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July 18, 2007

Astronomical Paris

Posted by John Baez

Today is my last day in Paris. With any luck, I’ll meet David Corfield in Delphi tomorrow.

As a kind of followup to my post on mathematical Paris, here is a tour of the Paris Observatory.

Posted at 3:35 PM UTC | Permalink | Followups (1)

July 16, 2007

Mathematical Imperatives

Posted by David Corfield

I like the way Yuri Manin generally throws a little ‘philosophy’ into his papers. In his Generalized operads and their inner cohomomorphisms with D. Borisov, they write:

One can and must approach operadic constructions from various directions and with various stocks of analogies. (p. 4)

That ‘must’ is interesting to think about. You might look to deontic logic for help, and be relieved that Kant’s Law (‘must implies can’) is satisified. But perhaps the more interesting question is ‘Must, or what will happen?’

Perhaps, something like: you’ll fail to understand operadic constructions fully, which would be failing in your duty as a mathematician.

Posted at 11:48 AM UTC | Permalink | Followups (3)

July 14, 2007

George Mackey

Posted by David Corfield

The latest edition of the Notices of the American Mathematical Society is out, and it contains reminiscences about the life and work of George Mackey.

For a long time I’ve been attracted by big mathematical visions. While I was PhD student I’d hunt out the informal writings of people like Atiyah and MacLane. But I think my favourite author at the time was Mackey, in particular the story of maths he had told in ‘The scope and history of commutative and noncommutative harmonic analysis’.

As Caroline Series puts it

I do not know any other writer with quite his gift of sifting out the essentials and exposing the bare bones of a subject. There is no doubt that his unique ability to cut through the technicalities and draw diverse strands together into one grand story has been a hugely wide and enduring influence. (p. 21)

Posted at 3:00 PM UTC | Permalink | Followups (2)

July 13, 2007

This Week’s Finds in Mathematical Physics (Week 254)

Posted by John Baez

In week254, learn about Witten’s new paper on 3d quantum gravity and the Monster group, mysterious relations between exceptional Lie superalgebras and the Standard Model of particle physics…

… and continue reading the Tale of Groupoidification.

Posted at 12:45 PM UTC | Permalink | Followups (31)

Breaking Out of the Box

Posted by David Corfield

While the Café’s gone a little quiet of late – and with two of its owners tripping off to Delphi soon while the other’s still on holiday, things can only get quieter – there are some interesting things happening abroad. In fact, walking in this morning, I was thinking up something to say about a vague sense I had from reading about canopolises, but when I reached the office and tuned into the blogosphere, there’s Noah Snyder clearly articulating the thought I’d had that things could be taken further.

The larger question is why did we ever restrict ourselves to ends of boxes when we could be letting the string ends of our n-categories wander about on the surfaces of spheres?

Posted at 8:05 AM UTC | Permalink | Followups (1)

July 9, 2007

Return of the Euler Characteristic of a Category

Posted by David Corfield

Tom Leinster has a follow up to The Euler characteristic of a category, which sparked a lively conversation here last October. The new one goes by the title The Euler characteristic of a category as the sum of a divergent series.

Abstract:

The Euler characteristic of a cell complex is often thought of as the alternating sum of the number of cells of each dimension. When the complex is infinite, the sum diverges. Nevertheless, it can sometimes be evaluated; in particular, this is possible when the complex is the nerve of a finite category. This provides an alternative definition of the Euler characteristic of a category, which is in many cases equivalent to the original one

Posted at 9:30 AM UTC | Permalink | Followups (28)

July 7, 2007

Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity

Posted by John Baez

I’m mainly here in Paris to talk about categories, logic and games with Paul-André Melliès of the Preuves, Programmes et Systèmes group at Université Paris 7. But, I was invited by Marc Lachièze-Rey of the AstroParticule et Cosmologie group to give a talk on physics.

So, I took this as an excuse to speak about the work of my student Derek Wise.

