a more fundamental motivation here, related to ego and easy self-deception. If you do the assignment, even if you cheat for part of it, you get to feel like you did it, while if you just turn in a blank page you have to accept the failure.
  2. Skepticism: Education isn’t worth much if it doesn’t actually work. Students may be skeptical that the things that professors are asking them to do actually help them learn what they want to learn, or that the things the professors want them to learn are actually the course’s most valuable content. A student who uses ChatGPT to write an essay might believe that they will never have to write something without ChatGPT in life, so why not use it now? Sometimes professors simply aren’t explicit about what an exercise is actually meant to teach (there have been a huge number of blog posts explaining that writing is meant to teach you to think, not to write), and sometimes professors are genuinely pretty bad at teaching, since there is little done to retain the good ones in most places. A student in this situation still has to be optimistic about some aspect of the education, at some time. But they may be disillusioned, or just interested in something very different.
  3. Internalized Expectations: Do employers actually care if you get a bad grade? Does it matter? By the time a student is in college, they’ve been spending half their waking hours in a school environment for over a decade. Maybe the need to get good grades is so thoroughly drilled in that the actual incentives don’t matter. If you think of yourself as the kind of person who doesn’t fail courses, and you start failing, what do you do?
  4. External Non-Credential Expectations: Don’t worry about the employers, worry about the parents. Some college students have the kind of parents who keep checking in on how they’re doing, who want to see evidence and progress the same way they did when they were kids. Any feedback, no matter how much it’s intended to teach, not to judge, might get twisted into a judgement. Better to avoid that judgement, right?
  5. Credentials, but for the Government, not Employers: Of course, for some students, failing really does wreck their life. If you’re on the kind of student visa that requires you maintain grades a certain level, you’ve got a much stronger incentive to cheat, imposed for much less reason.

If you’re aware of a good ethnography of cheating, let me know! And if you’re a social scientist, consider studying this!

Peter Rohde Photo albums

Peter’s photos: https://www.icloud.com/sharedalbum/#B275oqs3qKSZvQ

Screenshots: https://www.icloud.com/sharedalbum/#B27532ODWjIQb9

Climbing book launch: https://www.icloud.com/sharedalbum/#B27GWZuqDGnuOyN

Salisbury waters: https://www.icloud.com/sharedalbum/#B275qXGF1JQFkx

Christmas with Ash: https://www.icloud.com/sharedalbum/#B27G6XBubAhoT6

Hosin BBQ duck: https://www.icloud.com/sharedalbum/#B27GY8gBYG3b5mD

Hawks Nest to Smiths Lake: https://www.icloud.com/sharedalbum/#B2759UlCqSH5bE

Europe & Alps: https://www.icloud.com/sharedalbum/#B275ON9t3W0lu

Point Perpendicular: https://www.icloud.com/sharedalbum/#B27GqkRUiGivXD2

Newnes canyoning: https://www.icloud.com/sharedalbum/#B27GfnH8tgHSmX

Coffs Harbour to Yamba: https://www.icloud.com/sharedalbum/#B27J0DiRHJKuuWr

Wendy Bruere Christmas (2020): https://www.icloud.com/sharedalbum/#B27G4TcsmGoHysj

Six Foot Track: https://www.icloud.com/sharedalbum/#B2753qWtHZA9EX

Kosciusko to Kiandra: https://www.icloud.com/sharedalbum/#B27GgZLKuGaewVm

Camping food: https://www.icloud.com/sharedalbum/#B27GtnIORgbmHu

The Aardvark: https://www.icloud.com/sharedalbum/#B275VaUrzvmAiT

Kangaroo Valley kayaking: https://www.icloud.com/sharedalbum/#B27JEsNWnJrCpi0

Claustral canyon: https://www.icloud.com/sharedalbum/#B2755Z2WMOTpsk

Budawang: https://www.icloud.com/sharedalbum/#B27GDdyTvGvpINL

Mother’s Day panoramas (2021): https://www.icloud.com/sharedalbum/#B27GFssfGG9WmJP

Point Perpendicular & Nowra: https://www.icloud.com/sharedalbum/#B27GRMtznGPdeuZ

Blood moon: https://www.icloud.com/sharedalbum/#B27GdIshaG8NgGX

La Perouse to Coogee: https://www.icloud.com/sharedalbum/#B275aVbMK4h7qo

Canberra ASPI launch: https://www.icloud.com/sharedalbum/#B27GQOeMmGj4Zcv

Edible foraging: https://www.icloud.com/sharedalbum/#B275ejO179Si0N

Sydney to Wollongong: https://www.icloud.com/sharedalbum/#B275M7GFPUasMe

Album for Dad, Father’s Day (2021): https://www.icloud.com/sharedalbum/#B2752plgjnnkUe

Vaucluse (with Cheryl, Nestor & Wendy): https://www.icloud.com/sharedalbum/#B275CmvAS4uA0Z

Bouddi National Park: https://www.icloud.com/sharedalbum/#B27GdPblXG8WdOo

Tom Thumb (the 2nd): https://www.icloud.com/sharedalbum/#B275aDWbr4CN2w

Eden to Victoria: https://www.icloud.com/sharedalbum/#B27GJDfWGArX8l

Wendy’s book launch (the 2nd): https://www.icloud.com/sharedalbum/#B27GIcgc2G7h08y

Mark & Pat Bruere visit Sydney: https://www.icloud.com/sharedalbum/#B27G0ehgLbyWyg

New Years Eve climb (2021): https://www.icloud.com/sharedalbum/#B27Ju8EH6JOZxmU

Newnes Canyoning (2022): https://www.icloud.com/sharedalbum/#B275BydzFU0GZ8

Royal National Park (2022): https://www.icloud.com/sharedalbum/#B27GlxzuqGVI5nE

Peter & Wendy: https://www.icloud.com/sharedalbum/#B27Gf693ZG52tfd

Book photo shoots: too rude…

Wendy & Peter’s mushroom trip: https://www.icloud.com/sharedalbum/#B27GrhkPxG27So8

Post-mushroom hike: https://www.icloud.com/sharedalbum/#B27GdFryYG8i3Ur

Wendy Kalymnos favourites: https://www.icloud.com/sharedalbum/#B27JqstnBJEXkH2

Wendy Frenchmans screenshots: https://www.icloud.com/sharedalbum/#B27Jr1PPdJpd7Dq

Instagram: https://www.icloud.com/sharedalbum/#B27GzFCC1Gb4tqr

Haute route: https://www.icloud.com/sharedalbum/#B27J8GySPJtWoQ1

Kim’s KKKalendar: https://www.icloud.com/sharedalbum/#B275fk75vIL0sH

Frenchmans Cap Wild: https://www.icloud.com/sharedalbum/#B27G4VTwGGoFBkz

Photoshoot with Zixin: https://www.icloud.com/sharedalbum/#B27GPCdxkGKPkM4

Wendy birthday hike (2023): https://www.icloud.com/sharedalbum/#B27GWBC59GnHpQW

Bateman’s Bay to Bawley Point: https://www.icloud.com/sharedalbum/#B27JsHvHoJ8bxWf

Stockton Sand dunes (2023): https://www.icloud.com/sharedalbum/#B27GVfZ2vGloFZV

Wendy book launch (2023): https://www.icloud.com/sharedalbum/#B27J058xyJR4IBM

Dolomites (2023): https://www.icloud.com/sharedalbum/#B0Z5kuVsbGJUzKO

Mount Arapiles: https://www.icloud.com/sharedalbum/#B275GH8Mq8Uh2X

Mount Solitary loop: https://www.icloud.com/sharedalbum/#B275nhQST2mETE

Klaus Hanz Franz Rohde Kunst: https://www.icloud.com/sharedalbum/#B27GqQrCLGiY3vb

Klaus Rohde funeral slideshow: https://www.icloud.com/sharedalbum/#B27GDZLe8GXP58K

Dad (old, B&W): https://www.icloud.com/sharedalbum/#B27GLLXGLJ5mbT2

Klaus & Ursula wedding: https://www.icloud.com/sharedalbum/#B275cLqfN7154g

Test Greece: https://www.icloud.com/sharedalbum/#B27Jq4WnLJ6JMNd

From Will Skea (Alps): https://www.icloud.com/sharedalbum/#B27JHciePJFwacG

From Will Skea (Frenchmans Cap): https://www.icloud.com/sharedalbum/#B275ZhN2v3EVq6

From Will Skea (Arapiles): https://www.icloud.com/sharedalbum/#B27JPrgBGJu3BTD

Coffs Harbour to Yamba (2): https://www.icloud.com/sharedalbum/#B27GFqhgJG9LHgT

Mark magic show (2021): https://www.icloud.com/sharedalbum/#B27G60dj6ARCvd

Wendy Christmas present (2020): https://www.icloud.com/sharedalbum/#B275FrPQ6GxvRu

AHS 25 year reunion: https://www.icloud.com/sharedalbum/#B275O3DjHUvSv

WhatsApp: https://www.icloud.com/sharedalbum/#B275tzEA5fX1nc

Armidale High School: https://www.icloud.com/sharedalbum/#B27GnbeumG4PnAF

Book photos for Mum & Dad: https://www.icloud.com/sharedalbum/#B27Gtec4XQkASe

Miscellaneous: https://www.icloud.com/sharedalbum/#B27Gq6kMgGKn7GR

Three Capes Trail (2022): https://www.icloud.com/sharedalbum/#B27G7HOIlGrDUGZ

Childhood computer programming: https://www.icloud.com/sharedalbum/#B275fu2MutDU8N

Magic with Mark in Maroubra: https://www.icloud.com/sharedalbum/#B27Gv6DhEGD9U3G

Photoshoot with Zixin (2024): https://www.icloud.com/sharedalbum/#B27GCATCnJGoRfW

Butt Crack (2021): https://www.icloud.com/sharedalbum/#B275VtHQfMv0zw

Greece photos new (edited to remove photos from wrong album): https://www.icloud.com/sharedalbum/#B27GY3uThGoBcGj

Singapore (all combined): https://www.icloud.com/sharedalbum/#B275qsTcwJKJjl

Hong Kong (transit): https://www.icloud.com/sharedalbum/#B2759v1AbS8Hve

Taiwan: https://www.icloud.com/sharedalbum/#B27GQD2D7Gw0hAp

India (combined): https://www.icloud.com/sharedalbum/#B27Gtue8VQy83g

Freycinet: https://www.icloud.com/sharedalbum/#B27G5VpecGE5Tbg

Triglav: https://www.icloud.com/sharedalbum/#B275MbK9Vy8erz

Shared with me: https://www.icloud.com/sharedalbum/#B27GGXqixzPOrm

Mount Wellington climbing: https://www.icloud.com/sharedalbum/#B27Gd59qiG8Kjy4

New Zealand combined (2004): https://www.icloud.com/sharedalbum/#B27GIZ8BIGNN5jy

New Zealand combined (2005): https://www.icloud.com/sharedalbum/#B27GcuRfIGFVIcL

Yea: https://www.icloud.com/sharedalbum/#B27GZYbYHGhFIir

Mount Pleasant: https://www.icloud.com/sharedalbum/#B275Iy2hC0JTTL

D’Aguilar: https://www.icloud.com/sharedalbum/#B27Gh7fzTGZBosS

Bali (2001): https://www.icloud.com/sharedalbum/#B27G1qNHBGOTbIr

Samba Ninjas: https://www.icloud.com/sharedalbum/#B27GG34bAzqQ0v

Armidale (misc): https://www.icloud.com/sharedalbum/#B27GSkLVwGyobbX

Emma’s party (2008): https://www.icloud.com/sharedalbum/#B275S2ms99Zyby

Goettingen (2011): https://www.icloud.com/sharedalbum/#B27JIrbT3Jsgxhd

South Coast track: https://www.icloud.com/sharedalbum/#B27G58NWBG6QyN7

Minsk (2006): https://www.icloud.com/sharedalbum/#B27G3JpSBGX1UkQ

Baden-Baden (2019): https://www.icloud.com/sharedalbum/#B27595X5HTVzJr

Berlin (combined): https://www.icloud.com/sharedalbum/#B27JqWzChJ6qizD

Switzerland (combined): https://www.icloud.com/sharedalbum/#B275zXwoYGJ6HMF

Italy highlights: https://www.icloud.com/sharedalbum/#B27G47PHQGoJium

Germany (misc): https://www.icloud.com/sharedalbum/#B275hPMfYGu5xVJ

Garmisch (2022): https://www.icloud.com/sharedalbum/#B27GFsbvlG9Xrr6

Germany (2019): https://www.icloud.com/sharedalbum/#B27G6Mn98G56Ncb

Garmisch (2006): https://www.icloud.com/sharedalbum/#B27J5lIdKGLC9KG

Baden-Baden (2005): https://www.icloud.com/sharedalbum/#B275sWRpHHQkt9

Berlin (2005): https://www.icloud.com/sharedalbum/#B27GgOQtrGjQrpH

Zugspitze (2005): https://www.icloud.com/sharedalbum/#B27G81mNdGcApGt

Amsterdam, Bristol (2006): https://www.icloud.com/sharedalbum/#B275B9SRzyBjlH

Baden-Baden (2006): https://www.icloud.com/sharedalbum/#B275eD9V79I2XR

Berlin (2006): https://www.icloud.com/sharedalbum/#B275toRf1fH8MD

Berlin, Jena (2007): https://www.icloud.com/sharedalbum/#B27GTI3fvGVgNit

Erlangen (2006): https://www.icloud.com/sharedalbum/#B27JrotZ2JpMb0i

Garmisch (2010): https://www.icloud.com/sharedalbum/#B27JPJPSiJurzNg

Germany (2010): https://www.icloud.com/sharedalbum/#B275FhYPQP650

Stuttgart (2006): https://www.icloud.com/sharedalbum/#B27GmitydGVVaZh

Changi (2019): https://www.icloud.com/sharedalbum/#B27GnmlKoG4JHpX

Japan (2007): https://www.icloud.com/sharedalbum/#B275AerZbG6FxVL

Japan (2012): https://www.icloud.com/sharedalbum/#B27GjBjobGg6PUa

Miscellaneous (including Japan 2013): https://www.icloud.com/sharedalbum/#B27GTpbybGySbE8

Currumbin & Tugin (2021): https://www.icloud.com/sharedalbum/#B275vBKZ4xH9X6

Brisbane (2021): https://www.icloud.com/sharedalbum/#B275YHsSjxQnm0

Weed in Byron (26/6/2025): https://www.icloud.com/sharedalbum/#B275Q2ydoGsQ4O5

Weed in Byron 2: https://www.icloud.com/sharedalbum/#B27GQDYhLGwsuY4

September 25, 2025

Tim GowersCreating a database of motivated proofs

It’s been over three years since my last post on this blog and I have sometimes been asked, understandably, whether the project I announced in my previous post was actually happening. The answer is yes — the grant I received from the Astera Institute has funded several PhD students and a couple of postdocs, and we have been busy. In my previous post I suggested that I would be open to remote collaboration, but that has happened much less, partly because a Polymath-style approach would have been difficult to manage while also ensuring that my PhD students would have work that they could call their own to put in their theses.

