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Quaternions and Tensor Categories July 30, 2009

Posted by David Speyer in Algebraic Geometry, Category Theory, things I don't understand.
28 comments

As you can tell from the title of this post, I am trying to drag John Baez over to our blog.

Let Q be the ring of quaternions, i.e., \mathbb{R} \langle i,j,k \rangle with the standard relations. Let Q-mod be the category of left Q-modules. This has an obvious tensor structure (including duals), inherited from the category of \mathbb{R} vector spaces. Actually, that structure doesn’t quite work; I’ll leave to you good folks to work out what I should have said.

Let q=a+bi+cj+dk be a quaternion. Anyone who works with quaternions knows that there are two notions of trace. The naive trace, 4a, is the trace of multiplication by a on any irreducible Q-module, using the obvious tensor structure. But there is a better notion, the reduced trace, which is equal to 2a. Similarly, there is a naive norm, (a^2+b^2+c^2+d^2)^2, and there is a reduced norm a^2+b^2+c^2+d^2.

This all makes me think that there is a subtle tensor category structure on Q-mod, other than the obvious one, for which these are the trace and norm in the categorical sense. Can someone spell out the details for me?

By the way, a note about why I am asking. I am reading Milne’s excellent notes on motives, and I therefore want to understand the notion of a non-neutral Tannakian category (page 10). As I understand it, this notion allows us to evade some of the standard problems in defining characteristic p cohomology; one of which is the issue above about traces in quaternion algebras.

Topology that Algebra can’t see July 28, 2009

Posted by David Speyer in Algebraic Geometry, Number theory.
15 comments

Let X be an algebraic variety over \mathbb{C}; that is to say, the zero locus of a bunch of polynomials with complex coefficients. We will consider this zero locus as a topological space using the ordinary topology on \mathbb{C}. One of the main themes of algebraic geometry in the last century has been learning how to study the topology of X in terms of the algebraic properties of the defining equations.

In this post, I will explain that there are intrinsic limits to this approach; things that cannot be computed algebraically. In particular, I want to explain how from a categorical point of view, we can’t even compute the homology H_1( \ , \mathbb{Z}). And, even if you don’t believe in categories, you’ll still have to concede that we can’t compute \pi_1( \ ). This is a very pretty example and it should be more widely known.

Absolutely none of the ideas in this post are original; I think most of them are due to Serre. (Thanks to Attila Smith in comments for the reference.)

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RSI students blog July 22, 2009

Posted by David Speyer in Uncategorized.
1 comment so far

MIT runs a summer program called RSI where high school students come in and work on research with MIT professors and postdocs. There are research projects in all sorts of scientific fields, including many in mathematics.

This year, the RSI math students have decided to start a blog. About half their posts seem to be background to their research, like this post on classifying the representations of \mathfrak{sl}_2. The other half seem to be puzzles and Olympiad problems; for example, this post starts out by talking about the first US Olympiad problem ever, and moves to discussion of lattices and mobius functions.

Give them a visit at deltaepsilons.wordpress.com/!

Did someone break the arxiv? July 7, 2009

Posted by Scott Morrison in Uncategorized.
14 comments

Presumably this is a short-lived phenomenon, but right now it seems that all recent articles on the arxiv are unavailable. For example, if you go to http://arxiv.org/abs/0907.1051 you’ll get a message Paper 0907.1051 doesn’t exist. I haven’t looked through things thoroughly, but every link I followed from recent arxiv postings was also broken, while older (>2 years) links were fine.

I don’t see an announcement on the arxiv site. I wonder what’s happening?

Bleg: book recommendations for an undergraduate July 3, 2009

Posted by Ben Webster in blegs.
57 comments

Following Emily’s advice, I recently signed up to be mentor in the AWM Mentor Network. It’s been pretty good thus far (I recommend it to any of you who would like to do some menting), but I got a request from my mentee that I thought some of our audience might have better ideas about than me.

What math books would you suggest for relatively casual summer reading for an undergraduate math major finishing their third year? This is not the sort of thing I think about a lot, but I know a reasonable number of readers have a lot more experience with young mathematicians than I do.

Three geometric constructions of the irreducible representations of GL_n July 3, 2009

Posted by Joel Kamnitzer in Uncategorized.
10 comments

The past few weeks there has been a summer school and conference on geometric representation theory and extended affine Lie algebras at University of Ottawa. As part of this event, I gave a week long lecture series entitled “three geometric constructions of the irreducible representations of GL_n “. Specifically I discussed the Borel-Weil theorem, Ginzburg’s construction using Springer fibres, and the geometric Satake correspondence. I focused on GL_n to keep the root system combinatorics and the geometry as elementary as possible.

The typed lecture notes from my talk are now available. If you do read them, please let me know if you have any comments/corrections. (You can also find videos of the talks.)

The other lectures at the summer school were given by Neher, Kang, Wang, Savage, and Chari. I recommend reading their notes/watching their videos if you want to learn more about geometric representation theory, crystals, and affine Lie algebras.

Continued Fractions and Hyperelliptic Curves July 2, 2009

Posted by David Speyer in Algebraic Geometry, Number theory.
4 comments

I recently read a charming little paper: Quasi-elliptic integrals and periodic continued fractions, by van der Poorten and Tran. Most of us who have taken a number theory course of some kind learned how to solve Pell’s equation: x^2 - D y^2 =1 where D is a nonsquare positive integer. The usual method is to compute the continued fraction
\displaystyle{\sqrt{D} = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \frac{1}{\cdots}}}}.
One then defines the convergents of \sqrt{D} by
\displaystyle{x_0/y_0 = a_0}
\displaystyle{x_1/y_1 = a_0 + \frac{1}{a_1}}
\displaystyle{x_2/y_2 = a_0 + \frac{1}{a_1+\frac{1}{a_2}}} etcetera.

Then x_i^2 - D y_i^2 tends to be very small and, if you compute long enough, for some i you will have x_i^2 - D y_i^2=1.

What van der Poorten and Tran do is to ask what happens if D is not an integer, but a polynomial D(t) = t^{2g+2} + d_{2g+1} t^{2g+1} + \cdots + d_1 t + d_0. Before I get into details, I want to tell you about something gorgeous that I won’t explain at all. Using the methods in their paper, van der Poorten and Trap can discover identities like
\displaystyle{ \int \frac{3 x dx}{\sqrt{x^4+2x}} = \log \left( x^3+1+x \sqrt{x^4+2x} \right)}.
Isn’t that pretty?

It turns out that the continued fraction algorithm for \sqrt{D(t)} is actually much prettier than for integers. Everything should be understood in terms of the curve C cut out by y^2 = D(t). This is a curve of genus g, with two points at infinity. (One of these points is the limit of (t, \sqrt{D(t)}) and the other is the limit of (t, -\sqrt{D(t)}).) I’ll call these two points \infty_{+} and \infty_{-}. The theory is controlled by the line bundles \mathcal{O}(k \infty_+ + \ell \infty_-). In particular, there are nontrivial solutions to x(t)^2 - D(t) y(t)^2 =1 if and only if the continued fraction is periodic, if and only if \mathcal{O}(k \infty_+) = \mathcal{O}(k \infty_-) for some a >0.

Below the fold, I’ll explain what is meant by the continued fraction algorithm for an algebraic function, and tell you some of the other nice results from the paper.

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