Representation Theory and Number Theory Seminar, 2019-2020

Fall 2019: Monday, 3:00-4:00 PM, LCB 215

August 29:
Title: Brief Organizational Meeting

August 26:
Speaker: Stephen Miller, Rutgers
Title: An update on the sphere packing problem
Abstract: I will discuss recent work on the optimal arrangement of points in euclidean space. In addition to the solution to the sphere packing problem in dimensions 8 and 24 from 2016, the "Universal Optimality" conjecture has now been established in these dimensions as well. This shows that E8 and the Leech lattice minimize energy for any completely monotonic function of distance-squared. Previously there were no proofs in any dimension > 1 that a particular configuration minimizes energy for any nonzero example of such a potential. Beyond giving a new proof of the sphere packing results, Universal Optimality also gives information about long-range interactions. The techniques reduce the problem to a question in single-variable calculus, which is ultimately solved using modular forms and methods introduced by Viazovska. This will be a colloquium-style talk. (Joint work with Henry Cohn, Abhinav Kumar, Danylo Radchenko, and Maryna Viazovska)

September 9:
Speaker: Gil Moss, Utah
Title: The Whittaker model of Serre's universal unramified module
Abstract: Let F be a nonarchimedean local field with residue field of order q, and let l be a prime different from p. The mod-l representation theory of the F-points of reductive groups can exhibit very different behavior than the complex theory. We will discuss the interplay between Whittaker models and systems of spherical Hecke eigenvalues for representations of GL_n(F). The results can be applied toward an open conjecture in the theory of mod-l automorphic forms.

September 16:
Speaker: Ronno Das, University of Chicago
Title: Points and lines on cubic surfaces
Abstract: The Cayley-Salmon theorem states that every smooth cubic surface in CP^3 has exactly 27 lines. Their proof is that marking a line on each cubic surface produces a 27-sheeted cover of the moduli space M of smooth cubic surfaces. Similarly, marking a point produces a 'universal family' of cubic surfaces over M. One difficulty in understanding these spaces is that they are complements of incredibly singular hypersurfaces. In this talk I will explain how to compute the rational cohomology of these spaces. I'll also explain how these purely topological theorems have (via the machinery of the Weil Conjectures) purely arithmetic consequences: the average smooth cubic surface over a finite field F_q contains 1 line and q^2 + q + 1 points.

September 23:
Speaker: Stefan Patrikis, Utah
Title: Lifting irreducible Galois representations
Abstract: This will be a largely expository talk on "odd" Galois representations. I will begin with motivation from the theory of automorphic forms and representation theory of semisimple Lie groups, and then, with an emphasis on examples, I will discuss some recent results with Fakhruddin and Khare on lifting odd irreducible mod p Galois representations to geometric p-adic representations.

September 30:
Speaker: Sean Howe, Utah
Title: A(nother) conjecture about zeta functions, or, "it's zeta functions all the way down."
Abstract: We conjecture a unification of arithmetic and motivic/topological statistics over finite fields through a natural analytic topology on the ring of zeta functions. A key step will be to explain exactly what it means to evaluate the zeta function of a zeta function at a zeta function. This is joint work with Margaret Bilu.

October 21:
Speaker: Gordan Savin, Utah
Title: Exceptional Siegel-Weil formula
Abstract: Joint work with W.T. Gan

November 4:
Speaker: Petar Bakic, Utah
Title: The local theta correspondence
Abstract: The global theta correspondence is a standard tool in the study of automorphic forms. To understand it, one would like to have a description of its local variant, i.e. the local theta (aka Howe) correspondence. In the first part of the talk, I will go over the basic setup and the most important general results concerning theta correspondence. After that, I will discuss the recent work with M. Hanzer in which we give an explicit description of the local theta correspondence for dual pairs of Type I.

November 18:
Speaker: Andrea Dotto, University of Chicago
Title: Functoriality for Serre weights
Abstract: By work of Gee--Geraghty and myself, one can transfer Serre weights from the maximal compact subgroup of an inner form D* of GL(n) to a maximal compact subgroup of GL(n). Because of the congruence properties of the Jacquet--Langlands correspondence this transfer is compatible with the Breuil--Mezard formalism, which allows one to extend the Serre weight conjectures to D* (at least for a tame and generic residual representation). This talk aims to explain all of the above and to discuss a possible generalization to inner forms of unramified groups.

November 22, 3:30-4:30pm, LCB 225:
Speaker: Zheng Liu, UC Santa Barbara
Title: Doubling archimedean zeta integrals for symplectic and unitary groups
Abstract: In order to verify the compatibility between the conjecture of Coates--Perrin-Riou and the interpolation results of the p-adic L-functions constructed by using the doubling method, a doubling archimedean zeta integral needs to be calculated for holomorphic discrete series. When the holomorphic discrete series is of scalar weight, it has been done by Bocherer--Schmidt and Shimura. I will explain a way to compute this archimedean zeta integral for general vector weights by using the theory of theta correspondence.

November 25:
Speaker: Allen Moy, Hong Kong University of Science and Technology
Title: Decompositions of Euler-Poincaré presentations and resolutions
Abstract: Work of Bezrukavnikov-Kazhdan-Varshavsky uses an equivariant system of trivial idempotents of Moy-Prasad groups to obtain an Euler-Poincaré presentation of the r-depth Bernstein projector. Bestvina-Savin, generalizing earlier work of Schneider-Sthuler, showed this system of Moy-Prasad groups, allows a resolution of a smooth representation generated by its depth r-vectors. We report on work in progress with Gordan Savin establishing a direct sum decomposition of the equivariant system and therefore of the Euler-Poincaré presentation and the resolution.

Spring 2020: Monday, 3:00-4:00 PM, LCB 215

January 6:
Speaker: Peter Wear, UC San Diego

February 3:
Speaker: Mathilde Gerbelli-Gauthier, University of Chicago