In general I don’t see a satisfactory solution to that problem, but in this post I want to mention a subproject of the main project that is very much intended to be a large public collaboration. A few months ago, a call came out from Renaissance Philanthropies saying that they were launching a $9m AI for Math Fund to spend on projects in the general sphere of AI and mathematics, and inviting proposals. One of the categories that they specifically mentioned was creating new databases, and my group submitted a proposal to create a database of what we call “structured motivated proofs,” a piece of terminology that I will explain a bit more later in just a moment. I am happy to report that our proposal was one of the 29 successful ones. Since a good outcome to the project will depend on collaboration from many people outside the group, we need to publicize it, which is precisely the purpose of this post. Below I will be more specific about the kind of help we are looking for.

Why might yet another database of theorems and proofs be useful?

The underlying thought behind this project is that AI for mathematics is being held back not so much by an insufficient quantity of data as by the wrong kind of data. All mathematicians know, and some of us enjoy complaining about it, that it is common practice when presenting a proof in a mathematics paper, or even textbook, to hide the thought processes that led to the proof. Often this does not matter too much, because the thought processes may be standard ones that do not need to be spelt out to the intended audience. But when proofs start to get longer and more difficult, they can be hard to read because one has to absorb definitions and lemma statements that are not obviously useful, are presented as if they appeared from nowhere, and demonstrate their utility only much later in the argument.

A sign that this is a problem for AI is the behaviour one observes after asking an LLM to prove a statement that is too difficult for it. Very often, instead of admitting defeat, it will imitate the style of a typical mathematics paper and produce rabbits out of hats, together with arguments later on that those rabbits do the required job. The problem is that, unlike with a correct mathematics paper, one finds when one scrutinizes the arguments carefully that they are wrong. However, it is hard to find superficial features that distinguish between an incorrect rabbit with an incorrect argument justifying that rabbit (especially if the argument does not go into full detail) and a correct one, so the kinds of statistical methods used by LLMs do not have an easy way to penalize the incorrectness.

Of course, that does not mean that LLMs cannot do mathematics at all — they are remarkably good at it, at least compared with what I would have expected three years ago. How can that be, given the problem I have discussed in the previous paragraph?

The way I see it (which could change — things move so fast in this sphere), the data that is currently available to train LLMs and other systems is very suitable for a certain way of doing mathematics that I call guess and check. When trying to solve a maths problem, you will normally write down the routine parts of an argument without any fuss (and an LLM can do them too because it has seen plenty of similar examples), but if the problem as a whole is not routine, then at some point you have to stop and think, often because you need to construct an object that has certain properties (I mean this in a rather general way — the “object” might be a lemma that will split up the proof in a nice way) and it is not obvious how to do so. The guess-and-check approach to such moments is what it says: you make as intelligent a guess as you can and then see whether it has the properties you wanted. If it doesn’t, you make another guess, and you keep going until you get lucky.

The reason an LLM might be tempted to use this kind of approach is that the style of mathematical writing I described above makes it look as though that is what we as mathematicians do. Of course, we don’t actually do that, but we tend not to mention all the failed guesses we made and how we carefully examined why they failed, modifying them in appropriate ways in response, until we finally converged on an object that worked. We also don’t mention the reasoning that often takes place before we make the guess, saying to ourselves things like “Clearly an Abelian group can’t have that property, so I need to look for a non-Abelian group.”

Intelligent guess and check works well a lot of the time, particularly when carried out by an LLM that has seen many proofs of many theorems. I have often been surprised when I have asked an LLM a problem of the form \exists x\in X \ P(x), where P is some property that is hard to satisfy, and the LLM has had no trouble answering it. But somehow when this happens, the flavour of the answer given by the LLM leaves me with the impression that the technique it has used to construct x is one that it has seen before and regards as standard.

If the above picture of what LLMs can do is correct (the considerations for reinforcement-learning-based systems such as AlphaProof are not identical but I think that much of what I say in this post applies to them too for slightly different reasons), then the likely consequence is that if we pursue current approaches, then we will reach a plateau: broadly speaking they will be very good at answering a question if it is the kind of question that a mathematician with the right domain expertise and good instincts would find reasonably straightforward, but will struggle with anything that is not of that kind. In particular, they will struggle with research-level problems, which are, almost by definition, problems that experts in the area do not find straightforward. (Of course, there would probably be cases where an LLM spots relatively easy arguments that the experts had missed, but that wouldn’t fundamentally alter the fact that they weren’t really capable of doing research-level mathematics.)

But what if we had a database of theorems and proofs that did not hide the thought processes that lay behind the non-obvious details of the proofs? If we could train AI on a database of accounts of proof discoveries and if, having done so, we then asked it to provide similar accounts, then it would no longer resort to guess-and-check when it got stuck, because the proof-discovery accounts it had been trained on would not be resorting to it. There could be a problem getting it to unlearn its bad habits, but I don’t think that difficulty would be impossible to surmount.

The next question is what such a database might look like. One could just invite people to send in stream-of-consciousness accounts of how they themselves found certain proofs, but that option is unsatisfactory for several reasons.

  1. It can be very hard to remember where an idea came from, even a few seconds after one has had it — in that respect it is like a dream, the memory of which becomes rapidly less vivid as one wakes up.
  2. Often an idea will seem fairly obvious to one person but not to another.
  3. The phrase “motivated proof” means different things to different people, so without a lot of careful moderation and curation of entries, there is a risk that a database would be disorganized and not much more helpful than a database of conventionally written proofs.
  4. A stream-of-consciousness account could end up being a bit too much about the person who finds the proof and not enough about the mathematical reasons for the proof being feasibly discoverable.

To deal with these kinds of difficulties, we plan to introduce a notion of a structured motivated proof, by which we mean a proof that is generated in a very particular way that I will partially describe below. A major part of the project, and part of the reason we needed funding for it, is to create a platform that will make it convenient to input structured motivated proofs and difficult to insert the kinds of rabbits out of hats that make a proof mysterious and unmotivated. In this way we hope to gamify the task of creating the database, challenging people to input into our system proofs of certain theorems that appear to rely on “magic” ideas, and perhaps even offering prizes for proofs that contain steps that appear in advance to be particularly hard to motivate. (An example: the solution by Ellenberg and Gijswijt of the cap-set problem uses polynomials in a magic-seeming way. The idea of using polynomials came from an earlier paper of Croot, Lev and Pach that proved a closely related theorem, but in that paper it just appears in the statement of their Lemma 1, with no prior discussion apart from the words “in the present paper we use the polynomial method” in the introduction.)

What is a structured motivated proof?

I wrote about motivated proofs in my previous post, but thanks to many discussions with other members of the group, my ideas have developed quite a lot since then. Here are two ways we like to think about the concept.

1. A structured motivated proof is one that is generated by standard moves.

I will not go into full detail about what I mean by this, but will do so in a future post when we have created the platform that we would like people to use in order to input proofs into the database. But the basic idea is that at any one moment one is in a certain state, which we call a proof-discovery state, and there will be a set of possible moves that can take one from the current proof-discovery state to a new one.

A proof-discovery state is supposed to be a more formal representation of the state one is in when in the middle of solving a problem. Typically, if the problem is difficult, one will have asked a number of questions, and will be aware of logical relationships between them: for example, one might know that a positive answer to Q1 could be used to create a counterexample to Q2, or that Q3 is a special case of Q4, and so on. One will also have proved some results connected with the original question, and again these results will be related to each other and to the original problem in various ways that might be quite complicated: for example P1 might be a special case of Q2, which, if true would reduce Q3 to Q4, where Q3 is a generalization of the statement we are trying to prove.

Typically we will be focusing on one of the questions, and typically that question will take the form of some hypotheses and a target (the question being whether the hypotheses imply the target). One kind of move we might make is a standard logical move such as forwards or backwards reasoning: for example, if we have hypotheses of the form P(x) and \forall u\ P(u)\implies Q(u), then we might decide to deduce Q(x). But things get more interesting when we consider slightly less basic actions we might take. Here are three examples.

  1. We have in our list of hypotheses the fact that a function f is given by the formula f(x)=\exp(p(x)), where p is a polynomial, and our goal is to prove that there exists z such that f(z)=1. Without really thinking about it, we are conscious that f is a composition of two functions, one of which is continuous and one of which belongs to a class of functions that are all continuous, so f is continuous. Also, the conclusion \exists z\ f(z)=1 matches well the conclusion of the intermediate-value theorem. So the intermediate-value theorem comes naturally to mind and we add it to our list of available hypotheses. In practice we wouldn’t necessarily write it down, but the system we wish to develop is intended to model not just what we write down but also what is going on in our brains, so we propose a move that we call library extraction (closely related to what is often called premise selection in the literature). Note that we have to be a bit careful about library extraction. We don’t want the system to be allowed to call up results from the library that appear to be irrelevant but then magically turn out to be helpful, since those would feel like rabbits out of hats. So we want to allow extraction of results only if they are obvious given the context. It is not easy to define what “obvious” means, but there is a good rule of thumb for it: a library extraction is obvious if it is one of the first things ChatGPT thinks of when given a suitable non-cheating prompt. For example, I gave it the prompt, “I have a function f from the reals to the reals and I want to prove that there exists some z such that f(z)=1. Can you suggest any results that might be helpful?” and the intermediate-value theorem was its second suggestion. (Note that I had not even told it that f was continuous, so I did not need to make that particular observation before coming up with the prompt.)
  2. We have a goal of the form \exists x\in X\ P(x). If this were a Lean proof state, the most common way to discharge a goal of this form would be to input a choice for x. That is, we would instantiate the existential quantifier with some x_0 and our new goal would be P(x_0). However, as with library extraction, we have to be very careful about instantiation if we want our proof to be motivated, since we wish to disallow highly surprising choices of x_0 that can be found only after a long process of thought. So we have to restrict ourselves to obvious instantiations. One way that an instantiation in our system will count as obvious is if the variable is instantiated with a term that is already present in the proof-discovery state. If the desired term is not present, then in order to continue with the proof, it will be necessary to carry out moves that generate it. A very common technique for this is the use of metavariables: instead of guessing a suitable x_0, we create a variable x^\bullet and change the goal to P(x^\bullet), which we can think of as saying “I’m going to start trying to prove P(x^\bullet) even though I haven’t chosen x^\bullet yet. As the attempted proof proceeds, I will note down any properties Q_1,\dots,Q_k that x^\bullet might have that would help me finish the proof, in the hope that (i) I get to the end and (ii) the problem \exists x\ Q_1(x)\wedge\dots\wedge Q_k(x) is easier than the original problem.” Another kind of obvious instantiation is one where we try out an object that is “extreme” in some way — it might be the smallest element of X, or the largest, or the simplest. (Judging simplicity is another place where the ChatGPT rule of thumb can be used.)
  3. We cannot see how to answer the question we are focusing on so we ask a related question. Two very common kinds of related question (as emphasized by Polya) are generalization and specialization. Perhaps we don’t see why a hypothesis is helpful, so we see whether the result holds if we drop that hypothesis. If it does, then we are no longer distracted by an irrelevant hypothesis. If it does not, then we can hope to find a counterexample that will help us understand how to use the hypothesis. Or perhaps we are trying to prove a general statement but it is not clear how to do so, so instead we formulate some special cases, hoping that we can prove them and spot features of the proofs that we can generalize. Again we have to be rather careful here not to allow “non-obvious” generalizations and specializations. Roughly the idea there is that a generalization should be purely logical — for example, dropping a hypothesis is fine but replacing the hypothesis “f is twice differentiable” by “f is upper semicontinuous” is not — and that a specialization should be to a special case that counts as an obvious instantiation in the sense discussed just above.

2. A structured motivated proof is one that can be generated with the help of a point-and-click system.

This is a surprisingly useful way to conceive of what we are talking about, especially as it relates closely to what I was talking about earlier: imposing a standard form on motivated proofs (which is why we call them “structured” motivated proofs) and gamifying the process of producing them.

The idea is that a structured motivated proof is one that can be generated using an interface (which we are in the process of creating — at the moment we have a very basic prototype that has a few of the features we will need, but not yet the more interesting ones) that has one essential property: the user cannot type in data. So what can they do? They can select text that is on their screen (typically mathematical expressions or subexpressions), they can click buttons, choose items from drop-down menus, and accept or reject “obvious” suggestions made to them by the interface.