Derek just finished his thesis in June. This fall he’s going to U. C. Davis. He’s been talking to me about Cartan geometry and MacDowell–Mansouri gravity for several years now — but I’ve been keeping most of it secret, so nobody would scoop his thesis. It’s a great pleasure to finally say more about it!

Abstract and transparencies of the talk follow…

Posted at 9:55 PM UTC | Permalink | Followups (15)

Mathematical Paris

Posted by John Baez

I’m in Paris from July 1st to 19th. Just for fun, I’ve started taking pictures of streets named after mathematicians. Can you help me find more of these streets?

Posted at 9:51 AM UTC | Permalink | Followups (24)

July 6, 2007

Kernels in Machine Learning III

Posted by David Corfield

The use of kernel methods in machine learning often goes by the name nonparametric statistics. Sometimes this gets taken as though it’s the opposite of finite parametric statistics, but that’s not quite right. A typical piece of parametric statistics has a model like the two-parameter family of normal distributions, and the parameters of this model are then estimated from a sample.

What happens in the case of kernel methods in machine learning is that a model from a possibly infinite-dimensional family is specified by a function on the (finite) collection of training points. In other words you are necessarily restricted to a subfamily of models of dimension the size of the training set. So, in contrast to the usual parametric case, you are dealing with a data-dependent finite dimensional subfamily of models.

Posted at 10:47 AM UTC | Permalink | Post a Comment

July 5, 2007

Multiplicative Structure of Transgressed n-Bundles

Posted by urs

Remember the drama of the charged n-particle?

An n-particle of shape par propagating on target space tar and charged undern an n-bundle with connection given by the transport functor tra:tarnVect admits two natural operations: we may either quantize it. That yields the extended n-dimensional QFT of the n-particle, computing the n-space of its quantum states q(tra):parnVect.

But we may also, instead, transgress the n-bundle background field on target space to something on the particle’s configuration space.

For instance, a closed string (a 2-particle) charged under a Kalb-Ramond gerbe (a 2-bundle) gives rise to a line bundle (a 1-bundle) on loop space. I once described this in the functorial language used here in this comment.

But, and that’s the point of this entry here, these transgressed n-bundles have certain special properties: they are multiplicative with respect to the obvious composition of elements of the configuration space of the n-particle.

I have neither time nor energy at the moment to give a comprehensive description of that. What I do want to share is this:

With Bruce Bartlett I was talking, by private email, about the right abstract arrow-theoretic formulation to conceive multiplicative n-bundles with connection obtained from transgression on configuration spaces. It turns out that a n-transport functor is multiplicative if it is monoidal with respect to a certain natural variation of the concept of monoidal structure which is applicable for fibered categories.

In the file

The monoidal structure of the loop category

I spell out some key ingredients of how to conceive the situation here for the simple special case that we start with a 1-functor and transgress it to a “loop space”.

There is nothing particularly deep in there, but it did took us a little bit of thinking to extract the right structure here, simple as it may be. So I thought we might just as well share this with the rest of the world.

And, by the way, I will be on vacation in southern Spain until July 20.

Posted at 6:47 PM UTC | Permalink | Post a Comment

July 4, 2007

Supercategories

Posted by urs

Motivated by general questions in supersymmetric QFT, I would like to better understand some of the “arrow-theory” behind supersymmetry, finding a formulation which gives a systematic way to internalize the concept into various contexts. For instance, people have a pretty good purely arrow-theoretic understanding of finite-group QFT, such as Dijkgraaf-Witten theory. Can we understand how to superize this systematically, in a context where many of the standard tools one finds in the literature are simply not applicable?

In order to both motivate and further introduce the problem, I might, in a followup entry, start looking into the following

Exercise (The Willertonesque super 2-particle). Simon Willerton has demonstrated (see The Baby Version of Freed-Hopkins-Teleman) that the “quantum theory of the 2-particle propagating on a finite group” has a beautiful arrow-theoretic formulation:

Take the parameter space of the 2-particle to be the fundamental groupoid of the circle par:=Σ. Take its target space to be a finite group tar:=ΣG. Then configuration space is the groupoid conf:=Funct(Σ,ΣG):=ΛG, which plays the role of the loop group of the finite group G. The fact that this 2-particle is charged gives rise to a 2-vector bundle on this configuration space, and quantum states of the 2-particle are sections of this. In the simplest case (see the above entry for the more general case), this simply means that a state here is a representation ψ:ΛGVect of the configuration space groupoid on vector spaces.

