Mini-workshop on Geometric Methods in Number Theory and Representation Theory

Northwestern University
May 27-28, 2017



Date, Time and Venue

The mini-workshop is one of the scientific events for the Emphasis year in Algebraic Geometry, Number Theory and Physics, 2016-17. It will be held on Saturday and Sunday, May 27-28, 2017, at Harris Hall [Address: 1881 Sheridan Road, Evanston, IL]. Talks will be held in Harris L07, and breakfast/refreshments will be served outside the lecture room. Please check the page regularly for updates.

Speakers

Program

Saturday, May 27
9:00-9:30 Breakfast
9:30-10:30 Bhargav Bhatt Integral p-adic Hodge theory
11:00-12:00 Shou-Wu Zhang On the heights of Abelian varieties
12:00-2:00 Lunch break
2:00-3:00 Wei Ho Some geometric methods in arithmetic statistics
3:10-4:10 Ramin Takloo-Bighash Rational points on zero loci of Brauer elements
4:10-4:30 Coffee break
4:30-5:30 George Boxer On a vanishing conjecture for coherent cohomology of the Siegel threefold
Sunday, May 28
8:30-9:00 Breakfast
9:00-10:00 Xinwen Zhu Correspondences of Shimura varieties via the geometric Satake
10:15-11:15 Tsao-Hsien Chen Affine Matsuki correspondence for sheaves
11:30-12:30 Yiannis Sakellaridis Local and global harmonic analysis on stacks

 

Abstracts


Bhargav Bhatt, Integral p-adic Hodge theory

Abstract: Given a smooth projective variety with good reduction over a p-adic field, in recent joint work with Morrow and Scholze, we constructed a cohomology theory that interpolates between the etale cohomology of the generic fiber and the crystalline cohomology of the special fiber integrally. This leads to concrete consequences such as: the mod p Betti numbers of the generic fiber are bounded above by the de Rham Betti numbers of the special fibre. In my talk, I'll recall this story, and then explain how to formally deduce analogous results in the semistable case using "perfectoid stacks". (This is joint work with Matthew Morrow and Peter Scholze.)


George Boxer, On a vanishing conjecture for coherent cohomology of the Siegel threefold

Abstract: Calegari and Geraghty have recently constructed Taylor-Wiles systems out of Siegel modular forms of "low weight". In the especially interesting case which is relevant for the modularity conjecture for Abelian surfaces, their construction is conditional on a conjecture, which roughly says that no "sufficiently generic" systems of Hecke eigenvalues occur in the second cohomology group of a certain line bundle on the Siegel threefold. In the talk I will try to present the motivation for this conjecture, and then explain a proof, for a suitable notion of "sufficiently generic". This is joint work with Frank Calegari and Toby Gee.


Ramin Takloo-Bighash, Rational points on zero loci of Brauer elements

Abstract: We consider the problem of counting the number of rational points of bounded height in the zero-loci of Brauer group elements on semi-simple algebraic groups over number fields. We obtain asymptotic formulae for the counting problem for wonderful compactifications using the spectral theory of automorphic forms. Applications include asymptotic formulae for the number of matrices over Q whose determinant is a sum of two squares. These results provide a positive answer to some cases of a question of Serre concerning such counting problems. This is joint work with Daniel Loughran and Sho Tanimoto.


Tsao-Hsien Chen, Affine Matsuki correspondence for sheaves

Abstract: The Matsuki correspondence for a complex reductive group G is a remarkable bijection between H and K orbits on the flag manifold of G, where H is a real form of G and K is the complexification of the maximal compact subgroup of H. I will explain a version of Matsuki correspondence for the affine grassmannian Gr. The statement is an equivalence between two equivariant derived categories of Gr, one category involves the polynomial loop group of H; the other involves the loop group of K. This is a joint work with David Nadler.


Wei Ho, Some geometric methods in arithmetic statistics

Abstract: We will discuss some geometric techniques used in proving "arithmetic statistics" results, primarily using the case of Selmer groups for families of elliptic curves as a motivating example.


Yiannis Sakellaridis, Local and global harmonic analysis on stacks

Abstract: As the global and local conjectures of Arthur, Vogan, Gan--Gross--Prasad and others suggest, the Langlands program and its relative variants should be formulated not in terms of a single group or homogeneous space, but in terms of appropriate quotient stacks. I will explain what it means to perform harmonic analysis on quotient stacks (by defining appropriate Schwartz spaces), and what the "relative trace formula" of a quotient stack is (by regularizing orbital integrals -- at least, whenever this is possible).


Shou-Wu Zhang, On the heights of Abelian varieties

Abstract: I will talk about some new questions and results about the Faltings heights of abelian varieties of GL(2) type and CM type with applications to Diophantine geometry and class groups.


Xinwen Zhu, Correspondences of Shimura varieties via the geometric Satake

Abstract: I will explain a general strategy to construct cohomological correspondences between the mod p fibers of different Shimura varieties via the geometric Satake. In the special case of correspondences from a Shimura set to a Shimura variety, we obtain all the Tate classes in the middle cohomology of the latter, under a certain genericity condition. This is a joint work with Liang Xiao.


Registration and financial support

In order to get an accurate headcount, participants except speakers are asked to register by sending e-mails to GMNTRT@math.northwestern.edu. Please include your name, home institution, and tentative arrival and departure dates.

Partial financial support will be available for young participants who cannot find sufficient funding sources for attending the workshop, with preference given to graduate students. Those requesting financial support please register by 31 March, 2017. Please include your status (graduate student, post-doc, neither), affiliation, and PhD adviser if you are a graduate student.

If you are requesting travel support, please note that we can only reimburse air tickets that are issued by a US flag carrier.

Travel Information

The most convenient airport is the Chicago O'Hare International Airport (ORD).

Three taxi companies serve Evanston from the airport, and each offers a flat fare of $33 (+tip): 303 Taxi, American Taxi, Norshore Cab. Uber service can be requested at designated area (follow your Uber app) at ORD. Uber will take approximately 40 minutes and cost $30 - $40. Note that one can also take Uberpool from the airport, and it is sometimes much cheaper than Uber, if one needs to save money and has a bit more time.

For driving directions to Northwestern University, see here. For a link to the full campus map, please click here. Parking on campus is free on weekends.

There are a number of hotels in Evanston near Northwestern. A list and further visitor information may be found here.

Organizing Committee

Questions? e-mail GMNTRT@math.northwestern.edu


Department of Mathematics, Northwestern University