I’m honored to be visiting America’s virtuous neighbor to the north to speak in the PIMS – UBC Math Distinguished Colloquium on Friday, September 14, 2018.
Title: Birational geometry and algebraic cycles
Abstract: A fundamental problem of algebraic geometry is to determine which algebraic varieties are rational, that is, isomorphic to projective space after removing lower-dimensional subvarieties from both sides. We discuss the history of the problem. Some dramatic progress in the past 5 years uses a new tool in this context: the Chow group of algebraic cycles.
On the day before, Thursday, September 13, I’ll speak in the UBC Algebraic Geometry seminar.
Title: Hodge theory of classifying stacks
Abstract: The goal of this talk is to create a correspondence between the representation theory of algebraic groups and the topology of Lie groups. The idea is to study the Hodge theory of the classifying stack of a reductive group over a field of characteristic p, the case of characteristic 0 having been studied by Behrend, Bott, Simpson and Teleman. The approach yields new calculations in representation theory, motivated by topology.
The next Southern California Algebraic Geometry Seminar takes place at USC on Saturday, 14 April 2018. More information is available at the seminar webpage.
Our excellent slate of speakers is:
Ben Antieau (UIC)
Barbara Fantechi (Sissa)
Tommaso de Fernex (Utah)
Hiraku Nakajima (Kyoto)
Registration is free and very simple. Please register here.
Photo is by Julie Kitzenberger of a cat and pigeon who are friends. The 14th of April is pigeon day (25 Germinal) in the French Republican calendar.
Every year, the Distinguished Lecture Series (DLS) brings two to four eminent mathematicians to UCLA for a week or more to give a lecture series on their field, and to meet with faculty and graduate students.
C. David Levermore
I’ll be speaking (on rationality) at the AMS Fall Sectional Meeting at Riverside. Other speakers in the algebraic geometry session are Aravind Asok, Emily Clader, Omprokash Das, Humberto Diaz, Matthias Flach, Martin Gallauer, James McKernan, Wenhao Ou, Dustin Ross, and Alexander Soibelman. The session is organized by Humberto Diaz, Jose Gonzalez, and Ziv Ran.
The invited addresses are by Paul Balmer, Pavel Etingof, and Monica Vazirani.
Photo: NO YOU CAN’T HAVE MY LEAVES STOP ASKING, from reddit.
WAGS returns to UCLA.
The Fall 2017 edition of the Western Algebraic Geometry Symposium (WAGS) will take place the weekend of 14–15 October 2017 at IPAM on the UCLA campus, hosted by the UCLA Mathematics Department. Details are now on the conference website.
If you plan to attend but haven’t yet registered, please register. It’s free, and knowing who’s coming will allow us to ensure that:
- We have enough space.
- We have enough coffee.
- We have enough food.
- We have a name tag ready for you, so that the conference is successful in helping you meet fellow mathematicians and helping other mathematicians meet you.
- We can help our funder to demonstrate — with evidence — that they’re supporting a thriving enterprise.
Send questions to firstname.lastname@example.org. [UPDATE: I previously gave the wrong email address (starting fall17).]
Photo of the Powell Cat from the Daily Bruin. More about the Powell Cat on twitter.
My paper about the Tate conjecture for Bull. AMS is now available to view.
In it I survey the history of the Tate conjecture on algebraic cycles. The conjecture is closely intertwined with other big problems in arithmetic and algebraic geometry, including the Hodge and Birch–Swinnerton-Dyer conjectures. I conclude by discussing the recent proof of the Tate conjecture for K3 surfaces over finite fields.
After returning the proofs to the AMS, it occurred to me that it could be helpful to readers if I recommended some available related videos. I was too slow for the AMS’s speedy production, however, so I make the recommendations here.
F. Charles. K3 surfaces over finite fields: insights from complex geometry (2015).
K. Madapusi Pera. Regular integral models for orthogonal Shimura varieties and the Tate conjecture for K3 surfaces in finite characteristic (2012).
D. Maulik. Finiteness of K3 surfaces and the Tate conjecture (2012).
Photo is from foldedspace.org.
I’ve posted an updated version of a paper on the arXiv, Hodge theory of classifying stacks (initial version posted 10 March 2017). We compute the Hodge and de Rham cohomology of the classifying space BG (defined as sheaf cohomology on the algebraic stack BG) for reductive groups G over many fields, including fields of small characteristic. These calculations have a direct relation with representation theory, yielding new results there. The calculations are closely analogous to, but not always the same as, the cohomology of classifying spaces in topology. This is part of the grand theme of bringing topological ideas into algebraic geometry to build analogous organizing structures and machines.
In the first version of the paper, I defined Hodge and de Rham cohomology of classifying spaces by thinking of BG as a simplicial scheme. In the current version, I followed Bhargav Bhatt‘s suggestion to make the definitions in terms of the stack BG, with simplicial schemes as just a computational tool. Although I sometimes resist the language of algebraic stacks as being too abstract, it is powerful, especially now that so much cohomological machinery has been developed for stacks (notably in the Stacks Project). (In short, stacks are a flexible language for talking about quotients in algebraic geometry. There is a “quotient stack” [X/G] for any action of an algebraic group G on a scheme X.)
Using the language of stacks led to several improvements in the paper. For one thing, my earlier definition gave what should be considered wrong answers for non-smooth groups, such as the group scheme μp of pth roots of unity in characteristic p. Also, the paper now includes a description of equivariant Hodge cohomology for group actions on affine schemes, generalizing work of Simpson and Teleman in characteristic zero. There is a lot of room here for further generalizations (and calculations).
Photo: Susie the cat in Cambridge, November 2001.