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Equivariant cohomology wiki April 26, 2010

Posted by David Speyer in Algebraic Geometry, blog triumphalism.
1 comment so far

A few weeks ago, I went to a superb conference at AIM on Localization Techniques in Equivariant Cohomology. A frequent topic at that workshop was the lack of good surveys and references. Results from the 70s were being rediscovered, and no one knew where to start reading.

As a partial solution, the conference organizers have set up a wiki. There is a list of references, particularly on Schubert calculus and related problems, and of open problems. They are keeping editing restricted to selected users, but I imagine many of our readers would make excellent editors. Please take a look and, if you have anything to add, e-mail Rebecca Goldin or Julianna Tymoczko to become an editor.

The Weil conjectures : Curves April 19, 2010

Posted by David Speyer in Algebraic Geometry, characteristic p.
2 comments

Our goal for today is to prove the following theorem:

Theorem 1: Let X be a projective algebraic curve of genus g and F an endomorphism of degree q. Let H^* be a reasonable cohomology theory. Then the action of F^* on X has eigenvalues which are algebraic integers, with norm q^{1/2}.

For those who know the term, “reasonable cohomology theory” means “Weil cohomology theory”.

The consequence of this theorem, which does not mention cohomology, is

Theorem 2: Let X be a projective algebraic curve of genus g and F an endomorphism of degree q. Then there are algebraic integers \alpha_1, \alpha_2, …, \alpha_{2g} with norm q^{1/2} such that

\displaystyle{ \# \mathrm{Fix}(F^k) = q^k - \sum \alpha_i^k +1 }.

In a previous post, we established this in characteristic zero, by putting a positive definite hermitian structure on H^1(X, \mathbb{C}) such that q^{-1/2} F^* became unitary. But, as I discussed last time, we can’t define H^1(X, \mathbb{C}) when X has characteristic p. Instead, H^1(X) will be defined over some other field of characteristic zero, like \mathbb{Q}_{\ell}. We will therefore need to know that the eigenvalues of F are algebraic integers before we can even make sense of the statement that they have norm q^{1/2}.

It is possible to take the proof I present here and strip it down to its bare essentials, to give a proof of Theorem 2 which doesn’t even mention cohomology. See Hartshorne Exercise V.1.10. I am going to do the opposite; I will go slowly and focus on what each step is proving about H^1(X). The essential argument here is Weil’s, although I have modernized the presentation.

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MathOnline — a compendium of free online mathematical references April 16, 2010

Posted by David Speyer in Uncategorized.
4 comments

Andrea Ferretti has just revealed his new project: MathOnline is a searchable catalog of free online mathematical references. For example, here are all the online guides to Algebraic Number Theory. The focus is on expositions, not research (we already have the arXiv), and all material should be legal for redistribution. There is clearly a lot of work to be done filling in the catalog, but I think this site could become really useful. Head over there and add your favorite texts!

Cyclotomic integers, fusion categories, and subfactors (March) April 15, 2010

Posted by Noah Snyder in fusion categories, Number theory, Paper Advertisement, quantum algebra, subfactors.
1 comment so far

Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory to quantum algebra.

  1. A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
  2. Given a family of graphs G_n obtained from a graph G by attaching a chain of n edges to a chosen vertex, an effective bound on the greatest n so that G_n can be the principal graph of a subfactor.

Neither of these results look like they involve number theory. The connection comes from a result of Etingof, Nikshych, and Ostrik which says that the dimension of every object in a fusion category is a cyclotomic integer.

A possible subtitle to this paper is

What’s so special about (\sqrt{3} + \sqrt{7})/2?

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Reminder: BADMath is on April 24 April 12, 2010

Posted by David Speyer in Uncategorized.
4 comments

I just received an e-mail from the organizers of BADMath, a one day discrete math conference which travels around the Bay Area, asking me to remind interested parties that the registration deadline is coming up on Wednesday. This year, BADMath will be at Santa Clara university on Saturday, April 24. Registration is free but mandatory. There are often carpools from the large Bay Area universities; if anyone knows if such have been planned this year, please post them in the comments.

Photo of Santa Clara UniversityI won’t be making it to the West Coast this year, but past BADMath’s have been excellent. They usually have varied and fun speakers, and a lot of time for socialization. I’ve heard Dylan’s talk before, and can promise that it is very good; a mix of graph theory, algebraic topology, and algebraic geometry.

The Weil conjectures and the problem of coefficients April 12, 2010

Posted by David Speyer in Uncategorized.
4 comments

I have been discussing the Weil conjectures. In this post, I want to discuss one of the main difficulties in their proof. You should be able to follow this even if you have not been reading the earlier posts.