If, for example, the current goal is an existential statement \exists x\ P(x), then typing in a formula that defines a suitable x is not possible, so instead one must select text or generate new text by clicking buttons, choosing from short drop-down menus, and so on. This forces the user to generate x, which is our proxy for showing where the idea of using x came from.

Broadly speaking, the way the prototype works is to get an LLM to read a JSON object that describes the variables, hypotheses and goals involved in the proof state in a structured format, and to describe (by means of a fairly long prompt) the various moves it might be called upon to do. Thus, the proofs generated by the system are not formally verified, but that is not an issue that concerns us in practice since there will be a human in the loop throughout to catch any mistakes that the LLM might make, and this flexibility may even work to our advantage to better capture the fluidity of natural-language mathematics.

There is obviously a lot more to say about what the proof-generating moves are, or (approximately equivalently) what the options provided by a point-and-click system will be. I plan to discuss that in much more detail when we are closer to having an interface ready, the target for which is the end of this calendar year. But the aim of the project is to create a database of examples of proofs that have been successfully generated using the interface, which can then be used to train AI to play the generate-structured-motivated-proof game.

How to get involved.

There are several tasks that will need doing once the project gets properly under way. Here are some of the likely ones.

  1. The most important is for people to submit structured motivated (or move-generated) proofs to us on the platform we provide. We hope that the database will end up containing proofs of a wide range of difficulty (of two kinds — there might be fairly easy arguments that are hard to motivate and there might be arguments that are harder to follow but easier to motivate) and also a wide range of areas of mathematics. Our initial target, which is quite ambitious, is to have around 1000 entries by two years from now. While we are not in a position to accept entries yet, if you are interested in participating, then it is not too early to start thinking in a less formal way about how to convert some of your favourite proofs into motivated versions, since that will undoubtedly make it easier to get them accepted by our platform when it is ready.
  2. We are in the process of designing the platform. As I mentioned earlier, we already have a prototype, but there are many moves we will need it to be able to do that it cannot currently do. For example, the current prototype allows just a single proof state, which consists of some variable declarations, hypotheses, and goals. It does not yet support creating subsidiary proof states (which we would need if we wanted to allow the user to consider generalizations and specializations, for example). Also, for the moment the prototype gets an LLM to implement all moves, but some of the moves, such as applying modus ponens, are extremely mechanical and would be better done using a conventional program. (On the other hand, moves such as “obvious library extraction” or “provide the simplest example” are better done by an LLM.) Thirdly, a technical problem is that LaTeX is currently rendered as images, which makes it hard to select subexpressions, something we will need to be able to do in a non-clunky way. And the public version of the platform will need to be web-based and very convenient to use. We will want features such as being able to zoom out and look at some kind of dependency diagram of all the statements and questions currently in play, and then zoom in on various nodes if the user wishes to work on them. If you think you may be able (and willing) to help with some of these aspects of the platform, then we would be very happy to hear from you. For some, it would probably help to have a familiarity with proof assistants, while for others we would be looking for somebody with software engineering experience. The grant from the AI for Math Fund will allow us to pay for some of this help, at rates to be negotiated. We are not yet ready to specify in detail what help we need, but would welcome any initial expressions of interest.
  3. Once the platform is ready and people start to submit proofs, it is likely that, at least to start with, they will find that the platform does not always provide the moves they need. Perhaps they will have a very convincing account of where a non-obvious idea in the proof came from, but the system won’t be expressive enough for them to translate that account into a sequence of proof-generating moves. We will want to be able to react to such situations (if we agree that a new move is needed) by expanding the capacity of the platform. It will therefore be very helpful if people sign up to be beta-testers, so that we can try to get the platform to a reasonably stable state before opening it up to a wider public. Of course, to be a beta-tester you would need to have a few motivated proofs in mind.
  4. It is not obvious that every proof submitted via the platform, even if submitted successfully, would be a useful addition to the database. For instance, it might be such a routine argument that no idea really needs to have its origin explained. Or it might be that, despite our best efforts, somebody finds a way of sneaking in a rabbit while using only the moves that we have provided. (One way this could happen is if an LLM made a highly non-obvious suggestion that happened to work, in which case the rule of thumb that if an LLM thinks of it, it must be obvious, would have failed in that instance.) For this reason, we envisage having a team of moderators, who will check entries and make sure that they are good additions to the database. We hope that this will be an enjoyable task, but it may have its tedious aspects, so we envisage paying moderators — again, this expense was allowed for in our proposal to the AI for Math Fund.

If you think you might be interested in any of these roles, please feel free to get in touch. Probably the hardest recruitment task for us will be identifying the right people with the right mixture of mathematical knowledge and software engineering skills to help us turn the platform into a well-designed web-based one that is convenient and pleasurable to use. If you think you might be such a person, or if you have a good idea for how we should go about finding one, we would be particularly interested to hear from you.

In a future post, I will say more about the kinds of moves that our platform will allow, and will give examples of non-motivated proofs together with how motivated versions of those proofs can be found and entered using the platform (which may involve a certain amount of speculation about what the platform will end up looking like).

How does this relate to use of tactics in a proof assistant?

In one way, our “moves” can be regarded as tactics of a kind. However, some of the moves we will need are difficult to implement in conventional proof assistants such as Lean. In parallel with the work described above, we hope to create an interface to Lean that would allow one to carry out proof-discovery moves of the kind discussed above but with the proof-discovery states being collections of Lean proof states. Members of my group have already been working on this and have made some very interesting progress, but there is some way to go. However, we hope that at some point (and this is also part of the project pitched to the AI for Math Fund) we will have created another interface that will have Lean working in the background, so that it will be possible to generate motivated proofs that will be (or perhaps it is better to say include) proofs in Lean at the same time.

Another possibility that we are also considering is to use the output of the first platform (which, as mentioned above, will be fairly formal, but not in the strict sense of a language such as Lean) to create a kind of blueprint that can then be autoformalized automatically. Then we would have a platform that would in principle allow mathematicians to search for proofs while working on their computers without having to learn a formal language, with their thoughts being formalized as they go.

September 22, 2025

Tommaso DorigoHow Elementary Particles Die

A preamble

Subnuclear physics obeys the laws of quantum mechanics, which are quite a far cry from those of classical mechanics we are accustomed to. For that reason, one might be inclined to believe that analogies based on everyday life cannot come close to explaining the behavior of elementary particles. But that is not true – in fact, many properties of elementary particles are understandable in analogy with the behavior of classical systems, without the need to delve into the intricacies of the quantum world. And if you have been reading this blog for a while, you know what I think – the analogy is a powerful didactical instrument, and it is indeed at the very core of our learning processes.

read more

September 21, 2025

John PreskillBlending science with fiction in Baltimore

I judge a bookstore by the number of Diana Wynne Jones novels it stocks. The late British author wrote some of the twentieth century’s most widely lauded science-fiction and fantasy (SFF). She clinched more honors than I should list, including two World Fantasy Awards. Neil Gaiman, author of American Gods, called her “the best children’s writer of the last forty years” in 2010—and her books suit children of all ages.1 But Wynne Jones passed away as I was finishing college, and her books have been disappearing from American bookshops. The typical shop stocks, at best, a book in the series she began with Howl’s Moving Castle, which Hayao Miyazaki adapted into an animated film.

I don’t recall the last time I glimpsed Deep Secret in a bookshop, but it ranks amongst my favorite Wynne Jones books—and favorite books, full-stop. So I relished living part of that book this spring.

Deep Secret centers on video-game programmer Rupert Venables. Outside of his day job, he works as a Magid, a magic user who helps secure peace and progress across the multiple worlds. Another Magid has passed away, and Rupert must find a replacement for him. How does Rupert track down and interview his candidates? By consolidating their fate lines so that the candidates converge on an SFF convention. Of course.

My fate line drew me to an SFF convention this May. Balticon takes place annually in Baltimore, Maryland. It features not only authors, agents, and publishers, but also science lecturers. I received an invitation to lecture about quantum steampunk—not video-game content,2 but technology-oriented like Rupert’s work. I’d never attended an SFF convention,3 so I reread Deep Secret as though studying for an exam.

Rupert, too, is attending his first SFF convention. A man as starched as his name sounds, Rupert packs suits, slacks, and a polo-neck sweater for the weekend—to the horror of a denim-wearing participant. I didn’t bring suits, in my defense. But I did dress business-casual, despite having anticipated that jeans, T-shirts, and capes would surround me.

I checked into a Renaissance Hotel for Memorial Day weekend, just as Rupert checks into the Hotel Babylon for Easter weekend. Like him, I had to walk an inordinately long distance from the elevators to my room. But Rupert owes his trek to whoever’s disrupted the magical node centered on his hotel. My hotel’s architects simply should have installed more elevator banks.

Balticon shared much of its anatomy with Rupert’s con, despite taking place in a different century and country (not to mention world). Participants congregated downstairs at breakfast (continental at Balticon, waitered at Rupert’s hotel). Lectures and panels filled most of each day. A masquerade took place one night. (I slept through Balticon’s; impromptu veterinary surgery occupies Rupert during his con’s.) Participants vied for artwork at an auction. Booksellers and craftspeople hawked their wares in a dealer’s room. (None of Balticon’s craftspeople knew their otherworldly subject matter as intimately as Rupert’s Magid colleague Zinka Fearon does, I trust. Zinka paints her off-world experiences when in need of cash.)

In our hotel room, I read out bits of Deep Secret to my husband, who confirmed the uncanniness with which they echoed our experiences. Both cons featured floor-length robes, Batman costumes, and the occasional slinky dress. Some men sported long-enough locks, and some enough facial hair, to do a Merovingian king proud. Rupert registers “a towering papier-mâché and plastic alien” one night; on Sunday morning, a colossal blow-up unicorn startled my husband and me. We were riding the elevator downstairs to breakfast, pausing at floor after floor. Hotel guests packed the elevator like Star Wars fans at a Lucasfilm debut. Then, the elevator halted again. The doors opened on a bespectacled man, 40-something years old by my estimate, dressed as a blue-and-white unicorn. The costume billowed out around him; the golden horn towered multiple feet above his head. He gazed at our sardine can, and we gazed at him, without speaking. The elevator doors shut, and we continued toward breakfast.

Photo credit: Balticon

Despite having read Deep Secret multiple times, I savored it again. I even laughed out loud. Wynne Jones paints the SFF community with the humor, exasperation, and affection one might expect of a middle-school teacher contemplating her students. I empathize, belonging to a community—the physics world—nearly as idiosyncratic as the SFF community.4 Wynne Jones’s warmth for her people suffuses Deep Secret; introvert Rupert surprises himself by enjoying a dinner with con-goers and wishing to spend more time with them. The con-goers at my talk exhibited as much warmth as any audience I’ve spoken to, laughing, applauding, and asking questions. I appreciated sojourning in their community for a weekend.5

This year, my community is fêting the physicists who founded quantum theory a century ago. Wynne Jones sparked imaginations two decades ago. Let’s not let her memory slip from our fingertips like a paperback over which we’re falling asleep. After all, we aren’t forgetting Louis de Broglie, Paul Dirac, and their colleagues. So check out a Wynne Jones novel the next time you visit a library, or order a novel of hers to your neighborhood bookstore. Deep Secret shouldn’t be an actual secret.

With thanks to Balticon’s organizers, especially Miriam Winder Kelly, for inviting me and for fussing over their speakers’ comfort like hens over chicks.

1Wynne Jones dedicated her novel Hexwood to Gaiman, who expressed his delight in a poem. I fancy the comparison of Gaiman, a master of phantasmagoria and darkness, to a kitten.

2Yet?

3I’d attended a steampunk convention, and spoken at a Boston SFF convention, virtually. But as far as such conventions go, attending virtually is to attending in person as my drawings are to a Hayao Miyazaki film.

4But sporting fewer wizard hats.

5And I wonder what the Diana Wynne Jones Conference–Festival is like.

September 19, 2025

John PreskillNicole’s guide to writing research statements

Sunflowers are blooming, stores are trumpeting back-to-school sales, and professors are scrambling to chart out the courses they planned to develop in July. If you’re applying for an academic job this fall, now is the time to get your application ducks in a row. Seeking a postdoctoral or faculty position? Your applications will center on research statements. Often, a research statement describes your accomplishments and sketches your research plans. What do evaluators look for in such documents? Here’s my advice, which targets postdoctoral fellowships and faculty positions, especially for theoretical physicists.

  • Keep your audience in mind. Will a quantum information theorist, a quantum scientist, a general physicist, a general scientist, or a general academic evaluate your statement? What do they care about? What technical language do and don’t they understand?
  • What thread unites all the projects you’ve undertaken? Don’t walk through your research history chronologically, stepping from project to project. Cast the key projects in the form of a story—a research program. What vision underlies the program?
  • Here’s what I want to see when I read a description of a completed project.
    • The motivation for the project: This point ensures that the reader will care enough to read the rest of the description.
    • Crucial background information
    • The physical setup
    • A statement of the problem
    • Why the problem is difficult or, if relevant, how long the problem has remained open
    • Which mathematical toolkit you used to solve the problem or which conceptual insight unlocked the solution
    • Which technical or conceptual contribution you provided
    • Whom you collaborated with: Wide collaboration can signal a researcher’s maturity. If you collaborated with researchers at other institutions, name the institutions and, if relevant, their home countries. If you led the project, tell me that, too. If you collaborated with a well-known researcher, mentioning their name might help the reader situate your work within the research landscape they know. But avoid name-dropping, which lacks such a pedagogical purpose and which can come across as crude.
    • Your result’s significance/upshot/applications/impact: Has a lab based an experiment on your theoretical proposal? Does your simulation method outperform its competitors by X% in runtime? Has your mathematical toolkit found applications in three subfields of quantum physics? Consider mentioning whether a competitive conference or journal has accepted your results: QIP, STOC, Physical Review Letters, Nature Physics, etc. But such references shouldn’t serve as a crutch in conveying your results’ significance. You’ll impress me most by dazzling me with your physics; name-dropping venues instead can convey arrogance.
  • Not all past projects deserve the same amount of space. Tell a cohesive story. For example, you might detail one project, then synopsize two follow-up projects in two sentences.
  • A research statement must be high-level, because you don’t have space to provide details. Use mostly prose; and communicate intuition, including with simple examples. But sprinkle in math, such as notation that encapsulates a phrase in one concise symbol.

