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October 31, 2011

WZW Models in a Cohesive ∞-Topos

Posted by Urs Schreiber

A few hours back has started the

Workshop on Representation Theoretical and Categorical Structures In Quantum Geometry and Conformal Field Theory .

I have prepared some slides for my talk tomorrow, titled

Maybe you’d enjoy looking at it. I’d be interested in whatever comment you might have.

Where the previous talk focused on the fact that there is physics at all induced in a cohesive (,1)(\infty,1)-topos, this one looks more specifically at aspects of conformal field theory, namely at higher WZW theory, canonically existing in a cohesive \infty-topos.

Since my audience eats WZW models for breakfast, the slides do not explain what these are. They only give a little teaser for why they are of interest. Maybe I find the time to expand the nnLab entry WZW model. But at least a bunch of references is currently listed there already.

Posted at 5:12 PM UTC | Permalink | Followups (1)

Coalgebra Paper

Posted by David Corfield

That paper I was writing on coalgebra has finally appeared. Thanks again for the help I received back at these posts (I and II).

It’s good to see that defining entities coinductively continues to be a topic of interest, e.g., here, here and here.

Perhaps once the proof assistant Coq’s coinductive types are made more powerful, we’ll see results worth mentioning in a sequel.

Posted at 10:27 AM UTC | Permalink | Followups (14)

October 26, 2011

Prequantum Physics in a Cohesive ∞-Topos

Posted by Urs Schreiber

In a few hours starts the conference

Quantum Physics And Logic 2011 .

Since beamer-talks are being required, I have prepared some slides, titled

Maybe you enjoy having a look. Comments are most welcome.

Also Joost Nuiten, whose bachelor thesis I had the pleasure to advise this year, is giving a talk, on

Posted at 8:05 PM UTC | Permalink | Followups (109)

October 23, 2011

Measuring Diversity

Posted by Tom Leinster

Christina Cobbold and I wrote a paper on measuring biological diversity:

Tom Leinster and Christina A. Cobbold,
Measuring diversity: the importance of species similarity.
Ecology, in press (doi:10.1890/10-2402.1).

As the name of the journal suggests, our paper was written for ecologists — but mathematicians should find it pretty accessible too.

While I’m at it, I’ll mention that I’m coordinating a five-week research programme on The Mathematics of Biodiversity at the Centre de Recerca Matemàtica, Barcelona, next summer. It includes a one-week exploratory conference (2–6 July 2012), to which everyone interested is warmly welcome.

In a moment, I’ll start talking about organisms and species. But don’t be fooled: mathematically, none of this is intrinsically about biology. That’s why this post is called “Measuring diversity”, not “Measuring biological diversity”. You could apply it in many other ways, or not apply it at all, as you’ll see.

It’s an example of what Jordan Ellenberg has amusingly called applied pure math. I think that’s a joke in slightly poor taste, because I don’t want to surrender the term “applied math” to those who basically use it to mean “applied differential equations”. Nevertheless, I suspect we’re on the same side.

Posted at 11:10 PM UTC | Permalink | Followups (50)

A Math Puzzle Coming From Chemistry

Posted by John Baez

I posed this puzzle a while back over on Azimuth, and nobody solved it. Maybe it was too mathematical for most people there—it seems to be a problem in Klein geometry, actually. But also it’s a bit tricky: now that I think about it more, I’m not sure how to solve it either!

So, maybe you can help.

Posted at 11:37 AM UTC | Permalink | Followups (14)

October 19, 2011

Do You Know This Idempotent?

Posted by Tom Leinster

Here’s something extremely elementary. Given a map ff from a finite set to itself, the set of iterates

{f,f 2,f 3,} \{ f, f^2, f^3, \ldots \}

contains precisely one idempotent. What does this idempotent do?

It’s easy to say what its image is: it’s the set of periodic points. But I’m having a harder time understanding the idempotent itself — that is, how it acts on the set concerned.

Can you give an abstract account of where this idempotent comes from? Do you know an alternative way of computing it to the one below? Do you have any other insight?

Posted at 11:31 PM UTC | Permalink | Followups (36)

October 16, 2011

Spectra of Operators and Rings

Posted by Tom Leinster

There are many uses of the word ‘spectrum’ in mathematics, and most of them are related. (The main exception that I’m aware of is spectra in the sense of homotopy theory.) In particular, the spectrum of a linear operator on a finite-dimensional vector space — that is, its set of eigenvalues — can be seen as the spectrum of an associated commutative ring.

