I’m speaking at the AGNES workshop on the last weekend in April, at Stony Brook (click here for more workshop information).
Title: The Chow ring of a finite group
Abstract: A natural class of singular varieties consists of the quotients of vector spaces by linear actions of a finite group G. We can ask what the Chow group of algebraic cycles is, for such a quotient. By taking larger and larger representations of G, we can package these Chow groups into a ring, called the Chow ring of the classifying space of G, or (for short) the Chow ring of G. It maps to the cohomology ring of G, usually not by an isomorphism.
We present the latest tools for computing Chow rings of finite groups. These tools give complete calculations for all “small” groups and many other finite groups. A surprising point is that Chow rings become “wild,” in a precise sense, for some slightly larger finite groups.
The Spring 2014 workshop of AGNES (Algebraic Geometry Northeastern Series) is the weekend of Friday-Sunday, April 25-27, in the Simons Center at Stony Brook University. There are funds to support participants. The registration deadline is March 15th.
The list of confirmed and tentative speakers is as follows.
Ron Donagi (Pennsylvania)
Eugene Gorsky (Columbia)
Sandor Kovacs (Washington)
James McKernan* (UC San Diego)
Zsolt Patakfalvi (Princeton)
Michael Thaddeus (Columbia)
Burt Totaro (UCLA)
Cynthia Vinzant (Michigan)
*To be confirmed
The webpage for the workshop is at the following URL.
There is a registration form on the webpage. The deadline is March 15th, but particularly for funding purposes, participants are encouraged to register as soon as possible.
Additional details will be announced on the workshop webpage. We hope to see you at AGNES Spring 2014.
Organizers: Samuel Grushevsky, Radu Laza, Robert Lazarsfeld, Jason Starr
I’ve posted a new paper on the arXiv: Hodge structures of type (n,0,…,0,n).
The abstract: This paper determines all the possible endomorphism algebras for polarizable Q-Hodge structures of type (n,0,…,0,n). This generalizes the classification of the possible endomorphism algebras of abelian varieties by Albert and Shimura. As with abelian varieties, the most interesting feature of the classification is that in certain cases, every Hodge structure on which a given algebra acts must have extra endomorphisms.
Update (January 11, 2014): Now proofreading. I’ve found a typo in the title of Chapter 8. I misspelled ‘ring’ — pretty humbling. (To give due credit, my editor spotted the typo.)
Original (December 4, 2013):
More about the book at the CUP site (it’s not available yet — I’m checking the copyedited manuscript right now).
You can read the preface in manuscript here.
Table of contents
I am now halfway through teaching a graduate course on “Topological methods in algebraic geometry”. The idea is to give a quick, informal introduction to various topics which are important, but a little too advanced to appear in most first-year graduate courses. I hope to communicate a homotopy-theoretic point of view, which can be applied in many different ways.
The topics we have covered so far are:
- Classifying spaces in topology, principal G-bundles
- Spectral sequences
- Rational homotopy theory
- Topology of Lie groups
- Faithfully flat descent
To summarize some of the early parts of the course: every geometer will learn about the Chern classes of a complex vector bundle. My idea for the course was to put this construction in a broader context, by considering principal G-bundles for any group G instead of just the general linear group (or the unitary group). This brings up some of the main ideas of homotopy theory, since the classifying space for principal G-bundles is related to G by taking the loop space. Discussing spectral sequences and rational homotopy theory shows that we can compute a lot in homotopy theory. Using these general methods, we can get a rather strong understanding of the topology of all Lie groups. At this point, the connection with algebraic geometry becomes serious, since many properties of a compact Lie group are best understood by considering its complexification, which is a complex reductive group.
Grothendieck’s theory of faithfully flat descent is a beautiful topic of commutative algebra, which appears in surprisingly few books. It generalizes the idea of constructing sheaves or vector bundles by gluing together sheaves or vector bundles on open subsets. The theory should be useful for understanding principal bundles in algebraic geometry, equivariant Chow groups, and algebraic stacks in the rest of the course. And oh yes, I also want to discuss derived categories. We’ll see how it goes.
Here are the books I have suggested for the course so far.
- R. Bott and L. Tu. Differential forms in algebraic topology.
- J. Milnor and J. Stasheff. Characteristic classes.
- J. McCleary. A user’s guide to spectral sequences.
- Y. Félix, S. Halperin, and J.-C. Thomas. Rational homotopy theory.
- Y. Félix, J. Oprea, D. Tanré. Algebraic models in geometry.
- W. Waterhouse. Introduction to affine group schemes.
And anyone who’s been paying attention knows that important ingredients in the secret sauce can be found here:
- J.-P. Serre. Oeuvres: collected papers, 4 vols.