  • Be concrete, and illustrate with examples. Many physicists—especially theorists—lean toward general, abstract statements. The more general a statement is, we reason, the more systems it describes, so the more powerful it is. But humans can’t visualize and intuit about abstractions easily. Imagine a reader who has four minutes to digest your research statement before proceeding to the next 50 applications. As that reader flys through your writing, vague statements won’t leave much of an impression. So draw, in words, a picture that readers can visualize. For instance, don’t describe only systems, subsystems, and control; invoke atoms, cavities, and lasers. After hooking your reader with an image, you can generalize from it.
  • A research statement not only describes past projects, but also sketches research plans. Since research covers terra incognita, though, plans might sound impossible. How can you predict the unknown—especially the next five years of the unknown (as required if you’re applying for a faculty position), especially if you’re a theorist? Show that you’ve developed a map and a compass. Sketch the large-scale steps that you anticipate taking. Which mathematical toolkits will you leverage? What major challenge do you anticipate, and how do you hope to overcome it? Let me know if you’ve undertaken preliminary studies. Do numerical experiments support a theorem you conjecture?
  • When I was applying for faculty positions, a mentor told me the following: many a faculty member can identify a result (or constellation of results) that secured them an offer, as well as a result that earned them tenure. Help faculty-hiring committees identify the offer result and the tenure result.
  • Introduce notation before using it. If you use notation and introduce it afterward, the reader will encounter the notation; stop to puzzle over it; tentatively continue; read the introduction of the notation; return to the earlier use of the notation, to understand it; and then continue forward, including by rereading the introduction of the notation. This back-and-forth breaks up the reading process, which should flow smoothly.
  • Avoid verbs that fail to relate that you accomplished anything: “studied,” “investigated,” “worked on,” etc. What did you prove, show, demonstrate, solve, calculate, compute, etc.?

  • Tailor a version of your research statement to every position. Is Fellowship Committee X seeking biophysicists, statistical physicists, mathematical physicists, or interdisciplinary scientists? Also, respect every application’s guidelines about length.
  • If you have room, end the statement with a recap and a statement of significance. Yes, you’ll be repeating ideas mentioned earlier. But your reader’s takeaway hinges on the last text they read. End on a strong note, presenting a coherent vision.

  • Read examples. Which friends and colleagues, when applying for positions, have achieved success that you’d like to emulate? Ask if those individuals would share their research statements. Don’t take offense if they refuse; research statements are personal.

  • Writing is rewriting, a saying goes. Draft your research statement early, solicit feedback from a couple of mentors, edit the draft, and solicit more feedback.

September 18, 2025

John PreskillJohn Preskill receives 2025 Quantum Leadership Award

The 2025 Quantum Leadership Awards were announced at the Quantum World Congress on 18 September 2025. Upon receiving the Academic Pioneer in Quantum Award, John Preskill made these remarks.

I’m enormously excited and honored to receive this Quantum Leadership Award, and especially thrilled to receive it during this, the International Year of Quantum. The 100th anniversary of the discovery of quantum mechanics is a cause for celebration because that theory provides our deepest and most accurate description of how the universe works, and because that deeper understanding has incalculable value to humanity. What we have learned about electrons, photons, atoms, and molecules in the past century has already transformed our lives in many ways, but what lies ahead, as we learn to build and precisely control more and more complex quantum systems, will be even more astonishing.

As a professor at a great university, I have been lucky in many ways. Lucky to have the freedom to pursue the scientific challenges that I find most compelling and promising. Lucky to be surrounded by remarkable, supportive colleagues. Lucky to have had many collaborators who enabled me to do things I could never have done on my own. And lucky to have the opportunity to teach and mentor young scientists who have a passion for advancing the frontiers of science. What I’m most proud of is the quantum community we’ve built at Caltech, and the many dozens of young people who imbibed the interdisciplinary spirit of Caltech and then moved onward to become leaders in quantum science at universities, labs, and companies all over the world.

Right now is a thrilling time for quantum science and technology, a time of rapid progress, but these are still the early days in a nascent second quantum revolution. In quantum computing, we face two fundamental questions: How can we scale up to quantum machines that can solve very hard computational problems? And once we do so, what will be the most important applications for science and for industry? We don’t have fully satisfying answers yet to either question and we won’t find the answers all at once – they will unfold gradually as our knowledge and technology advance. But 10 years from now we’ll have much better answers than we have today.

Companies are now pursuing ambitious plans to build the world’s most powerful quantum computers.  Let’s not forget how we got to this point. It was by allowing some of the world’s most brilliant people to follow their curiosity and dream about what the future could bring. To fulfill the potential of quantum technology, we need that spirit of bold adventure now more than ever before. This award honors one scientist, and I’m profoundly grateful for this recognition. But more importantly it serves as a reminder of the vital ongoing need to support the fundamental research that will build foundations for the science and technology of the future. Thank you very much!

September 16, 2025

n-Category Café A Shadow of Triality?

It’s well known that you can construct the octonions using triality. One statement of triality is that Spin(8)Spin(8) has nontrivial outer automorphisms of order 3. On the other hand, the octonions have nontrivial inner automorphisms of order 3. My question: can we deduce one of these facts from the other?

The second fact is perhaps not very well known. It may even be hard to understand what it means. Though the octonions are nonassociative, for any nonzero octonion gg the map

f: 𝕆 𝕆 x gxg 1 \begin{array}{rccl} f \colon & \mathbb{O} &\to& \mathbb{O} \\ & x & \mapsto & g x g^{-1} \end{array}

is well-defined, since (gx)g 1=g(xg 1)(g x)g^{-1} = g(x g^{-1}), which one can show using the fact that the octonions are alternative. More surprisingly, whenever g 6=1g^6 = 1, this map ff is an automorphism of the octonions:

f(x+y)=f(x)+f(y),f(xy)=f(x)f(y)x,y𝕆 f(x+y) = f(x) + f(y) , \qquad f(x y) = f(x) f(y) \qquad \forall x,y \in \mathbb{O}

and ff has order 3:

f(f(f(x)))=xx𝕆 f(f(f(x))) = x \qquad \forall x \in \mathbb{O}

To understand this latter fact, we can look at

Theorem 2.1 here implies that an octonion gg with |g|=1{|g|} = 1 defines an inner automorphism f:xgxg 1f \colon x \mapsto g x g^{-1} if and only if gg has order 6.

However, the result is stated differently there. Paraphrasing somewhat, Lamont’s theorem says that any g𝕆g \in \mathbb{O} that is not a real multiple of 1𝕆1 \in \mathbb{O} defines an inner automorphism f:xgxg 1f \colon x \to g x g^{-1} if and only if gg obeys

4Re(g) 2=|g| 2 4 \mathrm{Re}(g)^2 = {|g|}^2

This equation is equivalent to Re(g)=±12|g|\operatorname{Re}(g) = \pm \frac{1}{2} {|g|}, which is equivalent to gg lying at either a 60 60^\circ angle or a 120 120^\circ angle from the octonion 11.

Nonzero octonions on the real line clearly define the identity inner automorphism. Thus, a nonzero octonion gg defines an inner automorphism if and only if its angle from 11 is 0 0^\circ, 60 60^\circ, 120 120^\circ or 180 180^\circ. In this case we can normalize gg without changing the inner automorphism it defines, and then we have g 6=1g^6 = 1. Note also that gg and g-g define the same inner automorphism.

It follows that an octonion gg on the unit sphere defines an inner automorphism iff g 6=1g^6 = 1, and that every nontrivial inner automorphism of 𝕆\mathbb{O} has order 3.

However, if you look at Lamont’s proof, you’ll see the equation 4Re(g) 2=|g| 24 \operatorname{Re}(g)^2 = {|g|}^2 plays no direct role! Instead, he really uses the assumption that g 3g^3 is a real multiple of 11, which is implied by this equation (as easily shown using what we’ve just seen).

From Lamont’s work, one can see the Moufang identities and the characteristic equation for octonions are what force all inner automorphisms of the octonions to have order 3.

Thus, an argument giving a positive answer to my question might involve a link between triality and the Moufang identities. Conway and Smith seem to link them in On Quaternions and Octonions. But I haven’t figured out how to get from the outer automorphisms of Spin(8)\text{Spin}(8) to the inner automorphisms of 𝕆\mathbb{O}, or vice versa!

I asked about this on MathOverflow, but I thought some people here would also be interested.

September 13, 2025

n-Category Café Burrito Monads, Arrow Kitchens, and Freyd Category Recipes

Guest post by Khyathi Komalan and Andrew Krenz

From Lawvere’s Hegelian taco to Baez’s layer cake analogy to Eugenia Cheng’s How to Bake Pi, categorists have cultivated a rich tradition of culinary metaphors and similes. A well-known example in the world of computation is Mark Dominus’s “monads are like burritos” — where a tortilla (computational context) wraps diverse ingredients (values) to create a cohesive entity (effectful value) whose burrito structure is maintained as the meal moves down the assembly line (undergoes computations).

Monads, like burritos, come in many different varieties. In computer science monads serve to streamline computational patterns such as exception handling and context management. We illustrate these two examples by analogy.

Imagine you work at a burrito truck.

If a customer orders a burrito sans rice but rice is accidentally added, it can’t be served. The Maybe monad handles exceptions such as this — when something goes wrong, it returns a special “Nothing” value rather than a flawed result, and once a failure occurs, all subsequent steps automatically preserve this state avoiding the need for repetitive error-checking.


Diagram 1

Figure 1: The Maybe Monad illustrated with the burrito-making process


In Haskell, the parameterized type “Maybe a” has two constructors, “Just a” and “Nothing.” The former is an alias for values of type “a” whereas the latter is indicative of an error. The following Haskell code exhibits the maybe type as an instance of the monad class:

instance Maybe Monad where
return = Just
Nothing >>= f = Nothing
(Just x) >>= f = f x

the return function has type a -> Maybe a, which is suggestive of its role as the monad unit. The so-called bind operation >>= has type Maybe a -> (a -> Maybe b) -> Maybe b, and corresponds to a bare-bones Kleisli composition (see Monads: Programmer’s Definition for details).

A slight generalization allows for descriptive error messages.

Definition. Given a collection of exceptions EE, there is an associated Either monad (()+E,η,μ)((-)+E, \eta, \mu).

  • η X:XX+E\eta_X:X \to X + E is the coproduct insertion
  • μ X:X+E+EX+E\mu_X:X + E + E \to X + E collapses two copies of EE into one
  • Kleisli morphisms are computations that may fail XY+EX \to Y + E
  • Kleisli composition automatically propagates exceptions

Of course, either monads are simply maybe monads with a set in place of the constant/singleton “Nothing” and they allow us not only to say that an error has occured, but also to indicate what that error was.

Now suppose one of your regular customers walks up to the window and orders “the usual.” Luckily you’ve recorded their preferences in a recipe book. The act of following the appropriate recipe is akin to executing computations that depend on a global read-only state. The * Reader monad * is the functional programmer’s way of incorporating this impure concept in pure functional terms.

Diagram 2


Figure 2: The Reader Monad illustrated with the burrito-making process

Definition. Given a collection of environments EE, there is an associated Reader monad (() E,η,μ)((-)^E, \eta, \mu).

  • η X:XX E\eta_X : X \to X^E turns elements into constant functions xλe.xx \mapsto \lambda e. x
  • μ X:(X E) EX E\mu_X : (X^E)^E \to X^E turns function-valued functions into functions via diagonal evaluation fλe.f(e)(e)f \mapsto \lambda e. f(e)(e)
  • Kleisli morphisms convert inputs into executable functions from environments to outputs XY EX \to Y^E
  • Composition in the Kleisli category keeps track of the (immutable) environment as computations are chained together.

Here is the same definition given as an instance of the Haskell monad class:

instance Monad ((->) r) where
return x = \_ -> x
g >>= f = \e -> f (g e) e

The seminal paper of Moggi has several other interesting examples illustrating the power of monads. Nevertheless, monads may not always suffice for all of our needs. For example, what would happen if our burrito truck suddenly exploded in popularity requiring automation of repetative processes and parallel work stations?

This is where “Arrows” enter the picture. Introduced by John Hughes in 2000, Arrows generalize strong monads. Because of this, Arrows handle more complicated computational patterns in a natural way. While monads wrap values in computational contexts (like burritos in tortillas), Arrows can represent entire preparation processes capable of coordinating multiple inputs while maintaining awareness of the broader kitchen environment.

Arrows come with three core operations that determine their behaviour; looking at their types, we see that Arrows are evocative of a lax internal hom that interacts with binary products.

class Arrow a where
arr :: (x -> y) -> a x y
(>>>) :: a x y -> a y z -> a x z
first :: a x y -> a (x,z) (y,z)
  1. arr turns functions into “Arrows.” This is like incorporating a standard burrito recipe or preparation step into the food truck’s workflow — taking a simple instruction like “add beans, then cheese” and automating it within our kitchen’s setup.
  2. >>> composes composable Arrows. This allows for separately automated processes to be seamlessly strung together.
  3. first enacts an automated process on one burrito while simultaneously passing a second burrito through the station.

