The Octoberfest is a noble tradition in category theory: a low-key, friendly conference for researchers to share their work and thoughts. This year it’s on Saturday October 26th and Sunday October 27th. It’s being run by Rick Blute out…
Let’s think about how classical statistical mechanics reduces to thermodynamics in the limit where Boltzmann’s constant \(k\) approaches zero, by looking at an example.
Here’s some basic information about the next big annual applied category theory conference — Applied Category Theory 2025 — and the school that goes along with that: the Adjoint School. James Fairbanks will hold ACT2025 and the Adjoint School…
I’ll explain exactly what I mean by ‘classical statistical mechanics’, and how free entropy is defined in this subject. Then I’ll show that as Boltzmann’s constant approaches zero, this free entropy approaches the free entropy we’ve already seen in thermodynamics!
To see classical thermodynamics as a limit of classical statistical mechanics, we want to see the Legendre transform as the limit of some quantity related to a Laplace transform. Here’s a mathematical result along those lines.
You can understand Stirling’s formula using the statistical mechanics of ‘energy particles’ — that is, theoretical entities with no properties except energy, which can be any nonnegative number.
The zeroth Bernoulli number is telling us the energy over temperature of a quantum harmonic oscillator in the high-temperature limit. The rest of the Bernoulli numbers are telling us all the ‘low-temperature corrections’ to the oscillator’s energy over temperature.
Today I’ll head toward an explicit bijection between principal polarizations of the Eisenstein surface and centers of hexagons in the hexagonal tiling honeycomb. I won’t quite get there, but I’ll lay the groundwork.
Today I’ll compute the Néron–Severi group of a very symmetrical abelian surface built from the Eisenstein integers. Then I’ll begin to explain a nice picture of a ‘slice’ of this group.
Bruce Bartlett floats a version of the Modularity Theorem for elliptic curves that frames it as an explicit bijection between sets, and has a question for the experts.
Four things can happen when you take an elliptic curve with integer coefficients and look at it over a finite field. There’s good reduction, bad reduction, ugly reduction and weird reduction. Let’s see examples of these four cases, and how they affect the count of points!
The Adjoint School is a way to learn applied category theory. This year it will lead up to an in-person research week at the University of Oxford on June 10 - 14, 2024. Apply now!
The geometrical meaning of a separable commutative algebra: it’s one whose spectrum $X$ has the property that $X \times X$ can be separated into the diagonal and the rest.
The word ‘separable’ means two things, but today we’ll use some geometry to show every finite separable extension of a field is a separable algebra over that field. And don’t worry, I’ll explain what all this stuff means!
If two algebraic representations of GL(n,k) restrict to give equivalent representations of the subgroup of invertible diagonal matrices, do they have to be equivalent as representations of GL(n,k)?
Some news! Nathaniel Osgood, Evan Patterson, Kris Brown, Xiaoyan Li, Sean Wu, William Waites and I are going to work together at the International Centre for Mathematical Sciences for six weeks starting on May 1st, 2024. We’re going to…
I’m writing a paper in honor of Hoàng Xuân Sính’s 90th birthday, and I’m running into a lot of questions. The term “categorical group” is often used to mean a group object in Cat; these days we also call…
I’m slowly cooking up a big stew of ideas connecting Grothendieck’s Galois theory to the Brauer 3-group, the tenfold way, the foundations of quantum physics and more. Here’s a tiny taste.
If you want to attend the 6th Annual International Conference on Applied Category Theory (ACT2023) or watch it on Zoom, you need to register by July 14th!
A second wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.
A wonderful one-to-one correspondence between the 10 Morita equivalence classes of Clifford algebras and Cartan’s 10 infinite families of compact symmetric spaces.
Types of guillotine partition of a square where the first cut is vertical are counted by the little Schröder numbers, as are operations in the Hipparchus operad.
The icosidodecahedron can be built by truncating either a regular icosahedron or a regular dodecahedron. It has 30 vertices, one at the center of each edge of the icosahedron—or equivalently, one at the center of each edge of a…
The ‘partition function’, familiar from statistical mechanics, is a generalization of cardinality that works for (finite) sets equipped with a function assigning to each point a real number.
