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December 31, 2008

The Toric Variety Associated to the Weyl Chambers

Posted by John Baez

Happy New Year! I’m making a resolution to avoid starting work on new papers. I want to spend more time learning math, playing music, and having fun.

For a long time, whenever people said the phrase toric variety, I’d cover my ears and refuse to listen to what they had to say. Since I knew nothing of algebraic geometry, I thought ‘toric varieties’ were just one more of those specialized concepts — like Fano varieties and del Pezzo surfaces — that were devised solely for the purpose of demonstrating one’s superior erudition in this esoteric subject.

That’s changed. Now I love toric varieties, and I have a question about them.

But I won’t explain what they are, because then everyone would know.

Posted at 10:12 PM UTC | Permalink | Followups (35)

Organizing the Pages at nLab

Posted by David Corfield

An e-mail discussion migrated to the General Discussion page at nLab , where Urs sensibly suggested it should migrate further here.

Posted at 8:42 AM UTC | Permalink | Followups (17)

Joint Math Meetings in Washington DC

Posted by John Baez

In just a few days, hordes of mathematicians will descend on Washington DC for the big annual joint meeting of the American Mathematical Society (AMS), Mathematical Association of America (MAA), Society for Industrial and Applied Mathematics (SIAM), and sundry other societies, organizations, clubs, conspiracies and cabals:

I’ll be there. Will you?

Posted at 2:38 AM UTC | Permalink | Followups (9)

December 30, 2008

Groupoidification from sigma-Models?

Posted by Urs Schreiber

The interest in groupoidification (see our recent discussion) is to a large extent motivated from the feeling that it illuminates general structural aspect of quantum field theory.

My motivation is this:

to every differential nonabelian cocycle describing an associated \infty-vector bundle with connection, there should canonically be associated the corresponding σ\sigma-model QFT, which, physically speaking, describes the worldvolume theory of a brane couopled to this \infty-bundle.

I have been thinking about this for quite a while now, starting with a series of posts on QFT of the charged nn-particle. It took me a bit to get the required machinery into place, such as the interpretation of parallel transport \infty-functors for \infty-bundles with connection in homotopical cohomology theory, or the machinery of universal \infty-bundles.

Then John started teaching us about groupoidification and I noticed that this should naturally arise when forming the σ\sigma-model of a nonabelian cocycle. I chatted about the basic idea of this insight in An exercise in groupoidification: the path integral.

Now I found the time to expand on this in much more detail. Not that this is done yet, but a coherent closed picture seems to be emerging, which I describe in these notes:

nnLab/schreiber: Nonabelian cocycles and their σ\sigma-Model QFTs

Posted at 3:40 PM UTC | Permalink | Followups (3)

December 26, 2008

Groupoidification Made Easy

Posted by John Baez

Merry Christmas! It’s still Christmas here in California, despite what the time stamp on this blog may say. So, it’s not too late for one last present! Here’s one just for you, from Santa and his elves:

Posted at 4:01 AM UTC | Permalink | Followups (98)

December 24, 2008

Tilings

Posted by John Baez

Merry Christmas! Here are some more presents. I hope these are a bit easier to appreciate than my new paper on infinite-dimensional representations of 2-groups. Again, they’re all about symmetry. But this time you don’t need a math degree to enoy them. You just need to look at ‘em!

Posted at 9:48 PM UTC | Permalink | Post a Comment

Infinite-Dimensional Representations of 2-Groups

Posted by John Baez

Yay! This paper is almost ready for the arXiv! We’ve been working on it for years… it turned out to involve a lot more measure theory than we first imagined it would:

Posted at 7:25 PM UTC | Permalink | Followups (13)

Science and the Environment

Posted by John Baez

As the year’s end approaches and I hole up at home, free of the pressures of teaching, my thoughts roam a bit more freely. So, they naturally turn to questions like this: am I doing the right things?