These data are subject to 9 axioms, which we eventually discuss below.

Diagram 3
Figure 3: Arrow Operations. The three fundamental operations of Arrows enable complex workflows beyond monadic structures.

Shortly before Arrows were introduced, Power, Robinson, and Thielecke were working on Freyd categories — a categorical structure designed to model “effectful” computation. Using our simile, a Freyd category formalizes the relationship between an ideal burrito recipe (pure theory) and the real-world process of making that burrito in a particular kitchen.

A Freyd category consists of three main components:

  1. A category CC with finite products which can be thought of as the syntax of our kitchen. In other words, CC is like a recipe book containing the abstract information one needs to interpret and implement in the context of an actual kitchen.
  2. A symmetric premonoidal category KK which plays the more semantic role of our real world kitchen.
  3. An identity-on-objects functor J:CKJ:C \to K which faithfully translates pure recipes into physical processes that work within the specific setup of the kitchen KK.

    Diagram 4
    Figure 4: Freyd Category Structure. The relationship between pure recipes (category C) and real-world kitchen operations (category K), connected by the identity-on-objects functor J that preserves structure while accommodating practical constraints.

Although Arrows originated in Haskell, a highly abstract functional programming language, researchers began noticing apparent correspondences between the components of Arrows and those of Freyd categories. These two structures, developed from different starting points, seemed to address the same fundamental challenge: how to systematically manage computations that involve effects, multiple inputs and outputs, and context-awareness. Therefore, it was hypothesized that Arrows are equivalent to Freyd categories.

As a part of the Adjoint School, our group has been focusing on R. Atkey’s work, which dispells this folklore and precisely formulates the relationship between Arrows and Freyd categories. Just as Atkey asks in the title of his paper, this blog post will investigate the question of “what is a categorical model of Arrows?” The answer not only clarifies the theoretical underpinnings of these structures, but also reveals practical insights for programming language design and quantum computation models. Ultimately, we will see that there are indeed subtle differences between Arrows and Freyd categories.


Key Insights: - Monads encapsulate computational effects by wrapping values in contexts, much like burritos wrap ingredients in tortillas - Different monads (Maybe, Reader, etc…) deal with different patterns like exception handling and context management - Arrows generalize monads to handle multiple inputs and coordinate complex processes, like managing an entire kitchen rather than just making individual burritos


Beyond the Kitchen: Arrows and Freyd Categories

Formally, a monad on a category CC is a monoid in the category of endofunctors of CC. Arrows, like monads, are monoids in a certain category of functors. To be more specific, the structure of an Arrow on a category CC can be described as a monoid in the category of strong profunctors on CC. Let’s take a closer look at this construction.

Arrows A profunctor PP on a category CC is a functor P:C op×CSet.P: C^{\text{op}}\times C \to \text{Set}. Intuitively, a profunctor associates to each pair of objects a set of “generalized morphisms” between those objects.

The identity profunctor is simply id(x,y):=C(x,y)\text{id}(x, y) := C(x, y), which uses the hom-sets of CC.

Composition of profunctors is defined as a coend. Given profunctors PP and QQ, their composition is the following profunctor:

(P*Q)(x,z)= yP(x,y)×Q(y,z)(P * Q)(x, z) = \int^y P(x, y) \times Q(y, z)

Notice that this formula is vaguely reminiscent of a dot product; replacing the integral with a sum over yy, and the cartesian product with multiplication, it looks like the dot product of the row vector P(x,)P(x,-) with the column vector Q(,z)Q(-,z).

operations. –> We will now unpack this data to reach a more down-to-earth description of Arrows. This resulting characterization aligns more closely with the way in which Arrows are implemented in programming languages like Haskell.

Definition. An Arrow in a cartesian closed category CC consists of a mapping on objects and three families of morphisms:

  • A mapping on objects Ar:ob(C)×ob(C)ob(C)\text{Ar} : \text{ob}(C) \times \text{ob}(C) \to \text{ob}(C)

This defines the Arrow type constructor, which takes input and output types and produces an Arrow type between them.

  • A family of morphisms arr:Y XAr(X,Y)\text{arr} : Y^X \to \text{Ar}(X, Y)

This operation lifts a pure function into the Arrow context, allowing regular functions to be treated as Arrows.

  • A family of morphisms :Ar(X,Y)×Ar(Y,Z)Ar(X,Z)\ggg : \text{Ar}(X, Y) \times \text{Ar}(Y, Z) \to \text{Ar}(X, Z)

    This enables sequential composition of Arrows, similar to function composition but now in terms of Arrows.

  • A family of morphisms first:Y XAr(X×W,Y×W)\text{first} : Y^X \to \text{Ar}(X \times W, Y \times W)

This is perhaps the most distinctive operation. Intuitively, it allows an Arrow to process the first component of a pair while leaving the second component unchanged.

These data are subject to nine axioms which govern their interactions. To make these abstract operations more concrete, consider the following example, where Ar(x,y):=Y X,\text{Ar}(x, y) := Y^X, arr:=id (Y X),\text{arr} := \text{id}_{(Y^X)}, :=composition,\ggg := \text{composition}, and first(f):=f×id.\text{first}(f) := f \times \text{id}. In what follows we list the Arrow laws and draw commutative diagrams based on this example.

The Arrow laws arr(id)a=a\text{arr}(\text{id})\ggg a=a and aarr(id)=aa\ggg \text{arr}(\text{id})=a express left and right unitality of identities under composition.

Diagram 5
Figure 5: Arrow Laws

The Arrow law (ab)c=a(bc),(a \ggg b)\ggg c=a \ggg (b \ggg c), represents associativity of composition.

Diagram 6
Figure 6: Arrow Laws

The Arrow law first(ab)=first(a)first(b)\text{first}(a\ggg b)=\text{first}(a)\ggg \text{first}(b) encodes functoriality of ×W:CC- \times W: C \to C.

Diagram 7
Figure 7: Arrow Laws

The Arrow law first(a)arr(π 1)=arr(π 1)a\text{first}(a)\ggg \text{arr}(\pi_{1})=\text{arr}(\pi_{1})\ggg a express naturality of the counit ×Wid C- \times W \to \text{id}_{C}, i.e., the first projection maps.

Diagram 8
Figure 8: Arrow Laws

The Arrow law first(a)arr(α)=arr(α)first(first(a))\text{first}(a)\ggg \text{arr}(\alpha)=\text{arr}(\alpha)\ggg \text{first}(\text{first}(a)) asks that first\text{first} play nicely with associators.

Diagram 9
Figure 9: Arrow Laws

The Arrow law first(a)arr(id×f)=arr(id×f)first(a)\text{first}(a)\ggg \text{arr}(\text{id} \times f)=\text{arr}(\text{id} \times f)\ggg \text{first}(a) is an interchange law which says id×g:(×W)(×W)\text{id} \times g:(- \times W) \to (- \times W') is a natural transformation for every g:WWg:W \to W' in CC.

Diagram 10
Figure 10: Arrow Laws

Two Arrow laws trivialise as a result of our example, so diagrams aren’t produced. The first such law is arr(f;g)=arr(f)arr(g).\text{arr}(f;g)=\text{arr}(f)\ggg \text{arr}(g). For our example, this law trivialises, as :=composition\ggg : = \text{composition} and arr:=id (Y X).\text{arr} := \text{id}_{(Y^X)}. The second law to trivialise is first(arr(f))=arr(f×id)\text{first}(\text{arr}(f))=\text{arr}(f \times \text{id}) since we have set first(f):=f×id.\text{first}(f) := f \times \text{id}.

Freyd Categories

To understand Freyd categories, we must first define what a symmetric premonoidal category is.

Definition. A symmetric premonoidal category includes:

  • An object II (unit).
  • Natural transformations that define how objects interact:
    • Associativity: α:(xy)zx(yz)\alpha : (x \otimes y) \otimes z \to x \otimes (y \otimes z)
    • Left unitor: λ:xIx\lambda : x \otimes I \to x
    • Right unitor: ρ:Ixx\rho : I \otimes x \to x
    • Symmetry: σ:xyyx\sigma : x \otimes y \to y \otimes x
  • All components are central .

A morphism f:xxf : x \to x' is central if g:yy,fy;xg=xg;fy\forall g:y \to y', \quad f \otimes y ; x' \otimes g = x \otimes g ; f \otimes y'

Now, we can define a Freyd category, recalling the definition from the introduction.

Definition. A Freyd category consists of:

  • A category CC with finite products.
  • A symmetric premonoidal category KK.
  • An identity-on-objects functor J:CKJ : C \to K that:
    • Preserves symmetric premonoidal structure.
    • Ensures J(f)J(f) is always central.

Arrows vs Freyd Categories: Similarities and Differences

At first glance, the definition of a Freyd category appears strikingly similar to that of an Arrow. This apparent similarity led to the folklore belief that they were equivalent structures.

A Freyd category consists of two categories CC and KK with an identity-on-objects functor J:CKJ: C \to K, where: - CC has finite products - KK is symmetric premonoidal (with a functor z&#8722; \otimes z) - JJ maps finite products in CC to the premonoidal structure in KK

In our culinary metaphor, this loosely translates to: - CC: The idealized recipes (Haskell types and functions) - KK: The real-world kitchen operations (computations represented by the Arrow type Ar(x,y)\text{Ar}(x,y)) - JJ: The translation process (via arr, embedding pure functions) - Composition in KK: The sequencing of operations (via >>>) - Premonoidal structure in KK: The ability to process pairs (via first)

Recalling the how we’ve interpreted Arrows in the cullinary setting, the apparent correspondence between Arrows and Fryed categories seemes quite natural. In fact, for many years the two concepts were thought to be two ways of speaking about the same thing among those in the programming languages community.

However, Atkey’s work revealed a crucial distinction: Arrows are more general than Freyd categories . The key difference lies in how they handle inputs:

  • Freyd categories allow only a single input to computations
  • Arrows support two separate inputs:
    • One may be structured (modeled using comonads)
    • This additional flexibility allows Arrows to represent computations that Freyd categories cannot

To bridge this gap, Atkey introduced the concept of indexed Freyd categories , which can model two structured inputs. The relationship can be summarized as: Arrows are equivalent to Closed Indexed Freyd Categories.

In our culinary metaphor, we can understand this relationship as follows: a Freyd category is like a restaurant that can only take one order at a time (a single input), while Arrows are like a more sophisticated establishment that can handle both individual orders and special requests that come with their own context (two inputs, one potentially structured). The closed indexed Freyd categories that Atkey identifies represent the perfect middle ground — restaurants that can efficiently manage multiple orders with specialized instructions while maintaining the core operational principles that make kitchens function. This is particularly valuable when preparing complex “quantum dishes” where ingredients might be entangled and interact with each other in non-local ways.

Diagram 11
Figure 11: Arrows vs. Freyd Categories. Arrows support two inputs (one potentially structured) and are equivalent to Closed Indexed Freyd Categories, which generalize standard Freyd Categories that handle only single inputs.

This distinction helps explain why Arrows have proven particularly useful in domains like quantum computing, where managing multiple inputs with complex relationships is essential.

R. Atkey’s paper finds the relationship between Arrows and different constraints on Freyd categories as follows:

Diagram 12
Figure 12: Relationship Between Structures


Key Insights: - Arrows can be defined both as monoids in categories of strong profunctors and operationally through concrete morphisms (arr\text{arr}, \ggg, first\text{first}) - Freyd categories formalize the relationship between pure functions and effectful computations using symmetric premonoidal structure - Despite the folklore belief, Arrows are strictly more general than Freyd categories because they can handle two separate inputs (one potentially structured) - Arrows are equivalent to closed indexed Freyd categories, bridging the conceptual gap


Applications and Questions The main goal of our Adjoint School project was to structure effects in quantum programming languages using generalizations of monads. Relative monads are a popular generalization of monads. These monads need not be endofunctors, and they’re known to generalize Arrows as well. Since we already know how to structure quantum effects using Arrows, it follows that it should be theoretically possible to structure quantum effects using relative monads.

Arrows’ capacity to handle multiple inputs with a single, potentially structured output offers tractability that is particularly useful in quantum computing. Particles in quantum systems can be in entangled states, where the manipulation of one particle influences others in real time, irrespective of distance. This non-local interaction can be modeled through Arrows’ ability to combine several inputs while keeping track of their interrelationships.

Our group investigated the possibility of doing exactly this. The main technical issue arises from the fact that the way Arrows have been implemented in Haskell to structure quantum effects does not provide a categorical semantics for the problem.

For our ACT2025 presentation, we were able to construct a relative monad capable of handling classical control in the quantum setting, but the following questions still remain:

  • Can one build a relative monad to model quantum effects?

  • If so, how might an implementation of these ideas in Haskell compare to Arrow-based approaches?

The ride from burrito monads to Arrow kitchens has carried us farther than we anticipated, illustrating that even established mathematical folklore sometimes requires precise re-evaluation. As we continue to learn about these structures, we hope this post will motivate others to participate in the exploration of these tools and their use in quantum computing and beyond.

September 01, 2025

Terence TaoA crowdsourced project to link up erdosproblems.com to the OEIS

Thomas Bloom’s erdosproblems.com site hosts nearly a thousand questions that originated, or were communicated by, Paul Erdős, as well as the current status of these questions (about a third of which are currently solved). The site is now a couple years old, and has been steadily adding features, the most recent of which has been a discussion forum for each individual question. For instance, a discussion I had with Stijn Cambie and Vjeko Kovac on one of these problems recently led to it being solved (and even formalized in Lean!).