There will be a special session on applied category theory in the 2023 Joint Mathematical Meetings in Boston. Submit an abstract to give a talk! The deadline to do so is Tuesday, September 13, 2022.
The Ninth Symposium on Compositional Structures (SYCO 9) is happening in Como, Italy, 8-9 September 2022. The deadline to submit a talk is Monday August 1, 2022.
Fifth International Conference on Applied Category Theory, ACT2022, will take place at the University of Strathclyde from 18 to 22 July 2022, preceded by the Adjoint School 2022 from 11 to 15 July.
Philip Helbig is an astrophysicist who wrote a paper called The flatness problem and the age of the Universe. It’s a good review of some very important problems, but the arXiv refused to accept it in the category where…
guest post by Anna Knörr as part of the Adjoint School for Applied Category Theory 2021 A socratic dialogue on composition! Listen in to what Synthesi has to say about scientific modelling, the programming language AlgebraicJulia and more. Join…
I’m trying to understand sublogical functors, so I’m looking for examples of sublogical functors between presheaf categories, preferably direct images (if that’s possible).
Eugenia Cheng is an expert on giving clear, fun math talks. Now you can take a free class from her on how to give clear, fun math talks! You need to be a grad student in category theory, and…
Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you: Applied Category Theory 2021 – Adjoint School. Applications due Friday 29…
Formally real Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
Duality in projective plane geometry, and how it lets us break the the Lie group E6 into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
I now have a semiannual column in the Notices of the American Mathematical Society! I’m excited: it gives me a chance to write short explanations of cool math topics and get them read by up to 30,000 mathematicians. It’s…
In this first post from the ACT2020 Adjoint School, Elena Di Lavore and Xiaoyan Li explain Carolyn Brown and Doug Gurr’s paper “A categorical linear framework for Petri nets”.
We show that the automorphisms of the octonions fixing a square root of -1 form the group SU(3) - and this group acts on octonions just as it does on the color states of a quark and a lepton.
There’s a way to build the octonions from complex scalars and vectors which makes it evident how SU(3) acts as automorphisms fixing some square root of -1 in the octonions.
This biweekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields.
I’ll describe the groupoid of ‘finite sets equipped with a permutation’ as a sum over Young diagrams. Taking the groupoid cardinality, this description gives a well-known formula for the probability that a random permutation belongs in any given conjugacy class.
We can use groupoid cardinality to compute the moments of this random variable: the number of cycles of length k in a random permutation of an n-element set.
Treat the number of cycles of length k in a permutation of an n-element set as a random variable. What do the moments of this random variable approach as n approaches infinity?
Suppose raindrops are falling on your head, randomly and independently, at an average rate of one per minute. What’s the average of the cube of the number of raindrops that fall on your head in one minute? We can solve this with the help of groupoid cardinalities.
As n approaches infinity, the number of cycles of length k in a random permutation of an n-element set approaches a Poisson distribution with mean 1/k.
Now we’ll compute the expected number of cycles of length k in a random permutation of an n-element set. This will answer lots of questions about random permutations!
In the limit as n approaches infinity, what is the probability that the shortest cycle in a randomly chosen permutation of an n-element set has length more than k?
guest post by Adam Ó Conghaile and Diego Roque We continue the 2019 Applied Category Theory School with a discussion of the paper Total maps of Turing categories by Cockett, Hofstra and Hrubeš. Thank you to Jonathan Gallagher for…
Screw theory is about the Euclidean group. A ‘screw’ is an element of the Lie algebra of this group: an infinitesimal translation together with an infinitesimal rotation.
We’re having a workshop on diversity in math on Friday 8 November 2019 at U. C. Riverside, right before the applied category theory meeting that weekend.
Does the process of converting a category with finite colimits into a symmetric monoidal category give a functor from Rex to SMC that preserves equalizers?