There is more structure here, but for the moment concentrate on this basic data. The point of this is that everything is purely combinatorial, well defined, and exhibits just the bare structure of the QFT here, stripped of all distracting technicalities.

The exercise is then: do the analogous discussion for the super 2-particle. Figure out what the super-parameter space of the super 2-particle in the above sense is, what its super-configuration space supergroupoid is and what its super-representations on supervector space are like.

Clearly the first step to make any progress at all here is to get a reasonable good understanding what supersymmetry really is, such as to apply it to this situation. So, this entry here is just about this question: What is the arrow theory of supersymmetry?

As an attempt to approach this exercise, I’ll introduce the concept of a supercategory, which is supposed to be to that of a supergroup like categories are to groups. I feel that this concept helps extracting some of essence of what is going on.

It turns out that this is closely related to another structure which has appeared in 2-dimensional QFT, that of G-equivariant categories.

It’s an exercise. I can’t be sure that I am on the right track. But I would like to share the following, as I proceed.

Posted at 9:37 PM UTC | Permalink | Followups (29)

The Inner Automorphism 3-Group of a Strict 2-Group

Posted by urs

David Roberts and I would like to share the following text:

D. Roberts, U.S.
The inner automorphism 3-group of a strict 2-group
(pdf)


Abstract

For any group G, there is a 2-group of inner automorphisms, INN(G). This plays the role of the universal G-bundle. Similarly, for every 2-group G (2 ) there is a 3-group INN(G (2 )) of inner automorphisms. We construct this for G (2 ) any strict 2-group, discuss how it can be understood as arising from the mapping cone of the identity on G (2 ) and show that it fits into a short exact sequence Disc(G (2 ))INN(G (2 ))ΣG (2 ) of strict 2-groupoids. We close by indicating how this makes INN(G (2 )) the universal G (2 )-2-bundle.

Posted at 11:56 AM UTC | Permalink | Followups (1)

June 28, 2007

Kernels in Machine Learning II

Posted by David Corfield

We’ve had a slight whiff of the idea that groupoids and groupoidication might have something to do with the kernel methods we discussed last time. It would surely help if we understood better what a kernel is. First off, why is it called a ‘kernel’?

This must relate to the kernels appearing in integral transforms,

(Tg)(u)= TK(t,u)g(t)dt,

where g, a function on the T domain, is mapped to Tg, a function on the U domain. Examples include those used in the Fourier and Laplace transforms. Do these kernels get viewed as kinds of matrix?

In our case these two domains are the same and K is symmetric. Our trained classifier has the form:

f(x)= i=1 mc iK(x i,x)

So the g(t) is i=1 mc iδ(x i,x). The kernel has allowed the transform of a weighted set of points in X, or, more precisely, a weighted sum of delta functions at those points, to a function on X.

Posted at 9:12 PM UTC | Permalink | Followups (21)

Conference on Categories in Geometry and Physics

Posted by urs

In September, the following conference will take place:

Categories in Geometry and Physics
September 24-28, 2007
at MedILS, Split, Croatia,

organized by Zoran Škoda and Igor Baković.

Requests for possible attendance should be promptly emailed to zskoda at irb.hr.

A web page for the conference will be out next week.

Posted at 2:04 PM UTC | Permalink | Followups (1)

This Week’s Finds in Mathematical Physics (Week 253)

Posted by John Baez

In week253, read about mysterious relations between the Standard Model, the SU(5) and SO(10) grand unified theories, the exceptional group E6, the complexified octonionic projective plane… and maybe even E8!

Posted at 7:55 AM UTC | Permalink | Followups (23)