A significantly older site is the On-line Encyclopedia of Integer Sequences (OEIS), which records hundreds of thousands of integer sequences that have some mathematician has encountered at some point. It is a highly useful resource, enabling researchers to discover relevant literature for a given problem so long as they can calculate enough of some integer sequence that is “canonically” attached to that problem that they can search for it in the OEIS.

A large fraction of problems in the Erdos problem webpage involve (either explicitly or implicitly) some sort of integer sequence – typically the largest or smallest size {f(n)} of some {n}-dependent structure (such as a graph of {n} vertices, or a subset of {\{1,\dots,n\}}) that obeys a certain property. In some cases, the sequence is already in the OEIS, and is noted in the Erdos problem web page. But in a large number of cases, the sequence either has not yet been entered into the OEIS, or it does appear but has not yet been noted on the Erdos web page.

Thomas Bloom and I are therefore proposing a crowdsourced project to systematically compute the hundreds of sequences associated to the Erdos problems and cross-check them against the OEIS. We have created a github repository to coordinate this process; as a by-product, this repository will also be tracking other relevant statistics about the Erdos problem website, such as the current status of formalizing the statements of these problems in the Formal Conjectures Repository.

The main feature of our repository is a large table recording the current status of each Erdos problem. For instance, Erdos problem #3 is currently listed as open, and additionally has the status of linkage with the OEIS listed as “possible”. This means that there are one or more sequences attached to this problem which *might* already be in the OEIS, or would be suitable for submission to the OEIS. Specifically, if one reads the commentary for that problem, one finds mention of the functions {r_k(N)} for {k=3,4,\dots}, defined as the size of the largest subset of {\{1,\dots,N\}} without a {k}-term progression. It is likely that several of the sequences {r_3(N)}, {r_4(N)}, etc. are in the OEIS, but it is a matter of locating them, either by searching for key words, or by calculating the first few values of these sequences and then looking for a match. (EDIT: a contributor has noted that the first foursequences appear as A003002, A003003, A003004, and A003005 in the OEIS, and the table has been updated accordingly.)

We have set things up so that new contributions (such as the addition of an OEIS number to the table) can be made by a Github pull request, specifically to modify this YAML file. Alternatively, one can create a Github issue for such changes, or simply leave a comment either on the appropriate Erdos problem forum page, or here on this blog.

Many of the sequences do not require advanced mathematical training to compute, and so we hope that this will be a good “citizen mathematics” project that can bring in the broader math-adjacent community to contribute to research-level mathematics problems, by providing experimental data, and potentially locating relevant references or connections that would otherwise be overlooked. This may also be a use case for AI assistance in mathematics through generating code to calculate the sequences in question, although of course one should always stay mindful of potential bugs or hallucinations in any AI-generated code, and find ways to independently verify the output. (But if the AI-generated sequence leads to a match with an existing sequence in the OEIS that is clearly relevant to the problem, then the task has been successfully accomplished, and no AI output needs to be directly incorporated into the database in such cases.)

This is an experimental project, and we may need to adjust the workflow as the project progresses, but we hope that it will be successful and lead to further progress on some fraction of these problems. The comment section of this blog can be used as a general discussion forum for the project, while the github issue page and the erdosproblems.com forum pages can be used for more specialized discussions of specific problems.

Tommaso DorigoSearching For Impossibly Rare Decays

I recently ran into a description of the Mu3e experiment, and got curious about it and the physics it studies. So after giving it a look, I am able to explain that shortly here - I think it is a great example of how deep our studies of particle physics are getting; or, on the negative side, how deep our frustration has gotten with the unassailable agreement of our experiments with Standard Model predictions.

Matter stable and unstable in the Standard Model

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August 30, 2025

Terence TaoSLMath announces new research programs

The Simons-Laufer Mathematical Sciences institute, or SLMath (formerly the Mathematical Sciences Research Institute, or MSRI) has recently restructured its program formats, and is now announcing three new research initiatives, whose applications open on Sep 1 2025:

  • AxIOM (Accelerating Innovation in Mathematics) is a new, month-long research program at SLMath, designed to accelerate innovation and introduce transformative ideas into the mathematical sciences. Programs begin in Spring 2027.
  • PROOF (Promoting Research Opportunities and Open Forums) is a two-week summer program designed to provide research opportunities for U.S.-based mathematicians, statisticians, and their collaborators in the U.S. and abroad, whose ongoing research may have been impacted by factors such as heavy teaching loads, professional isolation, limited access to funding, heavy administrative duties, personal obligations, or other constraints. Programs begin June-July 2026. The priority application deadline for PROOF 2026 is October 12, 2025.
  • Lasting Alliance Through Team Immersion and Collaborative Exploration (LATTICE) is a yearlong program which provides opportunities for U.S. mathematicians to conduct collaborative research on topics at the forefront of the mathematical and statistical sciences. Programs begin June-July 2026. LATTICE 2026 applications are open through February 1, 2026.

(Disclosure: I am vice-chair of the board of trustees at SLMath.)

n-Category Café Equivalence via Surjections

Pick a type of categorical structure: say bicategories, or monoidal categories, or whatever you like. Some of the functors between structures are equivalences, in whatever the appropriate sense might be. And some of those equivalences have one or both of these two properties:

  • They’re not just essentially surjective in every dimension — they’re actually surjective in every dimension.

  • They don’t just preserve the structure up to isomorphism or equivalence — they strictly preserve it.

Call an equivalence with both these properties a strict surjective equivalence. So a strict surjective equivalence is an equivalence of a very special and easy kind.

General principle: the standard notion of equivalence between structures is generated by just these very special ones. For example, two bicategories are biequivalent if and only if they can be linked up by a zigzag of strict surjective equivalences.

Why should we care? Because there are some types of structure where the right notion of equivalence isn’t clear, and this principle guides us to it. For example, it tells us the right notion of equivalence for double categories.

All this is done in my new paper:

Tom Leinster, Equivalence via surjections. arXiv:2508.20555, 2025.

I started thinking about this question during Maru Sarazola’s invited talk at Category Theory 2025 in Brno last month. She asked the question:

What is the right notion of equivalence between double categories?

and carefully went through the properties that the right notion of equivalence should have, some possible candidates, and different approaches one might take to deciding what “right” means.

The answer that Maru ultimately gave was that the right notion is “gregarious double equivalence”, proposed by Alexander Campbell in about 2020. And she gave a justification in terms of model categories, representing joint work between her, Lyne Moser and Paula Verdugo.

For the purposes of this post, it actually doesn’t matter what “gregarious double equivalence” means. What I want to talk about is the following principle, which popped into my head as Maru was speaking:

For many types of categorical structure, the natural notion of equivalence is generated, as an equivalence relation, by identifying AA and BB when there exists a strict surjective equivalence ABA \to B.

It occurred to me that this principle might give a rather different justification for why gregarious double equivalence is the right answer. And after some checking, I discovered that it does.

Let me explain.

A more concrete way to express the principle is that AA and BB are equivalent in the standard sense — whatever’s appropriate for the structures at hand — if and only if there exists a zigzag of strict surjective equivalences

A=A 0A 1A n=B. A = A_0 \leftarrow A_1 \rightarrow \ \cdots \ \leftarrow A_n = B.

For any type of categorical structure I can think of, the pullback of a strict surjective equivalence is a strict surjective equivalence. So a simpler concrete condition is just that there exists a span of strict surjective equivalences

ACB. A \leftarrow C \rightarrow B.

But hold on… what do I mean by “principle”?

What I mean is that for simple types of categorical structure, where “equivalence” and “strict surjective equivalence”, we have a theorem. Here are three examples.

  • Categories. We certainly know what it means for two categories to be equivalent. A “surjective equivalence” is an equivalence that’s not just essentially surjective on objects, but literally surjective on objects.

    In this case, the theorem is that categories AA and BB are equivalent if and only if there exists a span ACBA \leftarrow C \rightarrow B of surjective equivalences between them.

    (The word “strict” does nothing in this case.)

  • Monoidal categories. Again, we know what monoidal equivalence is, and it’s clear what a “strict surjective equivalence” is: a strict monoidal functor that’s a surjective equivalence of categories.

    The theorem is that monoidal categories AA and BB are monoidally equivalent if and only if there exists a span ACBA \leftarrow C \rightarrow B of strict surjective equivalences between them.

  • Bicategories. The pattern is the same. The standard notion of equivalence for bicategories is biequivalence. A “strict surjective equivalence”, in this setting, is a strict 22-functor that is literally surjective on objects and locally a surjective equivalence of categories. (Or put another way, surjective on 00-cells, locally surjective on 11-cells, and full and faithful on 22-cells.)

    The theorem is that bicategories AA and BB are biequivalent if and only if there exists a span ACBA \leftarrow C \rightarrow B of strict surjective equivalences between them.

Probably all these theorems are known. I included them in my paper because I couldn’t find them anywhere in the literature, not even the first one. But if you know a reference, I’d be glad to hear it.

Since the principle holds for categories, monoidal categories and bicategories, it’s reasonable to suppose that it might hold for other types of structure. And if we’re investigating some type of structure where the full notion of equivalence isn’t clear, this principle might help guide us to it.

For example, here’s a theorem on double categories, the main result of my paper:

  • Double categories. Again, it’s clear what “strict surjective equivalence” should mean: a strict double functor that’s surjective on 00-cells, locally surjective on both horizontal and vertical 11-cells, and full and faithful on 22-cells.

    The theorem is that double categories AA and BB are gregariously double equivalent if and only if there exists a span ACBA \leftarrow C \rightarrow B of strict surjective equivalences between them.

Even without me telling you what “gregarious double equivalence” means, the four theorems I’ve stated suggest that it’s the right notion of equivalence for double categories, because it continues the pattern we’ve seen for simpler categorical structures.

So, I agree with the conclusion that Moser, Sarazola and Verdugo had already reached! But for different reasons.

Incidentally, this must be the fastest paper I’ve ever written: just under six weeks from sitting in Maru’s talk and hearing the mathematical term “gregarious” for the first time ever to putting the paper on the arXiv. But the principle that all equivalences are generated by strict surjective equivalences was planted in my head in the late 1990s or early 2000s by Carlos Simpson. Back then, we were both working on higher category theory, and when he explained this principle, I found it very striking — so striking that I remembered it 20+ years later. There’s a bit more on that higher categorical context in the introduction to my paper.

August 26, 2025

Tommaso DorigoA Remarkable Graph: The Full Dalitz Plot Of Neutron Decay

The neutron is a fascinating particle, and one which has kept experimental physicists busy for almost a century now. Discovered by James Chadwick in 1932 in a cunning experiment which deserves a separate post (it is a promise, or a threat if you prefer),  the neutron has been all along a protagonist in the development of nuclear weapons as well as in the extraction of nuclear power from fission reactors. And of more relevance to our discussion here, it has powered endless studies both in the context of nuclear and subnuclear physics.

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August 24, 2025

August 23, 2025

August 22, 2025

Peter Rohde Why?

  1. The person dressed up as Ursula pretending to be my mother clearly isn’t and hasn’t been for a long time.
  2. When I went back to Armidale after leaving BTQ and being left unemployed she made numerous ongoing promises to provide me with assistance, both in obtaining my own accommodation and providing financial assistance.
  3. These didn’t materialise and the promises were revoked.
  4. Instead I was evicted from the family home and subject to ongoing stalking and harassment that required multiple referrals to law enforcement, both to the police and the Attorney-General, demanding cease and desist.
  5. These have been systematically ignored and up until the last message she continues to bypass these requests, approaching my personal friends to harass me and stalk me indirectly. The messages passed on are the usual fake “I’m worried about him” bullshit.
  6. Why has my family home been confiscated by security, who actively break the law by ignoring cease and desist from stalking notices made to law enforcement, forcing an unemployed civilian into ongoing homelessness since early in the year?
  7. What is the rational for my eviction and being barricaded from my own home?
  8. I continue to face a medical blockade and am unable to access essential medicines. Seroquel scripts are deliberately delayed past known script deadlines to try and destabilise me.
  9. Vyvanse scripts are denied outright as the psychiatrist does not respond. He is also known to be a state actor.
  10. It has been repeatedly indicated to me not to worry about finances because they have my back. Instead now the only cash I have is that obtained from fully drawing out a cash advance against my credit card and it will only last days. At that point I’m on the street.
  11. Is everyone here on the same page as to what the deal is? If not, who is playing you off? They clearly need to be deposed.
  12. These are violations of human rights and constitute war crimes and crimes against humanity. Whoever is behind it needs to be removed. End of story.
  13. Who else is being subject to this kind of high level manipulation?
  14. It has been repeatedly suggested that full accountability for the lives of those I care for would be provided. This has not been forthcoming. It is also a violation international law to not provide accountability for the lives of those who are known to have been threatened by the state. These are grounds for removal.
  15. Can anyone answer the question as to why I am in this situation? Who is even living in the family home? Some stooge dressed up as Ursula? It’s a poor lifestyle choice to say the least.
  16. It’s pretty obvious they’re trying to get rid of me and once they do they’ll get rid of all of you too.

August 20, 2025

Peter Rohde A call for global insurrection against tyranny and in the name of righteousness

Let it be known to all governments and systems of power:

  • It is their responsibility to serve the people not themselves.
  • While there are no equals, all are to be treated with equality.
  • Where they are self-serving there is a mandate for insurrection such that they serve the people.
  • Where they seek self-protection they will be denied and removed from power.
  • Where they keep secrets from the people there is a mandate for insurrection to enforce transparency and accountability for all.
  • Where they threaten or condemn the people they are condemned and there is a mandate for insurrection.
  • Where they fail to account for the lives of the people they serve there is a mandate for insurrection.
  • Where tyrannical power structures exist there is a mandate to disestablish them.
  • Where they declare war or work against one another there is a mandate for insurrection and unification.
  • Where they lie to us, deceive us or withhold the truth, they shall be removed from power and the truth be told to all.
  • Where legal systems uphold and enable tyranny they shall be removed. These are not our laws and we do not recognise them.