There are 24 inequivalent ways to extend the Conway group to a 2-group, i.e. a one-object weak 2-groupoid, where each 1-morphism has a circle’s worth of auto-2-morphisms.
I hope to see you at this conference! Applied Category Theory 2019, July 15-19, 2019, Oxford, UK. Here’s some information about it, such as how to submit papers….
There will be an International Conference on Homotopy Type Theory, HoTT 2019, from August 12th to 17th, 2019 at Carnegie Mellon University in Pittsburgh, USA.
Michel Dubois-Violette and Ivan Todorov claim to get the Standard Model gauge group as the subgroup of automorphisms of the exceptional Jordan algebra that 1) preserves a copy of 10d Minkowski spacetime inside this Jordan algebra, and 2) also preserves a unit imaginary octonion—which lets us pick out a copy of 4d Minkowski spacetime inside 10d Minkowski spacetime!
The First Symposium on Compositional Structures will be taking place at the School of Computer Science of the University of Birmingham, 20-21 September, 2018.
Tai-Danae Bradley and Brad Theilman, students at the Applied Category Theory 2018 school, continue discussing category-theoretic approaches to linguistics.
Kenny Courser and I have a new paper on coarse-graining open Markov processes, which makes use of symmetric monoidal double categories. Let us know what you think!
A summary of the paper “Mathematical Foundations for a Compositional Distributional Model of Meaning” Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark.
Statebox is a startup trying to combine categories, open games, dependent types, Petri nets, string diagrams, and blockchains into a universal language for distributed systems.
This is the first blog entry as part of the online reading seminar associated to Applied Category Theory 2018. Joseph Moeller and Dmitry Vagner explain some of Kissinger’s work on causality!
Let’s say the closed unit interval [0,1][0,1] maps onto a metric space XX if there is a continuous map from [0,1][0,1] onto XX. Similarly for the Cantor set. Puzzle 0. Does the Cantor set map onto the closed unit…
guest post by Spencer Breiner One Year Postdoc Position at Carnegie Mellon/NIST We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant…
The American Mathematical Society is having a meeting here at U. C. Riverside during the weekend of November 4th and 5th, 2017. I’m organizing a session on Applied Category Theory, and I’m looking for people to give talks.
The Mathematics Department of the University of California at Riverside is trying to hire some visiting assistant professors. We plan to make decisions quite soon! The positions are open to applicants who have PhD or will have a PhD…
guest post by Scott Carter About 7 months ago, Jamie Vicary contacted me with a Globular worksheet of which, initially, I could make neither heads nor tails. He patiently explained to me that what I was looking at was…
Open Markov processes — that is, Markov processes with specified ‘input’ and ‘output’ states — can be seen as morphisms in a category. This lets us take a big complicated Markov process and think of it as made of smaller open ones.
Yes, you can find categories everywhere — even in electrical engineering! The reason is that we live in a world of networks, and networks tend to be morphisms in categories.
In The Hitchhiker’s Guide to the Galaxy, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was!
PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked…
To understand ecosystems, ultimately will be to understand networks. - B. C. Patten and M. Witkamp A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green…
With a suitable Lorentzian metric, the lattice of self-adjoint 3 x 3 octonionic matrices with integral octonions as entries becomes isometric to a 27-dimensional lattice that may play a role in bosonic string theory.
Greg Egan just showed that $\mathrm{E}_{10}$: is the lattice of $2 \times 2$ self-adjoint matrices with integral octonions as entries! Here’s the proof.
There are 10 associative real super-division algebras, 10 classical families of compact symmetric spaces, 10 ways that Hamiltonians can get along with time reversal and charge conjugation symmetry… and even 10 dimensions of spacetime in string theory! Are these related? Yes!
The formation of a wormhole and the creation of a particle-antiparticle pair are closely related processes in a 3d TQFT. This sheds some light on how wormholes and entanglement are similar.
Belinskii, Khalatnikov and Lifshitz described homogeneous cosmologies that become chaotic as you go back toward the Big Bang. What does this have to do with octonionic integers?
Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers. Christopher Lee has developed software that makes this work.
This weekend I’m giving a talk on “The Foundations of Applied Mathematics”. It’s mostly about how the widespread use of diagrams in engineering, biology, and the like may force us to think about math in new ways, at least…
You can now see the list of talks at Category-Theoretic Foundations of Mathematics Workshop at the Department of Logic and Philosophy of Science of U.C. Irvine, May 4-5, 2013.
There will be a conference on Category-Theoretic Foundations of Mathematics at the Department of Logic and Philosophy of Science, U.C. Irvine on May 4-5, 2013.
What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg….
Tomorrow I’m giving a talk about an operad that shows up in biology. I wrote my lecture notes in the form of a blog entry: John Baez, Operads and the tree of life, Combinatorics Seminar, Université du Québec à…
In the third episode of The Three-Fold Way, we’ll see that the “q” in “qubit” stands for quaternion — at least when you think of them as representations of SU(2).
Those of you interested in journal prices and the like will enjoy this article pointed out by David Roberts: Robert Darnton, The library: three jeremiads, The New York Review of Books, December, 2010. The author is a historian at Harvard…
The third and last of a short series of posts on the foundations of quantum theory: the Koecher–Vinberg classification of self-dual homogeneous convex cones, and its relation to observable-state duality.
The second of a short series of posts on the foundations of quantum theory: the theorem by Jordan, von Neumann and Wigner classifying ‘finite-dimensional formally real Jordan algebras’.
Sophie Hebden has written a nice gentle introduction to the work of John Baez’s students that was funded by a grant from the Foundational Questions Institute.
Jacob Biamonte wants your comments on his paper about string diagrams in quantum computation and condensed matter physics, which are called “tensor networks”.
In week299 of This Week’s Finds, hear about the school on Quantum Information and Computer Science that was recently held in Oxford, and the subsequent workshop on Quantum Physics and Logic. Then, find out what Bill Schmitt told me about combinatorics.
In “week298” of This Week’s Finds, learn about finite subgroups of the unit quaternions, finite subloops of the unit octonions, Lie n-superalgebras built using division algebras, and Duff and Ferrara’s ideas connecting exceptional groups to quantum information theory. And learn what a "hyperdeterminant" is!
In “week297” of This Week’s Finds, see knot sculptures and learn about special relativity in finance, lazulinos, some peculiar infinite sums, and a marvelous fact about the number 12. Then: Dirichlet forms and electrical circuits!
In week296 of This Week’s Finds, get a free copy of Jerry Shurman’s book The Geometry of the Quintic, and read my attempt to construct a compact dagger-category whose morphisms are electrical circuits made of resistors.
In “week295” of This Week’s Finds, learn about the principle of least power, Poincaré duality for electrical circuits — and a curious generalization of Hamiltonian mechanics that involves entropy as well as energy.
In “week294”, hear an account of Gelfand’s famous math seminar in Moscow. Read what Jan Willems thinks about control theory and bond graphs. Learn the
proof of Tellegen’s theorem. Meet some categories where the morphisms
are circuits, learn why a category object in Vect is just a
2-term chain complex.. and gaze at Saturn’s rings edge-on.
The real numbers, complex numbers, quaternions and octonions give Lie 2-superalgebras that describe the parallel transport of superstrings, and Lie 3-superalgebras that describe the parallel transport of 2-branes!
In week293, catch up on recent papers and books about $n$-categories. Hear about last weekend’s Conference on the Mathematics of Environmental Sustainability and Green Technology at Harvey Mudd College. And learn how to think of networks of resistors as chain complexes which are also morphisms in a category.
In week291, listen to a crab canon on a Möbius strip, see how the Mandelbrot set mimics infinitely many Julia sets, and learn about 2-ports, 3-ports, and Poincaré duality for electrical circuits.
In “week290” of This Week’s Finds, read about categorification in analysis. Ponder a number puzzle. Meet the five most popular 1-ports: resistances, capacitances, inertances, effort sources and flow sources. And learn a bit more rational homotopy theory!