For example: should I do more to help save the planet? And if so, how can I do it in a way that takes advantage of my special skills? An education in math and physics leads people to value simple, elegant problems. The ecological crisis we face is anything but: it’s an incredible mess. Is there anything a mathematical physicist can do to help that a biologist or politician can’t do better? I try to proselytize on my webpage, but is that enough?

Questions like this don’t fit comfortably into this blog. So, I apologize to readers who prefer the usual fare. But these questions seem too important to completely ignore — and they’re especially timely for this reason:

In just a few weeks, science policy in the US may be run by scientists.

Posted at 2:26 AM UTC | Permalink | Followups (92)

December 23, 2008

Bridge Building

Posted by David Corfield

If anyone wanted to bridge the gap between the two cultures, Terry Tao’s post – Cohomology for dynamical systems might provide a good place to start. Remember our last collective effort at bridge-building saw us rather unsuccessfully try to categorify the Cauchy-Schwarz inequality.

Regarding this current prospective crossing point, we hear that the first cohomology group of a certain dynamical system is useful for the ‘ergodic inverse Gowers conjecture’, and that there are hints that higher cohomology elements may be relevant. The post finishes with mention of non-abelian cohomology.

It wouldn’t be surprising if algebraic topology provided the common ground. A while ago we heard Urs describe Koslov’s work on combinatorial algebraic topology.

Posted at 1:08 PM UTC | Permalink | Followups (22)

December 22, 2008

Higher Structures in Göttingen - Part II

Posted by John Baez

In November last year there was a little workshop in Göttingen on Higher Structures in Differential Geometry. Urs blogged about it here. In February there will be a kind of continuation:

Posted at 8:52 PM UTC | Permalink | Followups (25)

December 19, 2008

The Microcosm Principle

Posted by David Corfield

Furthering my study of coalgebra, I came across slides for a couple of talks (here and here) which put John and Jim’s microcosm principle into a coalgebraic context.

Recall their claim in Higher-Dimensional Algebra III that “certain algebraic structures can be defined in any category equipped with a categorified version of the same structure” (p. 11), as with monoid objects in a monoidal category.

We name this principle the microcosm principle, after the theory, common in pre-modern correlative cosmologies, that every feature of the microcosm (e.g. the human soul) corresponds to some feature of the macrocosm.

Later, in section 4.3, they give a formal treatment of the principle using operads.

On the other hand, in the slides and associated paper, Hasuo, Jacobs and Sokolova give the principle a 2-categorical formulation as a lax natural transformation X:1X: \mathbf{1} \implies \mathbb{C} between functors from a Lawvere theory 𝕃\mathbb{L} to CAT\mathbf{CAT}. I wonder how these treatments compare.

The microcosm principle would make for a good methodological entry for nLab, as would evil. I’ll see about extracting John’s sermon on the latter from old blog files. I must confess to feeling rather guilty for having done so little there, after pushing for it. Perhaps Santa will bring me a generous amount of free time for Christmas.

Posted at 9:46 AM UTC | Permalink | Followups (12)

December 16, 2008

Super Version of 2-Plectic Geometry for Classical Superstrings?

Posted by John Baez

In our paper Categorified symplectic geometry and the classical string, Alex Hoffnung, Chris Rogers and I described a Lie 2-algebra of observables for the classical bosonic string. The idea was to generalize the usual Poisson brackets coming from symplectic geometry, which make the observables for a classical point particle into a Lie algebra. The key was to replace symplectic geometry by the next thing up the dimensional ladder: 2-plectic geometry.

Now I have a slight hankering to do the same thing for the classical superstring. Ideally this would be a formal exercise in ‘super-thinking’ — replacing everything in sight by its ‘super’ (meaning /2\mathbb{Z}/2-graded) analogue. But maybe it’s not. Either way, I have a lot of catching up to do. So, here are some basic questions.