I’ll explain how that story goes. But let me state up front the question that I want to ask. Although it was motivated by my desire to link up the various notions of spectra, it doesn’t actually mention spectra at all.

Let VV be a finite-dimensional vector space over a field kk. Let TT be an operator on VV, in other words, a linear map VVV \to V. Write χ(x)k[x]\chi(x) \in k[x] for its characteristic polynomial. What is the significance of the ring

k[x]/(χ(x))? k[x]/(\chi(x))?

Let me explain the kind of answer I’m hoping for. Suppose we replace the characteristic polynomial χ\chi by the minimal polynomial mm. Then the ring

k[x]/(m(x)) k[x]/(m(x))

can be seen as the ring of polynomials in TT. Precisely: write End(V)End(V) for the kk-algebra of operators (endomorphisms) on VV, with composition as its product. The homomorphism k[x]End(V)k[x] \to End(V) sending xx to TT has (m(x))(m(x)) as its kernel, and the image is the subalgebra of End(V)End(V) generated by TT. Hence k[x]/(m(x))k[x]/(m(x)) is isomorphic to this subalgebra, which consists of the polynomials in TT.

But is there a nice way to think about k[x]/(χ(x))k[x]/(\chi(x))?

Posted at 1:58 PM UTC | Permalink | Followups (23)

October 11, 2011

Weak Systems of Arithmetic

Posted by John Baez

The recent discussion about the consistency of arithmetic made me want to brush up on my logic. I’d like to learn a bit about axioms for arithmetic that are weaker than Peano arithmetic. The most famous is Robinson arithmetic: Robinson arithmetic is also known as Q, after a Star Trek character who could instantly judge whether any statement was provable in this system, or not:

Instead of Peano arithmetic’s axiom schema for mathematical induction, Q only has inductive definitions of addition and multiplication, together with an axiom saying that every number other than zero is a successor. It’s so weak that it has computable nonstandard models! But, as the above article notes:

Q fascinates because it is a finitely axiomatized first-order theory that is considerably weaker than Peano arithmetic (PA), and whose axioms contain only one existential quantifier, yet like PA is incomplete and incompletable in the sense of Gödel’s Incompleteness Theorems, and essentially undecidable.

But there are many interesting systems of arithmetic between PA and Q in strength. I’m hoping that if I tell you a bit about these, experts will step in and tell us more interesting things—hopefully things we can understand!

Posted at 2:37 AM UTC | Permalink | Followups (63)

October 10, 2011

The 5th Scottish Category Theory Seminar

Posted by Tom Leinster

As surely as the mating season of the haggis comes round twice a year, it’ll soon be time for another Scottish Category Theory Seminar:

5th Scottish Category Theory Seminar

Informatics Forum, University of Edinburgh

Friday 25 November 2011, 2.00–5.40pm

The fifth meeting of the Scottish Category Theory Seminar will feature talks by the following speakers.

The meeting is open to all. There is no registration, but, if you intend to come, it would be helpful (but is not essential) to send an email to scotcats#cis.strath.ac.uk (changing # to @).

This meeting is generously supported by the Glasgow Mathematical Journal Trust. Its organizer is Alex Simpson. (We devolved organization of individual meetings to the respective locals, which should make it OK that I’m both an organizer of the series and a speaker at this meeting—right?)

Posted at 11:14 PM UTC | Permalink | Followups (1)

October 5, 2011

Theoretical Physics – Stack Exchange

Posted by Urs Schreiber

Since yesterday the site

Theoretical Physics – Stack Exchange

is in what is called public beta phase after building up speed in a private beta phase for about a month. This is about having questions and answers at research level in theoretical physics.

So far it has been working out all right. Which is not trivial, given the otherwise very low ratio of decent research-level theoretical physics discussion on the web compared to the impressive amount of decent research-level mathematics discussion on the web.

Posted at 11:57 PM UTC | Permalink | Followups (1)

October 3, 2011

Zalamea on Sheaf Logic

Posted by David Corfield

Zachary Luke Fraser writes to me of a seminar which was given by Fernando Zalamea last week at the Jan van Eyck Academie in Maastricht on 29 September. If the webpage for the seminar – Sheaf Logic & Philosophical Synthesis – isn’t working, you can access an audio file of it.

From the webpage, you’ll also be able to find a link to a draft of Zachary’s English translation of a few chapters from Zalamea’s Synthetic Philosophy of Contemporary Mathematics. Now we can read his favourable opinions on my own book in English.

Zalamea’s work has cropped up before at the Café, and earlier.

Posted at 2:43 PM UTC | Permalink | Followups (2)