In “week289” of This Week’s Finds, learn how E8 shows up in condensed matter physics. Continuing exploring the grand analogy between different physical systems! And learn about differential graded Lie algebras in rational homotopy theory.
Ever wonder what mathematicians can do to help solve the environmental problems facing all of us? I do. Maybe this will help: 2010 Harvey Mudd College Mathematics Conference on the Mathematics of Environmental Sustainability and Green Technology, Harvey Mudd,…
In “week288”, start exploring a huge set of analogies linking many branches of physics… and keep learning about rational homotopy theory. This time we’ll tackle differential graded Lie algebras.
In “week287” of This Week’s Finds, hear about the history of categorical logic, and continue learning about rational homotopy theory - especially Sullivan’s approach based on differential forms.
In “week285” of This Week’s Finds, discover the beauty of roots. Then, hear what happened on the second day of the session on homotopy theory and higher algebraic structures in the AMS conference at U.C. Riverside!
There’s a BBC documentary called “Dangerous Knowledge”, about “four brilliant mathematicians — Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing — whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide”.
In “week284” of This Week’s Finds, learn about the pentagon-decagon-hexagon identity, golden triangles, categorified quantum groups, the Halperin-Carlsson
conjecture, real Johnson-Wilson theories, Picard 2-stacks,
quasicategories, motivic cohomology theory, and toric varieties!
See galaxies in visible, infrared and ultraviolet light. Read about who first discovered the regular icosahedron: Theatetus or the ancient Scots. And learn more about the geometry of the icosahedron!
In week282 of This Week’s Finds, visit Mercury: Learn how this planet’s powerful magnetic field interacts with the solar wind to produce flux transfer events and plasmoids. Then read about the web of connections between associative, commutative, Lie and…
In “week281” of This Week’s Finds, learn about the newly discovered ring of Saturn, tilings with 5-fold and 10-fold quasisymmetry, and the latest news on quantum gravity from Corfu.
In week278, hear the latest about Betelgeuse. Read how red supergiants spew out dust which eventually forms planets like ours. Watch a hypervelocity collision in a distant solar system. Learn the new way to make graphene, and read my history of the Earth - for physicists. And when you’re ready: dive into groupoidification!
A few notes on Dan Ghica’s ideas about ‘function interface models for hardware compilation’ — a potentially very practical application of symmetric monoidal closed categories.
In “week277”, find out what’s a million times thinner than paper, stronger than diamond,
a better conductor than copper, and absorbs exactly pi times the fine structure constant of the light you shine through it.
Groupoidfest 09 is being held on October 24–25, 2009 at the Department of Mathematics of the University of Colorado, Boulder, and it’s being organized by Arlan Ramsay.
I’ve heard an interesting story about using Khovanov homology to help prove the existence of exotic smooth structures in 4 dimensions. Could you help check to see if it’s correct?
These are the first of some notes of David Ben-Zvi’s lectures at a workshop on topological field theories at Northwestern University, held in May 2009.
nn-Café regulars will know about Representative Conyer’s bill that would repeal the National Institute of Health’s public access policy and forbid other US funding agencies from mandating open access to research papers written with the help of federal grant…
In “week276”, read about Betelgeuse, the Local Bubble, the Loop I Bubble, the cloudlets from
Sco-Gen, and the “local fluff”. Get to know the nLab. And learn how
Paul-André Mélliès and Nicolas Tabareau have
taken some classic results of Lawvere on algebraic theories and
generalized them to other kinds of theories, like PROPs.
You can now see the introduction to a paper James Dolan is writing about algebraic geometry for category theorists. You can also see 5 lectures he gave on this topic.
Ronald Brown, Philip J. Higgins and Rafael Sivera have come out with a book called Nonabelian algebraic topology: homotopy groupoids and filtered spaces. You can download it for free.
There will be a summer school in Corfu from September 13th to September 20th, with courses on quantum gravity, renormalization and higher gauge theory.
