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 n-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 n-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.