The main claim of the Weil conjectures can be summarized as follows:

Theorem Let X be a smooth projective variety of dimension d over \mathbb{F}_q. Then there are complex numbers \alpha_{i,j}, for 0 \leq i \leq 2d and 1 \leq j \leq b_i(X), and obeying |\alpha_{i,j}| = q^{i/2}, such that

\displaystyle{ \# X(\mathbb{F}_{q^k}) = \sum_{i=1}^{2d} (-1)^i \sum_{j=1}^{b_i} \alpha_{i,j}^k.}

The analogous claim in characteristic zero, which we proved last time, is

Theorem Let X be a smooth projective variety over \mathbb{C}, with \omega \in H^2(X) the hyperplane class. Let F be an endomorphism of \omega such that F^* \omega = q \omega. Then there are complex numbers \alpha_{i,j} as above, such that

\displaystyle{ \# (\mbox{fixed points of}\ F^k) = \sum_{i=1}^{2d} (-1)^i \sum_{j=1}^{b_i} \alpha_{i,j}^k.}

The way we proved this was to essentially establish:

Theorem Let X, F and \omega be as above. Then there is a positive definite Hermitian inner product on H^i(X, \mathbb{C}) such that q^{-i/2} F^* : H^i \to H^i is a unitary operator.

The required theorem then follows from the Lefschetz trace formula, and the fact that the eigenvalues of a unitary matrix have norm 1.

If we wanted to prove the characteristic p theorem in the same way, we might hope to follow the same process: define H^i(X, \mathbb{C}), place a Hermitian structure on it, and prove a Lefschetz trace formula.

There are cohomology theories with Lefschetz trace formulas. But they do not have coefficients in \mathbb{C}. Their coefficient ring is \mathbb{Q}_{\ell} (etale cohomology) or the ring W(\mathbb{F}_q) of Witt vectors (various p-adic theories).

In particular, while we can talk about the eigenvalues of F^*: H^i(X) \to H^i(X), there will be no natural way to embed these eigenvalues in \mathbb{C}. There is an unnatural way: If we could prove that the characteristic polynomial of F^* has coefficients in \mathbb{Q}, then we could consider the roots of the same polynomial in \mathbb{C}. It would be enough to prove that these complex roots have norm q^{i/2}. But it is unclear how to show that the polynomial has coefficients in \mathbb{Q}, let alone how to bound the roots. In particular, it is unclear what could be the analogue of a Hermitian structure for a \mathbb{Q}_{\ell} vector space.

In future posts, we will tackle these problems. For now, below the fold, I reproduce an argument of Serre showing that there is no way to define a reasonable cohomology theory for finite characteristic varieties, with coefficients in \mathbb{R}. (And therefore, not in any subring of \mathbb{R} either.)

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What’s with locality and subdomain specific British blogs? April 5, 2010

Posted by Ben Webster in blog triumphalism, Blogroll.
21 comments

A week or so ago, I happened across the London Number Theory Blog. I thought to myself “Huh, that’s an interesting idea; a blog specifically for workers on a particular subject in a particular city. Seems a little funny, but it could have some interesting uses.” Indeed, while there are some points of general interest on there, it does seem largely aimed at people who are participating their study group (which is great! more study groups should have blogs like this).

Then, just today, I found the Edinburgh Mathematical Physics Group blog. And then I started to wonder…one such blog seems like a curiosity, but two starts to feel like a trend. Was this driven by some kind of external event for just a weird coincidence?

Characteristic zero analogues of the Weil conjectures: higher dimension April 5, 2010

Posted by David Speyer in Algebraic Geometry, characteristic p, complex analysis.
11 comments

In our previous post, we proved

Theorem Let X be a smooth projective curve over \mathbb{C} and F an endomorphism of degree q > 0. The eigenvalues of F on H^1(X) have norm q^{1/2}.

Today, we would like to generalize this to varieties of higher dimension. The obvious guess is

Nontheorem Let X be a smooth projective variety, over \mathbb{C}, of dimension d. Let F be an endomorphism of X of degree q^d. The eigenvalues of F on H^r(X) have norm q^{r/2}.

This is not a theorem! I believe it is Serre who first figured out how to fix and prove this result. That is the topic of today’s post.
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More on that webinar April 3, 2010

Posted by Ben Webster in blog triumphalism, crazy ideas, Math Overflow.
8 comments

So, I attended the webinar I mentioned in the previous post; it was an interesting experience. (more…)

“Webinar” on MathOverflow April 2, 2010

Posted by Ben Webster in Uncategorized.
6 comments

Anton Geraschenko’s doing a webinar on MathOverflow Saturday (that is, today) at 11am West Coast/2pm East Coast time. From the little I know this will involve a short presentation by him about the site, and a Q&A period, through a Java based web meeting interface. I’m having a lot of trouble seeing anything about the actual format on the website, and Anton’s announcement is pretty terse, but it sounds like it should be an interesting discussion about math on the internet, and if nothing else a good way to find out about this “Elluminate” platform they seem to be using.

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