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March 31, 2009

Categories, Quanta and Concepts at the Perimeter Institute

Posted by John Baez

Many customers here at the nn-Café believe — do I dare say know? — that category theory provides crucial clues about the foundations of quantum theory. This summer there will be a workshop at the Perimeter Institute devoted to pursuing these clues:

Posted at 4:05 PM UTC | Permalink | Followups (20)

March 27, 2009

First-Order Logical Duality

Posted by David Corfield

This is the title of the PhD thesis in which Henrik Forssell presents

…an extension of Stone Duality for Boolean Algebras from classical propositional logic to classical first-order logic. The leading idea is, in broad strokes, to take the traditional logical distinction between syntax and semantics and analyze it in terms of the classical mathematical distinction between algebra and geometry, with syntax corresponding to algebra and semantics to geometry.

Posted at 10:15 AM UTC | Permalink | Followups (10)

March 25, 2009

The Algebra of Grand Unified Theories III

Posted by John Baez

If you’re a phenomenologist, here’s a question: is the nonsupersymmetric SO(10)SO(10) GUT still experimentally viable? I get the impression that it is.

If you’re a follower of Husserl or Heidegger, here’s a word of reassurance: I don’t mean that kind of ‘phenomenologist’. I mean an particle physicist who is not an experimentalist, but is nonetheless deeply concerned with experimental data.

And if you’re a physicist of a more theoretical or mathematical sort — the sort perhaps more likely to frequent the nn-Café — please ask your phenomenologist pals to stop by and pass on their wisdom!

Posted at 7:54 PM UTC | Permalink | Followups (81)

March 23, 2009

The Borromean Link Configuration

Posted by David Corfield

I’ve been asked by Alan Calvatti, Kuolab for Imaging Informatics, UCSD Radiology, to post here a question readers may be able to answer.

I’m looking for a categorical formalization of a concept that abstractly may be captured by the following expression [Borromean Link Configuration or BLC]

[BLC]:aPbbPcaPc¬Rabc, [BLC]: a P b \wedge b P c \wedge a P c \wedge \not R a b c,

where RR is a ternary relation and PP is a binary relation that intuitively represents the “boundary of RR”, for some objects aa, bb, cc.

Posted at 1:03 PM UTC | Permalink | Followups (49)

March 22, 2009

Unitary Representations of the Poincaré Group

Posted by John Baez

John Huerta’s paper on grand unified theories made some people eager to discuss aspects of group representation theory that this paper deliberately avoided.

For example, in relativistic quantum mechanics, we classify particles as unitary irreducible representations (or ‘irreps’) of a group like this:

G×PG \times P

Here GG is a compact Lie group depending on the theory of physics we happen to be studying, called the ‘internal symmetry group’. For example, the Standard Model has G=G = U(1) ×\times SU(2) ×\times SU(3).

PP, on the other hand, is the Poincaré group! This is, roughly speaking, the group of symmetries of Minkowski spacetime. So, this is the same regardless of our theory, unless we posit extra dimensions or something funky like that.

A unitary irrep of G×PG \times P is always built by tensoring a unitary irrep of GG with one of PP. So, the project of classifying particles splits into two parts: one depending on GG, one depending on PP.

In this thread let’s talk about the second part! If we do, we’ll learn why physicists classify particles according to their mass and spin (or more precisely, helicity). So, when we hear them mutter something about a ‘massless left-handed spin-1/2 particle’, we’ll know which representation of the Poincaré group they’re talking about.

Posted at 11:08 PM UTC | Permalink | Followups (146)

Open Access at MIT

Posted by John Baez

This Thursday, MIT followed the initiative of that lesser-known school up the river and decided to insist that all their research be made freely accessible online!

Posted at 7:20 AM UTC | Permalink | Followups (13)

March 20, 2009

Twisted Differential String- and Fivebrane-Structures

Posted by Urs Schreiber

Last summer at the Hausdorff institute in Bonn # we had started working on the following; now it is finally converging to something. Maybe somebody is interested in having a look.

Hisham Sati, U. S., Jim Stasheff,
Twisted differential String- and Fivebrane-Structures
(pdf, nnLab)

Posted at 9:42 PM UTC | Permalink | Followups (76)

March 16, 2009

The Algebra of Grand Unified Theories II

Posted by John Baez

I started out using this blog entry as a plea for people to help improve the following paper… but now it’s a lot better, so we put it on the arXiv:

This fills in the details for John Huerta’s talk.

Posted at 1:20 AM UTC | Permalink | Followups (107)

March 13, 2009

Geometric Help Wanted

Posted by David Corfield

I’m giving a talk to some philosophers and would like to get across in a non-technical way the idea that geometry has gone, and continues to go, through profound changes. I wanted to touch on the Erlangen Program, and perhaps illustrate its idea with the following example:

Elementary plane geometry and the projective investigation of a quadric surface with reference to one of its points are one and the same.