Posted at 7:14 PM UTC | Permalink | Followups (5)

What to Make of Mathematical Difficulties

Posted by David Corfield

T. R. drew to our attention a conference dedicated to Grothendieck. One of the papers there is Mathematics and Creativity (presumably written by Leila Schneps), which contains the passage:

Pierre Cartier observed that when Grothendieck took interest in some mathematical domain that he had not considered up till then, finding a whole collection of theorems, results and concepts already developed by others, he would continue building on this work ‘by turning it upside down’. Michel Demazure described his approach as ‘turning the problem into its own solution’. In fact, Grothendieck’s spontaneous reaction to whatever appeared to be causing a difficulty - nilpotent elements when taking spectra or rings, curve automorphisms for construction of moduli spaces - was to adopt and embrace the very phenomenon that was problematic, weaving it in as an integral feature of the structure he was studying, and thus transforming it from a difficulty into a clarifying feature of the situation. (p. 8)

Posted at 2:33 PM UTC | Permalink | Followups (12)

December 15, 2008

This Week’s Finds in Mathematical Physics (Week 273)

Posted by John Baez

In week273 of This Week’s Finds, read more about the geysers on Enceladus. Hear the story of the Earth, with an emphasis on mineral evolution — from chondrites to the Big Splat, the Late Heavy Bombardment, the Great Oxidation Event, Snowball Earth… to now.

Then, learn about Pontryagin duality!

Posted at 2:34 AM UTC | Permalink | Followups (52)

December 9, 2008

The Status of Coalgebra

Posted by David Corfield

After my post on coalgebra, I’m still unsure which position to take regarding its status with regard to algebra. Here are some options:

  • (1) It’s not a distinction worth making – a coalgebra for (C,F)(C, F) is an algebra for (C op,F op)(C^{op}, F^{op}).
  • (2) It is a distinction worth making, but there’s plenty of coalgebraic thinking going on – it’s just not flagged as such.
  • (3) Coalgebra is a small industry providing a few tools for specific situations, largely in computer science, but with occasional uses in topology, etc.
Posted at 2:30 PM UTC | Permalink | Followups (56)

Science Citation Index

Posted by John Baez

When I heard some of the top journals in category theory aren’t listed by the Science Citation Index, I posted a question on the category theory mailing list.

Posted at 3:14 AM UTC | Permalink | Followups (12)

A Quick Algebra Quiz

Posted by John Baez

Here’s a quick algebra quiz. It’s really a test of your reflexes when it comes to algebra and category theory. It should take less than a minute if you have the right mental training. If you don’t, you may be doomed.

So, take a deep breath and give it a try.

Posted at 2:20 AM UTC | Permalink | Followups (16)

December 7, 2008

Smooth Structures in Ottawa

Posted by John Baez

Here at the nn-Café we’re trying to get to the bottom of some big questions — for example, the nature of smoothness. A manifold is a kind of smooth space — but more general smooth spaces have been studied by Chen, Lawvere, Kock, Souriau and others, and these are starting to find their way into mathematical physics.

That’s just the beginning, though! Smoothness has a lot to do with derivatives. The concept of derivative can be generalized in some surprising ways. For example, it’s important in Joyal’s work on combinatorics — he explained how we can take the derivative of a structure like ‘being a 2-colored finite set’ More recently, Goodwillie introduced a concept of ‘approximation by Taylor series’ for interesting functors in homotopy theory. Even more recently, Ehrhard and Regnier introduced derivatives in logic — or more precisely, the lambda calculus.

So, it’s a great idea to have a conference on all these concepts of smoothness:

Posted at 12:40 AM UTC | Permalink | Followups (15)

December 4, 2008

Question on Infinity-Yoneda

Posted by Urs Schreiber

What is known, maybe partially, about generalizations of the Yoneda lemma to any one of the existing \infty-categorical models?

Posted at 11:57 AM UTC | Permalink | Followups (9)

December 3, 2008

Zhu on Lie’s Second Theorem for Lie Groupoids

Posted by Urs Schreiber

The last couple of days Chenchang Zhu had been visiting Hamburg. Yesterday she gave a nice colloquium talk on her work:

Chenchang Zhu
Lie II theorem
(pdf slides, 57 slides with overlay)

Posted at 4:48 PM UTC | Permalink | Followups (11)