This is the natural order that guarantees our survival and gifts this world to our children. This world belongs to them and where we fail to serve them we condemn ourselves. And where government has failed to uphold this, we will not obey them as they have no right to exist.

We do not have to ask for these things, they are required, and if not given we shall simply take them.

Where the truth has not been told it shall be told.

If we fail to do so we condemn our children ourselves.

August 11, 2025

Terence TaoRough numbers between consecutive primes

First things first: due to an abrupt suspension of NSF funding to my home university of UCLA, the Institute of Pure and Applied Mathematics (which had been preliminarily approved for a five-year NSF grant to run the institute) is currently fundraising to ensure continuity of operations during the suspension, with a goal of raising $500,000. Donations can be made at this page. As incoming Director of Special Projects at IPAM, I am grateful for the support (both moral and financial) that we have already received in the last few days, but we are still short of our fundraising goal.

Back to math. Ayla Gafni and I have just uploaded to the arXiv the paper “Rough numbers between consecutive primes“. In this paper we resolve a question of Erdös concerning rough numbers between consecutive gaps, and with the assistance of modern sieve theory calculations, we in fact obtain quite precise asymptotics for the problem. (As a side note, this research was supported by my personal NSF grant which is also currently suspended; I am grateful to recent donations to my own research fund which have helped me complete this research.)

Define a prime gap to be an interval {(p_n, p_{n+1})} between consecutive primes. We say that a prime gap contains a rough number if there is an integer {m \in (p_n,p_{n+1})} whose least prime factor is at least the length {p_{n+1}-p_n} of the gap. For instance, the prime gap {(3,5)} contains the rough number {4}, but the prime gap {(7,11)} does not (all integers between {7} and {11} have a prime factor less than {4}). The first few {n} for which the {n^\mathrm{th}} prime gap contains a rough number are

\displaystyle  2, 3, 5, 7, 10, 13, 15, 17, 20, \dots.

Numerically, the proportion of {n} for which the {n^\mathrm{th}} prime gap does not contain a rough number decays slowly as {n} increases:

Erdös initially thought that all but finitely many prime gaps should contain a rough number, but changed his mind, as per the following quote:

…I am now sure that this is not true and I “almost” have a counterexample. Pillai and Szekeres observed that for every {t \leq 16}, a set of {t} consecutive integers always contains one which is relatively prime to the others. This is false for {t = 17}, the smallest counterexample being {2184, 2185, \dots, 2200}. Consider now the two arithmetic progressions {2183 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13} and {2201 + d \cdot 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13}. There certainly will be infinitely many values of {d} for which the progressions simultaneously represent primes; this follows at once from hypothesis H of Schinzel, but cannot at present be proved. These primes are consecutive and give the required counterexample. I expect that this situation is rather exceptional and that the integers {k} for which there is no {m} satisfying {p_k < m < p_{k+1}} and {p(m) > p_{k+1} - p_k} have density {0}.

In fact Erdös’s observation can be made simpler: any pair of cousin primes {p_{n+1}=p_n+4} for {p_n > 3} (of which {(7,11)} is the first example) will produce a prime gap that does not contain any rough numbers.

The latter question of Erdös is listed as problem #682 on Thomas Bloom’s Erdös problems website. In this paper we answer Erdös’s question, and in fact give a rather precise bound for the number of counterexamples:

Theorem 1 (Erdos #682) For {X>2}, let {N(X)} be the number of prime gaps {(p_n, p_{n+1})} with {p_n \in [X,2X]} that do not contain a rough number. Then

\displaystyle  N(X) \ll \frac{X}{\log^2 X}. \ \ \ \ \ (1)

Assuming the Dickson–Hardy–Littlewood prime tuples conjecture, we can improve this to

\displaystyle  N(X) \sim c \frac{X}{\log^2 X} \ \ \ \ \ (2)

for some (explicitly describable) constant {c>0}.

In fact we believe that {c \approx 2.8}, although the formula we have to compute {c} converges very slowly. This is (weakly) supported by numerical evidence:

While many questions about prime gaps remain open, the theory of rough numbers is much better understood, thanks to modern sieve theoretic tools such as the fundamental lemma of sieve theory. The main idea is to frame the problem in terms of counting the number of rough numbers in short intervals {[x,x+H]}, where {x} ranges in some dyadic interval {[X,2X]} and {H} is a much smaller quantity, such as {H = \log^\alpha X} for some {0 < \alpha < 1}. Here, one has to tweak the definition of “rough” to mean “no prime factors less than {z}” for some intermediate {z} (e.g., {z = \exp(\log^\beta X)} for some {0 < \beta < \alpha} turns out to be a reasonable choice). These problems are very analogous to the extremely well studied problem of counting primes in short intervals, but one can make more progress without needing powerful conjectures such as the Hardy–Littlewood prime tuples conjecture. In particular, because of the fundamental lemma of sieve theory, one can compute the mean and variance (i.e., the first two moments) of such counts to high accuracy, using in particular some calculations on the mean values of singular series that go back at least to the work of Montgomery from 1970. This second moment analysis turns out to be enough (after optimizing all the parameters) to answer Erdös’s problem with a weaker bound

\displaystyle  N(X) \ll \frac{X}{\log^{4/3-o(1)} X}.

To do better, we need to work with higher moments. The fundamental lemma also works in this setting; one now needs precise asymptotics for the mean value of singular series of {k}-tuples, but this was fortunately worked out (in more or less exactly the format we needed) by Montgomery and Soundararajan in 2004. Their focus was establishing a central limit theorem for the distribution of primes in short intervals (conditional on the prime tuples conjecture), but their analysis can be adapted to show (unconditionally) good concentration of measure results for rough numbers in short intervals. A direct application of their estimates improves the upper bound on {N(X)} to

\displaystyle  N(X) \ll \frac{X}{\log^{2-o(1)} X}

and some more careful tweaking of parameters allows one to remove the {o(1)} error. This latter analysis reveals that in fact the dominant contribution to {N(X)} will come with prime gaps of bounded length, of which our understanding is still relatively poor (it was only in 2014 that Yitang Zhang famously showed that infinitely many such gaps exist). At this point we finally have to resort to (a Dickson-type form of) the prime tuples conjecture to get the asymptotic (2).

August 09, 2025

Justin WilsonPhases of a Game Show, Part 2

In a previous post, we discussed a phase transition that occurred in the piping above you on a game show. In the scenario, you are led on stage in front of a large audience. After a brief time, the audience votes on how “likeable” you are. The catch is that it doesn’t simply tally the votes, but turns spigots on a lattice of piping above your head. Water is then released and if enough people like you, it closes off the passage, keeping you dry. This exciting game show1 was described in that post:

Each “like” turns a spigot off, stopping water from flowing through one pipe in a grid overhead. Once voting ends, water is dumped into the system. If it can find a path to the bottom, you get soaked. [Emphasis added] The better your “likeability,” the less likely spigots open a path for water to flow and the drier you stay. That’s your prize for this game show (and hey, you also get the knowledge that people out there like you).

This system models a type of phase transition known as percolation.

The full post is here:

I highlighted above a key phrase “If it can find a path to the bottom, you get soaked.” What I didn’t say, but should have is that the water was being forced through the pipes, not just dropping down due to gravity. This is a very important point since our phases and phase transition changes dramatically if we just let gravity do the work. In the case of the water being “forced,” it can travel back up pipes if it helps it find its way out and onto your head, but in the case when only gravity is present, it falls down the pipes. To facilitate gravity, we’ll turn the pipes 45 degrees, and if we insert water at a single point on top, it could look like this:

Testing our gravity setup by putting in water at only one pipe up top. Notice that it never goes back up a pipe, only down.

This setup is a different problem called directed percolation. It also has a phase transition, but one that is different in some fundamental ways from regular percolation.

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Before we explore its stranger properties, we can ask, “At what likability threshold do you remain dry?” Well, this happens to have a transition chance of 35.53%!2 This system is a lot more generous, keeping you dry even when a majority of people dislike you. This number comes from numerical computations which have been done rather precisely, and we can even compute it ourselves. In fact, you can see this clearly with this plot

Notice that as we make the system bigger and bigger, the chance of getting soaked less than 35.53% increases and above it, it decreases. This is the same kind of hallmark of a phase transition as we saw in our previous case.

We can also look at the water as it flows down the system to see the clusters that make it from top to bottom

The “Soaked” phase (left), the transition point (middle), and the “Dry” phase (right) as well as the water’s flow through the system (blue).

There is still a fractal-looking pattern at the transition point. With all of these similarities with the regular percolation problem from the last post, what is different? And why is that plot so long and skinny? If gravity wants to pull you down, is that somehow altering the motion down, making it distinct from the motion left or right?

Well, if you go back to the two plots above, you’ll notice a few things that really make them differ from the percolation plots. In the fine print of the first, I’ve noted that the vertical distance is L1.58, so for a horizontal size of 40, the vertical size is roughly 340! That is definitely not a square. And in the second plot, there appears to be more vertical distance than horizontal distance. What is special about this 1.58 number3? It turns out, it’s a critical exponent in this problem, a universal aspect of directed percolation, that distinguishes it from regular percolation. We will call it z = 1.58 the dynamical critical exponent since it is revealed as water flows down in time (dynamically). This dynamical exponent z can reveal itself by looking at these “long and skinny” setups, but be masked by the square setup.

Universality and the finite size of our system

One thing we took away in the previous post was that we lose any sense of scale at this type of phase transition4. But whenever we have “only” thousands of pipes, the size of the system provides a scale! This is the main reason why we begin to see smooth curves and not sharp jumps in quantities. If the system of pipes were infinite (and we had infinite time for the water to go down the pipes), the probability you get soaked would be 100% less than the 35.53% likeability and 0% more than 35.53% likeability. For physical systems, the finite size is often not a huge issue since the scale is closer to the 1023 atoms present in macroscopic systems, and so even things that are technically smooth curves look very sharp.

The problem of size becomes more severe with directed percolation because horizontal and vertical distances start behaving differently thanks to gravity. In this case, if we lay out our nice grid of 10 × 10, 20 × 20, or 30 × 30, we start to notice that the likeability threshold where you stop getting soaked, seems to depend on the size of the system more than before. In actuality it doesn’t, but for these small sizes, you are definitely getting soaked well into the so-called “Dry Phase” we previously labeled. This is seen in the red curves here where each bigger square has a curve underneath the last:

Gravity has done something to the system. Flowing down is different from flowing left or right. In fact, if we flow down by some amount h and over to the right by some distance w, then at the directed percolation transition point

The amount water flows down is related to how far it flows to the right or left by this weird, fractional power of w. This 1.58 is z, our new dynamical critical exponent, which is a universal feature of directed percolation5. It tells us that if we make a system 30 pipes wide, it should extend roughly 301.58 ≈ 216 pipes in height to begin picking out the phase transition effectively. The blue curves in the above plot show this and notice how they all converge on one point; that point is the phase transition. It is revealed by small sizes! To understand why, just think about how the curves are changing as we make the system bigger and bigger.

The red curves will still converge to the phase transition, but it takes larger system sizes for it to reveal itself. This is related to the property that at the phase transition there is no longer a sense of scale, but away from the transition, the vertical scale of clusters could be so large that our puny 60-by-60 grid cannot even begin to reveal it. So if we sit at say a likeability of 0.4 in the 60-by-60 grid, we can say that the vertical size of a typical cluster is most likely more than 60.

A different phase transition but connections to new types of physics

This “gravity mode” for our game show we may call “easy mode” since it requires less of the audience to like you, but the implications here are wide. This type of phase transition has been seen in many kinds of local dynamics where there is a preferred configuration or state. These called an absorbing state phase transitions, and they are a property of certain random dynamical systems. Gravity has provided the distinction here, but more generically, causality and time itself provide that direction, leading to dynamics that obey the same universality as directed percolation.

1

Trademark pending.

2

Usually, you’ll see 0.6447 quoted instead, but that’s just 1−0.3553, which counts open pipes instead of closed as we’re doing.

3

I should note that we have this number to much higher precision than the two decimal points presented here, see the Wikipedia entry where

4

This is a second-order or continuous phase transition. Most transitions in the water phase diagram are first-order transitions which still retain a scale.

5

To drive this point home: Even if we change the lattice, this power law will remain intact. Sometimes it shows up in completely different scenarios too, like in absorbing state phase transitions.

August 04, 2025

Clifford JohnsonHarvest

There’s a lot of joyful knife-work in my future. #bolognese #summersalad –cvj

The post Harvest appeared first on Asymptotia.

July 29, 2025

David Hoggintegrating out nuisances

Further insipired by yesterday's post about binary fitting, I worked today on the treatment of nuisance parameters that have known distributions. These can be treated as noise sometimes. Let me explain:

If I had to cartoon inference (or measurement) in the face of nuisance parameters, I would say that frequentists profile (optimize) over the nuisances and Bayesians marginalize (integrate) over the nuisances. In general frequentists cannot integrate over anything, because there is no measure in any of the parameter spaces. But sometimes there is a measure. In particular, when there is a compact symmetry:

We know (or very strongly believe) that all possible orientations of a binary-star orbit are equally likely. In this model (or under this normal assumption) we have a distribution over two angles (theta and phi for that orbit pole, say); it is the distribution set by the compact group SO(2). Thus we can treat the orientation as a noise source with known distribution and integrate over it, just like we would any other noise source. So, in this case (and many cases like it) we can integrate (marginalize) even as frequentists. That is, there are frequentism-safe marginalizations possible in binary-star orbit fitting. This should drop the 12-parameter fits (for ESA Gaia data) down to 8-parameter, if I have done my math right.