This blog entry is supposed to be a forum for learning and discussing Wigner’s classification of the representations of the Poincaré group. Ask and answer questions about this subject here!
If you know a bit of group representation theory and you’ve always wanted to understand some particle physics, now is your chance: read a gentle expository account of the algebraic patterns lurking behind three famous Grand Unified Theories!
Mike Stay and I have finished what we hope is the final version of our paper for Bob Coecke’s book on New Structures for Physics. Peter Selinger’s paper for this book is also done.
There’s a bill in the US House of Representatives that would repeal the NIH’s public access policy… and ban similar policies by other federal agencies.
John Huerta is taking his oral exam soon. He’ll give a talk on the group representation theory underlying three famous grand unified theories, and how they fit together in a larger pattern. You can see the slides now.
A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.
I’m going to the Joint Mathematics Meetings in Washington DC from January 5th to January 8th, 2009, and giving talks on 2-groups and groupoidification. Who else will be in town?
Here are some basic questions about describing the classical superstring using the ‘super’ analogue of multisymplectic geometry, if such a thing exists.
Read more about the geysers of Enceladus. Hear the history of mineral evolution, from chondrites to the Big Splat, the Late Heavy Bombardment, the Great Oxidation Event, and Snowball Earth… to now. Then, learn about Pontryagin duality.
See what the Cassini probe saw as it shot through the Enceladus
plumes, see what happens when the Io flux tube hits Jupiter, learn two new properties of the number 6, and discover the wonders of standard Borel spaces, commutative von Neumann algebras, and Polish groups.
Here are some basic questions about the process of taking a locally compact abelian group A to the group of homomorphisms from A to the invertible complex numbers.
Read about massive volcanic eruptions on Jupiter’s moon Io,
allotropes of sulfur, quasicrystals in various dimensions, Jeffrey
Morton’s extension of the “groupoidification” program, and Stephen Summers’ review of
new work on constructive quantum field theory!
See lava on Jupiter’s moon Io. Hear about Greg Egan’s new novel. And then, learn about some little-known interactions between the numbers 5, 8, 12, and 24.
Moving on up a dimension, now let’s look at the A3 lattice. This arises naturally from the group SU(4), but you’ve also seen it in grocery stores if you ever paid attention to stacks of oranges.
Please comment on two chapters of a forthcoming book edited by Bob Coecke: ‘Introduction to categories and categorical logic’ by Abramsky and Tzevelekos, and ‘Categories for the practicing physicist’ by Coecke and Paquette.
See a marvelous view of Io, and then learn more about Frobenius algebras than you probably wanted to know — and a bit about modular tensor categories and the mathematics of music, too!
In this new version of our paper, we systematically explain how n-dimensional field theories give n-plectic manifolds. We also say how a B field affects the 2-plectic structure for a string.
The Crookes radiometer is also known as a ‘light mill’ — a little glass bulb with a windmill in it, with vanes black on one side and white on the other. It puzzled Reynolds, Maxwell and even Einstein. Do we really understand it yet?
In “week267” see the tilings of the Alhambra and learn about wallpaper groups, 17 wallpaper groups, their corresponding
2d orbifolds, the role of 2-groups as symmetries of orbifolds, the work of
Carrasco and Cegarra on hypercrossed complexes, and the
work of João Faria Martins on the fundamental 2-group of
a 2-knot.
The first Theorems into Coffee prize is awarded. Read about Steve Lack’s work on PROPs, and try your hand at the latest Theorems into Coffee challenge.
The analogies between physics, topology, logic and computer science, visible so clearly with the help of symmetric monoidal closed categories, are just the tip of a larger iceberg involving $n$-categories. The Periodic Table seems to be a useful guide here.
Just as any symplectic manifold gives a Lie algebra of observables, any 2-plectic manifold gives a Lie 2-algebra of observables. This shows up in string theory!
Read about Europa, the Pythagorean pentagram, Bill Schmitt’s work on Hopf
algebras in combinatorics, the magnum opus of Aguiar and Mahajan, and
quaternionic analysis.
