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…
The University of San Diego mathematics department is hiring! We have two assistant professor positions open this year; these are starting tenure-track positions. Applications must be through the USD web site; the deadline is November 15, 2021. The University…
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…
Over at 3Blue1Brown Grant Sanderson is asking the provocative question “Why aren’t you making math videos?” Certainly for many of us there are reasons which include things such as lack of time or energy. However, if you have time,…
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…
Job announcement for a postdoctoral position in higher category theory, homotopy type theory, especially as related to quantum logic or quantum field theory at Johns Hopkins.
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…
Guest post by Bruce Bartlett The sunny campus of Stellenbosch University in South Africa is hiring! We’re looking to make a permanent appointment of a mathematician at Lecturer or Senior Lecturer level. Job description and application webpage. Deadline 30…
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.
New paper: a linear analogue of Joyal’s proof of Cayley’s tree formula, showing how many nilpotent operators there are on a vector space of finite cardinality.
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.
A pair of plausible-sounding and occasionally-cited claims about presentable objects are in fact false; though in practice it doesn’t matter a whole lot if we are more careful.
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.
I want to tell you about Elmendorf’s theorem on equivariant homotopy theory. This theorem played a key role in a recent preprint I wrote with Hisham Sati and Urs Schreiber: John Huerta, Hisham Sati, Urs Schreiber - Real ADE-equivariant…
Interpreting linear logic into intuitionistic logic via a Chu construction automatically produces many concepts of traditional constructive mathematics.
The models of an (n+1)-theory are the semantic n-theories in that (n+1)-theory, when it is regarded as an “n-doctrine”, i.e. a language in which to write n-theories.
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!
Guest post by Chris Kapulkin Two years ago, I wrote a post for the n-Cafe, in which I sketched how to make precise the claim that intensional type theory (and ultimately HoTT) is the internal language of higher category…
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…
This post explains the meaning of “the comprehension construction,” the title of a recent paper by Riehl and Verity (https://arxiv.org/abs/1706.10023).
What is the value of the whole in terms of the values of its parts? This apparently very general question, when made precise in a reasonable way, has a surprisingly specific answer.
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.
A family of entropy measures for probability distributions on metric spaces. Or viewed another way: how to measure biological diversity while taking into account the varying similarities between species.
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…
A family of q-deformations of Shannon entropy, interpreted as expected surprise. Plus, a simple characterization theorem for these “surprise entropies”, and another for relative surprise entropy.
An expository summary of Hyland and Power’s “The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads” for the Kan Extension Seminar II.
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…
Progress on the problem of handling logical uncertainty: how do we formally describe our uncertainty about mathematical facts such as the digits of pi or the truth of the Riemann hypothesis?
Magnitude is a highly informative invariant of metric spaces. Here are some of the highlights of the theory, now collected together in a survey paper by Mark Meckes and me.
There’s a very general notion of endomorphism object that specializes to endomorphism monads, endomorphism operads, and the codensity monad. Has anyone seen it?
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.
A little bird tells me that Macquarie University is hiring (even more) category theorists! Specifically, they are offering two-year research fellowship positions, details of which can be found here. Macquarie University, which is in greater Sydney, is the home…
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!
A summary of what we know and don’t know about intensional type theory (HoTT) as an internal language for higher categories (guest post by Chris Kapulkin).
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…
1. Reflexive completion. 2. Mathematicians accepting and declining overtures from GCHQ. 3. The Euler characteristic of an algebra. 4. Review of Nick Gurski’s higher categories book. 5. Are lectures the best way to teach?
Bishop sets and Klein geometries are both quotient constructions, and can both be realized by HITs, in contrast to the univalence/extensionality based approach common in set theory.
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.
FINAL CFP and EXTENDED DEADLINE: SoTFoM II `Competing Foundations?’, 12-13 January 2015, London. The focus of this conference is on different approaches to the foundations of mathematics. The interaction between set-theoretic and category-theoretic foundations has had significant philosophical impact, and…
Guest post by Bruce Bartlett I recently put an article on the arXiv: Bruce Bartlett, Quasistrict symmetric monoidal 2-categories via wire diagrams. It’s about Chris Schommer-Pries’s recent strictification result from his updated thesis, that every symmetric monoidal bicategory is…
Many mathematicians are uncomfortable about the NSA/GCHQ’s surveillance of entire populations, but fundamentally aren’t that bothered. Does it really matter?