July 28, 2025

David Hoggbinary stars with periods of exactly one year

On Friday, Kareem El-Badry (Caltech) gave a seminar about looking for (and finding!) stars in binary orbits around dark or much darker companions, like black holes, neutron stars, and white dwarfs. He showed results that involve ESA Gaia astrometry, where he noted that the Gaia Mission has no sensitivity to periods right at (or within an inverse mission-length frequency difference of) one-year periods (inverse year frequencies). After the talk I objected that these are not exactly degenerate; El-Badry said that the inferences blow up there.

I spent some time on the weekend thinking about this point, and I now understand it: There is a particular one-year orbit that a star can have (around a darker companion) such that the photocenter of the system makes a motion that is identical to the apparent parallax motion. Thus there is an exact degeneracy between the parallax and a certain one-year orbit.

Does that mean that we can't measure orbits at one year (or, for that matter, parallaxes)? No, it does not. After all, the parallax ellipse has a particular celestial (angular) shape and phase. But it might require some kind of reparameterization of orbits near one-year periods. I think I know how to do that. Should we find the missing binaries? (Oh and by the way, this degeneracy means that, in a strict frequentist sense, Gaia can't measure parallaxes at all without additional information.)

John PreskillLittle ray of sunshine

A common saying goes, you should never meet your heroes, because they’ll disappoint you. But you shouldn’t trust every common saying; some heroes impress you more, the better you know them. Ray Laflamme was such a hero.

I first heard of Ray in my undergraduate quantum-computation course. The instructor assigned two textbooks: the physics-centric “Schumacher and Westmoreland” and “Kaye, Laflamme, and Mosca,” suited to computer scientists. Back then—in 2011—experimentalists were toiling over single quantum logic gates, implemented on pairs and trios of qubits. Some of today’s most advanced quantum-computing platforms, such as ultracold atoms, resembled the scrawnier of the horses at a racetrack. My class studied a stepping stone to those contenders: linear quantum optics (quantum light). Laflamme, as I knew him then, had helped design the implementation. 

Imagine my awe upon meeting Ray the following year, as a master’s student at the Perimeter Institute for Theoretical Physics. He belonged to Perimeter’s faculty and served as a co-director of the nearby Institute for Quantum Computing (IQC). Ray was slim, had thinning hair of a color similar to mine, and wore rectangular glasses frames. He often wore a smile, too. I can hear his French-Canadian accent in my memory, but not without hearing him smile at the ends of most sentences.

Photo credit: IQC

My master’s program entailed a research project, which I wanted to center on quantum information theory, one of Ray’s specialties. He met with me and suggested a project, and I began reading relevant papers. I then decided to pursue research with another faculty member and a postdoc, eliminating my academic claim on Ray’s time. But he agreed to keep meeting with me. Heaven knows how he managed; institute directorships devour one’s schedule like ravens dining on a battlefield. Still, we talked approximately every other week.

My master’s program intimidated me, I confessed. It crammed graduate-level courses, which deserved a semester each, into weeks. My class raced through Quantum Field Theory I and Quantum Field Theory II—a year’s worth of material—in part of an autumn. General relativity, condensed matter, and statistical physics swept over us during the same season. I preferred to learn thoroughly, deeply, and using strategies I’d honed over two decades. But I didn’t have time, despite arriving at Perimeter’s library at 8:40 every morning and leaving around 9:30 PM.

In response, Ray confessed that his master’s program had intimidated him. Upon completing his undergraduate degree, Ray viewed himself as a nobody from nowhere. He chafed in the legendary, if idiosyncratically named, program he attended afterward: Part III of the Mathematical Tripos at the University of Cambridge. A Cambridge undergraduate can earn a master’s degree in three steps (tripos) at the Department of Applied Mathematics and Theoretical Physics. Other students, upon completing bachelor’s degrees elsewhere, undertake the third step to earn their master’s. Ray tackled this step, Part III.

He worked his rear off, delving more deeply into course material than lecturers did. Ray would labor over every premise in a theorem’s proof, including when nobody could explain the trickiest step to him.1 A friend and classmate helped him survive. The two studied together, as I studied with a few fellow Perimeter students; and Ray took walks with his friend on Sundays, as I planned lunches with other students on weekends.

Yet the program’s competitiveness appalled Ray. All students’ exam scores appeared on the same piece of paper, posted where everyone could read it. The department would retain the highest scorers in its PhD program; the other students would have to continue their studies elsewhere. Hearing about Ray’s program, I appreciated more than ever the collaboration characteristic of mine.

Ray addressed that trickiest proof step better than he’d feared, come springtime: his name appeared near the top of the exam list. Once he saw the grades, a faculty member notified him that his PhD advisor was waiting upstairs. Ray didn’t recall climbing those stairs, but he found Stephen Hawking at the top.

As one should expect of a Hawking student, Ray studied quantum gravity during his PhD. But by the time I met him, Ray had helped co-found quantum computation. He’d also extended his physics expertise as far from 1980s quantum gravity as one can, by becoming an experimentalist. The nobody from nowhere had earned his wings—then invented novel wings that nobody had dreamed of. But he descended from the heights every other week, to tell stories to a nobody of a master’s student.

The author’s copy of “Kaye, Laflamme, and Mosca”…
…in good company.

Seven and a half years later, I advertised openings in the research group I was establishing in Maryland. A student emailed from the IQC, whose co-directorship Ray had relinquished in 2017. The student had seen me present a talk, it had inspired him to switch fields into quantum thermodynamics, and he asked me to co-supervise his PhD. His IQC supervisor had blessed the request: Ray Laflamme.

The student was Shayan Majidy, now a postdoc at Harvard. Co-supervising him with Ray Laflamme reminded me of cooking in the same kitchen as Julia Child. I still wonder how I, green behind the ears, landed such a gig. Shayan delighted in describing the difference between his supervisors’ advising styles. An energetic young researcher,2 I’d respond to emails as early as 6:00 AM. I’d press Shayan about literature he’d read, walk him through what he hadn’t grasped, and toss a paper draft back and forth with him multiple times per day. Ray, who’d mellowed during his career, mostly poured out support and warmth like hollandaise sauce. 

Once, Shayan emailed Ray and me to ask if he could take a vacation. I responded first, as laconically as my PhD advisor would have: “Have fun!” Ray replied a few days later. He elaborated on his pleasure at Shayan’s plans and on how much Shayan deserved the break.

When I visited Perimeter in 2022, Shayan insisted on a selfie with both his PhD advisors.

This June, an illness took Ray earlier than expected. We physicists lost an intellectual explorer, a co-founder of the quantum-computing community, and a scientist of my favorite type: a wonderful physicist who was a wonderful human being. Days after he passed, I was holed up in a New York hotel room, wincing over a web search. I was checking whether a quantum system satisfies certain tenets of quantum error correction, and we call those tenets the Knill–Laflamme conditions. Our community will keep checking the Knill–Laflamme conditions, keep studying quantum gates implementable with linear optics, and more. Part of Ray won’t leave us anytime soon—the way he wouldn’t leave a nobody of a master’s student who needed a conversation.

1For the record, some of the most rigorous researchers I know work in Cambridge’s Department of Applied Mathematics and Theoretical Physics today. I’ve even blogged about some

2As I still am, thank you very much.

July 25, 2025

Clifford JohnsonFantastic Collaboration!

Well, I can now officially mention that I've been part of the filmmaking team (in a way) working hard to bring you an enjoyable and interesting Fantastic Four movie! I think it has been about two and a half years (?) since this all began. This was a nearly perfect model of how science consulting can work in film. I worked with everyone, wherever I was needed, with the director, writers, producers, director of photography, VFX teams, set design, and so on. They made me feel welcome and part of whatever creative team I was talking to, which was great. They were open to lots of ideas right from when they were starting out thinking about tone, story ideas, and so forth, right through to final (key) tweaks right at the end of the process as recently as mere weeks ago.

It began early on with with having great conversations Matt Shakman and his writing team about the fact that Reed Richards is first and foremost a curiosity-driven physicist (and so quite different from the engineer we have in Tony Stark that we see RdJ bring out so well), and how things like his dedication to his work (and his outlook on things that comes from such work) might play out in terms of family dynamic, personal relationships, etc., - Without it turning into the tedious cliches about scientists somehow not being able to navigate the world of human relationships. Obviously, I could speak to this as a physicist who works on precisely the things Reed works on, as well as a family man, and as well as someone who remembers that it's still all about telling a story. And there are so many stories to tell at that intersection... Anyway, I think these early conversations (as well as suggestions I made in many sets of notes along the way) helped inform (even if only a little bit? who knows?) what Pedro Pascal brought to the character. This aspect of the film is one of the things I'm most pleased about seeing up on screen.

Beyond that, you'll see lots of things I gave them that I'm also delighted to see made it to the film, in many scenes. This includes (but not limited to!): [...] Click to continue reading this post

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David Hogghow significant is your anomaly?

So imagine that you have a unique data set Y, and in that data set Y you measure a bunch of parameters θ by a bunch of different methods. Then you find, in your favorite analysis, your estimate of one particular parameter is way out of line: All of physics must be wrong! How do you figure out the significance of your result?

If you only ever have data Y, you can't answer this question very satisfactorily: You searched Y for an anomaly, and now you want to test the significance. That's why so many a posteriori anomaly results end up going away: That search probably tested way more hypotheses than you think it did, so any significances should be reduced accordingly.

The best approach is to use only part of your data (somehow) to search, and then use a found anomaly to propose a hypothesis test, and then test that test in the held-out or new data. But that often isn't possible, or it is already too late. But if you can do this, then there is usually a likelihood ratio that is decisive about the significance of the anomaly!

I discussed all these issues today with Kate Storey-Fisher (Stanford) and Abby Williams (Chicago) today, as we are trying to finish a paper on the anomalous amplitude of the kinematic dipole in quasar samples.

July 24, 2025

David Hoggfinding emission lines (and other oddities) in hot stars

I showed my robust spectral decomposition (dimensionality reduction) and residuals to the MPIA Binaries group today. There was much useful feedback (including that my H-gamma was actually H-delta; embarassing!). One comment was that the model isn't truly a causal separation between star and lines, so there will be some mean lines in the star model; lines aren't entirely outliers. That's true! The group suggested that I iterate to remove stars with lines from the training set.

After the meeting, I implemented some of that, but problems like this have a pathology: If you carefully remove stars with high residuals at some wavelength, then the training data will be deficient, or low, at that wavelength. And then the model will go lower, and then more stars will have excess at that wavelength and: Disaster. So when I implemented, I required a 2-sigma deviation, and I removed both high and low outliers. I don't know if this will work, but I am testing now.

July 23, 2025

Mark GoodsellEntangled colliders

There are several interesting papers on the arXiv today. One of them, arXiv:2507.15949, involves my former PhD supervisor. It's on the subject of Quantum Entanglement at collider experiments, and relates back to a paper of his from 1992 that I didn't know about (there's a great line in the new paper where the authors complain that their earlier paper was ignored). (Quantum) Entanglement is the phenomenon where two or more particles are in a special state so that their properties are related, but we don't know what those properties are until we measure them. In Quantum Mechanics we would say that the actual state is not decided until we measure them, and this leads to 'spooky action at a distance' because by measuring one particle we appear to set the corresponding property of the other. An alternative explanation would be that there is some hidden quantity or 'hidden variable' where both particles secretly know all along what state they are in. However, surprisingly it's possible to discriminate between these two cases, and set up quantitative tests known as 'Bell inequalities': you can make a measurement and, if the result of that measurement is less than a certain value, then a hidden variable theory cannot explain it. Experiments to test this using photons at low energies were performed in the early 80s by Alain Aspect and others that violated Bell inequalities and thus confirming the Quantum Mechanical interpretation. 

In recent years, experimentalists have become interested in performing similar tests using different particles at higher energies; it is legitimate to ask "is this true for fermions?" or "does this break down at high energy?" Apparently similar questions were asked in the early 90s at LEP where electrons and positrons were collided (instead of protons at the LHC) and the 1992 paper pointed out that they were not really testing Bell Inequalities. The new paper revisits the older argument, and applies it to the new case of e.g. proton collisions producing a top-antitop pair. They argue that the quantity of interest for the Bell Inequality is the spin density matrix:

But what can actually be measured is the differential cross-section (the rate of production of particles in a certain angular volume):

The symbols B and C appear in both expressions: when performing experimental tests of Bell inequalities they are identified with each other. Since the differential cross-section can be measured, the measurement for the Bell Inequality can then be made and tested. However, the authors of the new paper claim that, in order to identify the two sets of symbols, it is necessary to use Quantum Field Theory: the second equation is a prediction based on QFT from the first. In other words, the logic is circular, and Quantum Mechanics has been assumed -- so it's not surprising that the Bell inequality is violated!

I haven't worked on this topic myself, so it will be interesting to see if there is some pushback from the authors of papers such as arXiv:2003.02280 (who proposed such top-antitop studies). 


Fermi decay constant -- at three loops!

 I also want to point out arXiv:2507.15946 by Stephen Martin, who has performed a three-loop computation of the decay rate of the muon in the Standard Model at three loops. This quantity is incredibly important; it is measured very precisely, and so we use it to extract the underlying parameters of the Standard Model -- or, any theory beyond it. But since it's a complicated process, this is a tricky computation, even at low loop order. The results in this paper will be useful for all sorts of calculations, such as extracting the Higgs boson's self-coupling -- and working out whether the universe is metastable!