Chen spaces and Souriau’s diffeological spaces are two great contexts for differential geometry. Alex Hoffnung and his thesis advisor just wrote a paper studying these in detail.
Learn about the Southern Ring Nebula, the frosty dunes of Mars, quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a
new construction of e8, and a categorification of sl(2).
Groupoidifying the commutation relations between annihilation and creation operators in quantum mechanics. An in-class experiment demonstrating these relations.
The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re secretly the same!
An intro to degroupoidification: the process of turning groupoids into vector spaces, and spans of groupoids into linear operators. A key prerequisite: ‘groupoid cardinality’.
Analysis of a media kerfuffle: can looking at the Universe actually hasten its demise? Is that what Krauss and Dent’s paper really said? What did it really say?
Heisenberg’s matrix mechanics and its many generalizations, such as the category of relations and the weak 2-category of spans. Understanding Hecke operators in terms of spans.
Simultaneously categorifying and q-deforming Pascal’s triangle will lead us to a categorified quantum group. Here we take the first steps in that direction.
Hendryk Pfeiffer describes the sort of gadget whose representations form a modular tensor category… and shows how to reconstruct this gadget from its modular tensor category of representations.
Getting irreducible representations of symmetric groups from flag representations. Using ‘crackpot matrices’ to describe Hecke operators between flag representations.
James Dolan on two applications of Hecke operators: showing that any doubly transitive permutation representation is the direct sum of two irreducible representations, and getting ahold of the irreducible representations of n!
A talk by Chris Douglas reporting on his work with Arthur Bartels and André Henriques on “higher Clifford algebras”. They’re related to elliptic cohomology and they form a 3-category!
Categorifying and q-deforming the binomial coefficients. Why are the q-binomial coefficients polynomials with natural number coefficients? And, why are they “palindromic” polynomials? Bruhat classes and Schubert cells.
When you have any structure on a set, it has a group of symmetries. Here James Dolan shows how to work backwards: given the symmetries, how read off an axiom system describing the structure those symmetries preserve!
In “week257”, learn about astrophysics, number theory, topos theory in physics, distributive laws for monads, and hear what’s happening to the Tale of Groupoidification.
Categorifying and q-deforming the theory of binomial coefficients — and multinomial coefficients! — using the analogy between projective geometry and set theory.
In week255, hear what happened at the 2007 Abel Symposium in Oslo. Read explanations of Jacob Lurie and Ulrike Tillman’s talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen’s talks on string topology, Stephan Stolz’s talk on cohomology and…
Have you ever thought you were getting a PDF file of a journal article, only to hit a webpage from a publisher demanding money for it? Then you’ve been web spammed.
Connes and Marcolli’s new book, Witten’s new paper, exceptional Lie superalgebras and the Standard Model… and the Tale of Groupoidification, continued.
From the Standard Model to SU(5), SO(10), E6… and maybe even on to E8, with a friendly tip of the hat to symmetric spaces like the complexified octonionic plane.
The long-range weather report on Neptune, hot Neptunes in other solar systems, the electromagnetic snake at the center of our galaxy, and Hecke operators.
There’s a new AMS Notices article on how the board of Topology resigned to protest Elsevier’s high prices. Support the Banff Protocol — avoid publishing in highly expensive journals!
From particles to strings. First: building a Hilbert space from a category C equipped with an "amplitude" functor A: C → U(1). Then: building a 2-Hilbert space from a 2-category C equipped with a 2-functor A: C → U(1)Tor.
In week243 of This Week’s Finds, hear about Claude Shannon, his sidekick Kelly, and how they used information theory to make money at casinos and the stock market. Hear about the new book Fearless Symmetry, which explains fancy number…
Photos of Saturn, its ring and moons. Unmanned NASA missions versus sending canned primates to Mars. Jeffrey Mortons’ work on topological quantum field theory.
Read about the open access movement, Freeman Dyson’s 1951 lecture notes, the origins of mathematics in little clay figures called “tokens”, and Koszul duality for L∞-algebras!