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!
Summarizes the approach to coherence theorems expressed as a rectification for pseudoalgebras for a 2-monad using codescent objects, following Steve Lack.
Summarizes “A Classification of Accessible Categories,” which characterizes those categories that are locally presentable or accessible relative to a sound limit doctrine.
Describes Kelly’s “Elementary observations on 2-categorical limits” and the general theory of weighted limits and colimits, which are described here in a special case.
Summarizes the paper of Kelly and Street “Review of the elements of 2-categories” containing common background material for papers in the Sydney Category Seminar Lecture Notes 420
Summarizes Lawvere’s “Metric Spaces, Generalized Logic, and Closed Categories” which explores applications of enriched category theory to metric topology.
An emerging pattern in algebra and topology leads to a new notion of finitely generated FI-modules, which capture the representation stable sequences that arise in practice.
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.
A friendly reminder: applications for the Kan Extension Seminar are due at the end of the week. More information can be found in the initial announcement and on the seminar website. For those who don’t enroll, watch this space….
The enriched version of the algebraic small object argument produces the mapping (co)cylinder factorizations for chain complexes of modules over a commutative ring.
Find out what algebraic varieties, convex sets, linear subspaces, real numbers, logical theories and extension fields have in common with formal concepts.
Much like folks around here, Alistair Savage has been thinking about categorification for some time now and has together with Erhard Neher announced an upcoming opportunity for graduate students and postdocs to take part in a winter school as…
When you spell a word down the phone, some letters are more easily confused than others. What’s a good measure of your communication power that takes these confusions into account?
The axiom of choice (including the law of excluded middle) is equivalent to the statement that H^1 of all discrete sets with values in all groups vanishes.
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.
I just returned from a month at Hong Kong University, visiting James Fullwood, an algebraic geometer who likes to think about the mathematics of string theory. There, I gave a colloquium on G2 and the rolling ball, a paper…
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…
The category-theoretic scone or “gluing construction” packages the type-theorist’s method of “logical relations” to prove canonicity and parametricity properties of type theory.
The magnitude function is a mysterious invariant of graphs. It’s closely related to invariants that have proven meaningful and useful in other branches of mathematics. But uncovering its meaning for graphs poses a challenge.
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.
Continuing the series of what type theory can do for philosophy, let us take a look at the infamous question of the title. To forestall criticism, let me say straight away that I’m not proposing in these posts that…
With a type-theoretic foundational system, we can describe constructions on objects with internalized universal properties, including the hom-functors of higher groupoids.
An extension of the monadicity theorem using F-categories to detect lax, pseudo, and colax morphisms provides new insight into the fundamental nature of such weak morphisms.
A contribution for a book on mathematical aspects of QFT, on extended (“multi-tiered”) prequantum Chern-Simons theory formulated in terms of higher geometry.
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.
Enriched indexed categories, indexed over a base category and enriched over an indexed monoidal category, simultaneously generalize indexed categories, internal categories, and enriched categories.
Here at the Australian National University, I’ve started running a seminar with Mathew Langford. Mat is a PhD student working on geometric analysis, specifically something called extrinsic curvature flow, but we both love mathematical physics, so we decided to…
Every functor can be made into a reflection or coreflection by changing its domain. There is a formal and unenlightening proof, but for some examples we can find “natural” proofs which are actually informative.
A revolutionary paper by Press and Dyson describes surprising and exciting new strategies for the Iterated Prisoner’s Dilemma – but they don’t “beat” TIT FOR TAT unless we change the way we measure victory.
The internal homotopy type theory of the (infinity,1)-topos of simplicial objects gives us a “directed homotopy type theory” to talk about (infinity,1)-categories.
I just looked at the front page of The Guardian, a quality British newspaper that’s especially widely read online… and I was amazed to see that their second-leading story was on The Cost of Knowledge: The news is that…
Most kinds of exact completion, including categories of sheaves, are a special case of a reflection from certain sites into higher-ary exact categories.
Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory at the Fields Institute, May 7-9, 2012
Below are various recent job offers in maths that colleagues are asking me to circulate. One in Liverpool on motives. One in Prague, on geometry and algebra. One in Erlangen, on higher categories and TQFT. One in Adelaide, on…
An internal axiomatization of factorization systems, subtoposes, local toposes, and cohesive toposes in homotopy type theory, using “higher modality” and codiscreteness.
A web-based journal for publications in mathematics and physics for topics that are usefully viewed from the point of view of higher category theory and homotopy theory.
Categories of spaces with discrete and codiscrete objects are closely related to fibrations and opfibrations; the “scone” freely adds both codiscrete objects and cartesian arrows.
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 another thread Tom Leinster would like to learn what a sigma-model in quantum field theory is. Here I want to explain this in a way that will make perfect sense to Tom, and hopefully even intrigue him. To…
A homotopical extension of the notion of “inductive definition” allows us to construct CW complexes and mimic other constructions from homotopy theory in type theory.
There is to be a Symposium – Sets Within Geometry – held in Nancy, France on 26-29 July, 2011. Confirmed speakers are: FW Lawvere (Buffalo), Yuri I. Manin (Bonn and IHES), Anders Kock (Aarhus), Christian Houzel (Paris), Colin McLarty…
I’m sinking in a sea of administrative duties at the moment, so for a bit of sanity I thought I’d jot down the glimmer of a thought I had. We spoke back here about the term model for a…
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…
guest post by David Roberts This post is about my forthcoming paper, extracted from chapter 1 of my thesis: Internal categories, anafunctors and localisations and is also a bit of a call for examples from nn-category cafe visitors (see…
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”.
This week at MPI Bonn is (or has been) taking place a conference in honor of Alan Carey’s 60th birthday. on “noncommutative geometry and index theory, statistical models, geometric issues in quantum field theory. Hamiltonian anomalies and bundle n-gerbes”….
The calculus of exact squares for computing with Kan extensions isn’t well-known in some circles, but it has the advantage of generalizing well to derivators and (∞,1)-categories.
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.
By using homotopy 2-categories and derivators, we can squeeze a lot of information out of (infinity,1)-categories using only comparatively easy 2-categorical machinery.
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!
An easier-to-understand description of the left adjoint to the homotopy coherent nerve, due to Dugger and Spivak, enables us to make explicit computations of its hom-spaces, and better understand the relationship between quasicategories and simplicial categories.
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.
The still-hypothetical notion of “extraordinary 2-multicategory” generalizes a 2-category just enough to include extraordinary natural transformations.
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 nCafé is currently affected by a bug that prevents comments to be posted. Here is a link to discussion on the nForum to alleviate this problem for the moment.
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.
I have just returned from attending three days of the Final Workshop of the Newton Institute Non-Abelian Fundamental Groups in Arithmetic Geometry Programme. I was kindly invited by Café visitor Minhyong Kim, whose lecture we discussed a while ago. While…
Structural set theories such as ETCS provide an alternate foundation for mathematics, which is arguably closer to mathematical practice than ZF-like “material” set theories.
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!
A clever Yoneda argument shows that modulo size concerns, if completeness lifts to categories of algebras, then so do all individual limits. It’s interesting to compare how different approaches to size deal with this.
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”.
(Strong) Feferman set theory provides a set-theoretic foundation for category theory that avoids many of the problems with other approaches such as Grothendieck universes.
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!
An enhanced structure on a 2-category, called a “proarrow equipment,” lets us define weighted limits and develop a good deal of “formal category theory.”
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.
Over in a discussion at Math Overflow I was reminded about Halmos’ great article on writing mathematics, which I highly recommend to all graduate students (or anyone else, for that matter). P. R. Halmos, How to write mathematics, L’Enseignement…
Guest post by Emily Riehl A popular slogan is that (∞,1)(\infty,1)-categories (also called quasi-categories or ∞\infty-categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to…
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 week in our Journal Club on [[geometric ∞\infty-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for…
Preliminaries for the discussion of geometric infinity-function theory: higher categories, higher sheaves, higher algebra, higher traces and what it all means.
A place to discuss and learn about the work by Ben-Zvi/Francis/Nadler on geometric infinity-function theory and its application in infinity-quantum field 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?