Posted at 9:39 AM UTC | Permalink | Followups (24)

The Okubo Algebra

Posted by John Baez

While studying the algebra of Grand Unified Theories and the role of division algebras in supersymmetric Yang–Mills theory, I bumped into a curious entity called the ‘Okubo algebra’.

If you know anything about it, tell me!

Posted at 1:56 AM UTC | Permalink | Followups (30)

March 12, 2009

This Week’s Finds in Mathematical Physics (Week 274)

Posted by John Baez

In week274 of This Week’s Finds, see a gorgeous view of a Martian crater. Learn about infinite-dimensional 2-vector spaces, and representations of ‘2-groups’ on these. Get a hint of how representations of the Poincaré 2-group arise from pictures like this:

And learn about Mackey’s classic work on unitary group representations!

Posted at 8:51 PM UTC | Permalink | Followups (13)

March 10, 2009


Posted by David Corfield

When I first read in John’s writings about 2-groups I naïvely imagined that people would rapidly find analogues for everything done with ordinary groups. Hence my call for a Klein 2-geometry in 2006.

Later in that year, I’m to be found on the recently formed Café musing:

I presume people are wondering about which equivalents of features of group representation theory might be found? Are there ‘locally compact 2-groups’, and if so, are there Haar measure and Peter-Weyl theorem equivalents? For finite 2-groups, is there an equivalent of orthogonality in the character table? In general, is there an equivalent of the adjunction between the restricting and inducing functors? What about the branching rules? There must be dozens more questions like these.

Posted at 12:12 PM UTC | Permalink | Followups (17)

March 8, 2009

Banning Open Access II

Posted by John Baez

Remember John Conyer’s Fair Copyright in Research Works Act, which would ban the National Institute of Health from making taxpayer-funded research freely accessible, and also ban other federal agencies from adopting open-access policies?

Now Conyers has made a hilarious argument in favor of this bill, over at the Huffington Post.

Posted at 3:08 AM UTC | Permalink | Followups (17)

March 5, 2009

Super-Yang-Mills Theory

Posted by John Baez

I’m trying to learn the basics of supersymmetric Yang–Mills theory, and I’m stuck on what should be an elementary calculation. I was never all that good at index-juggling, and years of category theory have destroyed whatever limited abilities I had.

So, this post is basically a plea for help from experts. But I like to pay off my karmic debts in advance. So, I’ll start by explaining some basic stuff to the nonexperts out there.

(Warning: by ‘nonexpert’ I mean ‘a poor schlep like me, who knows some quantum field theory but always avoided supersymmetry’.)

Posted at 8:33 PM UTC | Permalink | Followups (18)

Question on Homotopical Structure on SimpSet

Posted by Urs Schreiber

I’d like to better understand how the category SSet with its classical model structure sits inside SSetSSet equuipped with Joyal’s model structure.

More concretely, I’d like to get a better formal idea of the following situation:

in the standard model structure on SSetSSet, for every fibrant object XKanSSetX \in Kan \subset SSet (=\infty-groupoid) the object [Δ 1,X][\Delta^1, X] is a path object for XX. So KanKan equipped with [Δ 1,][\Delta^1, -] is a category of fibrant objects in this sense with a functorial assignment of path objects.

In the Joyal model structure, for every fibrant object XWeakKanSSetX \in WeakKan \subset SSet (= quasi-category) the object [Δ 1,X][\Delta^1, X] clearly plays the role of the right directed path space object for XX, it still factors the diagonal as X[Δ 1,X]X×X, X \to [\Delta^1, X] \to X \times X \,, but it is no longer a path object in the standard model-theoretic sense, as X[Δ 1,X]X \to [\Delta^1,X] need not be a weak equivalence anymore: Δ 1\Delta^1 is a directed interval object.

I am thinking that there should be a good and nice relaxation of the axioms of category of fibrant objects such that WeakKanWeakKan becomes an example and such that the inclusion KanWeakKanKan \hookrightarrow WeakKan becomes an inclusion of categories of fibrant objects in the relaxed sense.

There are some obvious guesses for how to try to relax the path space object axiom. But I am not entirely sure yet what the good way to do it really is. Has anyone thought about this?

Posted at 8:13 PM UTC | Permalink | Followups (14)

March 2, 2009

The Stabilizer of a Subcategory

Posted by David Corfield

A very long time ago, John gave us a definition of a stabilizer of an object in a category on which some 2-group is acting, having had a moment of insight near the Nine Zigzag Bridge in Shanghai.

Posted at 11:26 AM UTC | Permalink | Followups (40)

Two Streams in the Philosophy of Mathematics

Posted by David Corfield

A conference with this title will be held on 1-3 July 2009 at the University of Hertfordshire, Hatfield, UK. A description of the rationale for the conference is here. Mathematicians are more than welcome to participate. The deadline for abstracts is 30 April.

Update: I forgot to include a link to the conference homepage.

Posted at 10:09 AM UTC | Permalink | Followups (18)