Might the cohomology of dynamical systems provide a meeting ground for researchers on the ‘combinatorics’ side of mathematics, and those on the ‘theory-building’ side?
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.
To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one…
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.
The ‘field with one element’ has been honoured by a great accolade. As announced here, it has been awarded a blog all to itself. Not bad for an entity with dubious existence credentials….
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.
On local nets constructed from transport 2-functors and examples relating to lattice models, Hopf spin chains, asymptotic inclusion of subfactors. And some remarks on the relation between conformal nets and vertex operator algebras.
A quick review of Landsman’s result on strict deformation quantization of Poisson manifolds dual to Lie algebroids: the quantum algebra is nothing but the groupoid algebra of the Lie groupoid integrating the Lie algebroid.
Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
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.
Mathematics exams for 16 year olds are getting easier, it is claimed. It’s fairly easy to check for yourself. Take a look at the Arithmetic, Algebra and Geometry papers from 1959 and compare with a contemporary specimen GCSE paper. Even…
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.
I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le…
An article which discusses lifts through the 7-fold connected cover of the structure group of the tangent bundle in the context of electric-magnetic duality in string theory.
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).
A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-particles charged under them.
Groupoidifying the commutation relations between annihilation and creation operators in quantum mechanics. An in-class experiment demonstrating these relations.
On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
Associated L-infinity structures are obtained from Lie action infinity-algebroids, leading to a concept of sections and covariant derivatives in this context.
On how integration and transgression of differential forms is realized in terms of inner homs applied to transport n-functors and their corresponding Lie oo-algebraic connection data.
The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re secretly the same!
On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
An intro to degroupoidification: the process of turning groupoids into vector spaces, and spans of groupoids into linear operators. A key prerequisite: ‘groupoid cardinality’.
On the general idea of transgression of n-connections and on the underlying machinery of generalized smooth spaces and their differential graded-commutative algebras of differential forms.
The application Alexandre Borovik and I submitted to the John Templeton Foundation as part of their funding of the core theme of infinity was successful. We intend to discuss and disseminate ideas via a blog – A Dialogue on Infinity….
I’ve come across something promissing for the Progic project. Apparently there is a way to complete the analogy: propositional logic : predicate logic :: Bayesian networks: ? The answer, it is claimed, is ‘probabilistic relational models’. Now before we…
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?
On the notion of concordance of 2-bundles and, more generally, on a notion of omega-anafunctor and a possible closed structure on the category of omega-categories with omega-anafunctors between them.
I would like to announce that we in the Centre for Reasoning here in Canterbury are launching a new MA course for September 2008. As you can see, this offers the chance to select from four core modules: Logical reasoning,…
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!
More by Todd Trimble on the duality between symmetry and structure: that is, between groups of transformations of finite sets and complete axiomatic theories.
Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
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.
On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan’s old article on this issue in particular.
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.
Nils Baas on higher order structures, Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
My colleague here in Canterbury Jon Williamson is part of an international research group, progicnet, whose aim is to find a good integration of probability theory and first-order logic. For one reason or another, some technical projects get counted…
What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
A review of elements of the Batalin-Vilkovisky formalism, with an eye towards my claim that this describes configuration spaces which are Lie n-algebroids.
Back from Tuscany, I find two e-mails requests awaiting me. First, and I’m now very late on this story, Alexandre Borovik asked me to draw attention to the plight of a Mathematical Summer School held in Turkey. Second, Tim Porter…
Hendryk Pfeiffer asked me to forward the following question to the Café. Dear nn-category people, I have a question about tensor categories on which I would appreciate comments and references. As probably several people are interested in this, I…
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…
The concept of an “Adinkra” - a graph used to describe representations of N-extended d=1 supersymmetry algebras - remarkably resembles some categorical structures which appear in the context of supersymmetry.
Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
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.
We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
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.
How transformations of extended d-dimensional quantum field theories are related to (d-1)-dimensional quantum field theories. How this is known either as twisting or as, in fact, holography.
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.
A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
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!
For some grand theory building and an answer to the question ‘What is the field with one element?’, see Nikolai Durov’s New Approach to Arakelov Geometry.
Are we doing our job as broadcasters well? Max Tegmark has a new paper out on the physical universe as an abstract mathematical structure. Not a whiff of categories, let alone nn-categories. Tegmark has read some of philosophy of…
In Brussels, we heard from Koen Vervloesem about attempts towards better automated theorem provers. Readers of my book will know that I devoted its second chapter to automated theorem provers, to provide a relief against which to consider ‘real…
From an interview with Gian-Carlo Rota and David Sharp: Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that…
In Brussels, Brendan Larvor took us through a range of options for those of us who want our philosophy of mathematics to take serious notice of the history of mathematics. A distinction he relied upon was one Bernard Williams introduced…
A list of some papers involved in the historical development of the idea of expressing bundles with connection in terms of their parallel transport around loops.
On the ArXiv today, Yuri Manin has one of those wide-ranging overviews of the life of mathematics: Mathematical knowledge: internal, social and cultural aspects. One comment - When Poincaré said that there are no solved problems, there are only problems…
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.
I’m having a spot of bother getting a paper published. It’s about the philosopher Michael Friedman’s treatment of mathematics in his Dynamics of Reason. I’d be grateful for any comments from the Café clientele.
Having noticed (e.g., here and here) that what I do in my day job (statistical learning theory) has much to do with my hobby (things discussed here), I ought to be thinking about probability theory in category theoretic terms….
The next day I set off East to Jena, following the path taken by Carnap, and by my host, David Green, a British mathematician who works on the cohomology of finite groups. While in Wuppertal, David had become interested…
Last week I gave a couple of talks in Germany. Thursday saw me in the town of Wuppertal, famous for its Schwebebahn, a railway built above the river Wupper, which snakes its way through the middle of the town. As…
Café regular John Armstrong has a blog. It goes by the name of The Unapologetic Mathematician. A subtle allusion to Hardy’s A Mathematician’s Apology, playing cleverly on the two meanings of apology?…
It’s worth taking a look at an interview Mikio Sato gave to Emmanuel Andronikof in 1990, published in February’s Notices of the American Mathematical Society. Sato is famous for algebraic analysis, D-modules, and the like, about which I know next…
Twice in recent days I have confronted the possibility of experiencing a kind of alienation due to interviews. First, my co-author Darian Leader and I were interviewed by the New Scientist about our book Why Do People Get Ill?. A…
As neither John nor Urs has announced it, readers might like to find out about their motivations for starting and running this blog in an interview they gave to Bruce Bartlett, available in written form and also as an MP3…
Continuing our earlier discussion about duality, it’s worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between ‘formal’ and ‘concrete’ duality. Formal duality concerns mere arrow reversal in the relevant diagrams,…
One of the reasons I have an interest in what we find out about mechanics in different rigs is that many machine learning algorithms are expressible in thermodynamic form, as the tutorial, Energy-Based Models: Structured Learning Beyond Likelihoods, by…
I’m in one of those phases where everywhere I look I see the same thing. It’s Fourier duality and its cousins, a family which crops up here with amazing regularity. Back in August, John wrote: So, amazingly enough, Fourier duality…
To keep me from brooding on the pleasure I’m missing out on by not being with my Café co-hosts in Toronto, let me try out a blog post. In just about every academic endeavour to which I’ve applied myself, I…
There’s nothing quite like a research proposal to give you a sense of some of the big stories out there. Try Geometry and Quantum Theory for what’s happening in Holland of relevance to the Café. From a couple of years…
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.
In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences. If I point out to you three…
A question by Bruce Bartlett about categories of algebras, algebras as categories and the possible implications for non-commutative algebraic geometry.
The following “guest post” is by David Roberts: At the Australian Mathematical Society’s annual meeting in Sydney, Brian Wang gave the talk Gerbes, D-branes and loop group representations. This is of particular interest to at least one blogger here, so…
Marni Sheppeard reports from the AustMS2006 conference, which, as anyone who knows about Australian mathematics might expect, is holding a category theory session. Dominic Verity is giving one of the talks, in which he considers the raison d’être for higher…
Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are only a handful of participants’ notes available (Scott Morrison’s are…
Here are some notes for my talk at the Berlin workshop. Fortunately I was upgraded to a 45-minute talk. Even so, I didn’t manage to reach the last part where I discuss David Carr’s ideas. I would be interested in…
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!