Avraham Aizenbud
Title: Point-wise surjective presentations of stacks, or why I am not afraid of (infinity) stacks anymore?
Abstract: Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms source, target :Mor →Ob, inversion: Mor →Mor, composition :Mor
In joint work with Nir Avni (in progress) we show that any algebraic groupoid has a presentation such that this functor will be essentially surjective for many fields (and under some assumptions on the stack, for any field). The results are also extended to Henselian rings. Despite the title, the talk will be about usual stacks and not infinity-stacks, yet in some of the proofs it is more convenient to use the language of higher categories and I'll try to explain why. No prior knowledge of infinity stacks will be assumed, but a superficial acquaintance with usual stacks will be helpful.
Slides Chat
Alexander Braverman
Title: Generalizing local geometric class field theory
Abstract: Let K be the field of Laurent power series over complex numbers. Local geometric class field theory (due to G.Laumon) is an equivalence of categories between D-modules on K* (considered as an ind-scheme over complex numbers) and quasi-coherent sheaves on the moduli space of local systems of rank 1 on the formal punctured disc.
We present a several series of very surprising conjectures generalizing this statement (the LHS side of the conjectures involves things like D-modules on K or D-modules on GL(n,K)); quite remarkably the conjectures although being completely mathamatical follow from quantum field theory considerations (more precisely, they follow a certain interpretation of the so called 3-dimensional mirror symmetry due to T.Dimfote, D.Gaiotto, J. Hilburn, P.Yoo and others; this was the way they were historically discovered). The plan of the talk is as follows:
- Review of local geometric class field theory
- Formulation of the above conjectures
- Discussion of known cases
- If time permits: some hints about the motivation from physics
Vladimir Berkovich
Title: Hodge theory for non-Archimedean analytic spaces.
Abstrarct: In a work in progress, I’ve extended the classical construction of the integral vanishing cycles complexes for schemes of finite type over the ring of convergent power series with complex coefficients to so called special formal schemes over the completion of that ring. This allows one to define integral “etale” cohomology groups for a class of non-Archimedean analytic spaces (that includes compact ones) over the fraction field of the above completion. They are finitely generated abelian groups which give rise to all l-adic etale and de Rham cohomology groups of the space. In the talk I’ll explain a theorem which states that the integral “etale” cohomology groups of a proper non-Archimedean analytic space are provided with a mixed Hodge structure functorial in the space. Furthermore, if the space is the non-Archimedean analytification of the generic fiber of a proper scheme over the ring of convergent power series, it is the limit mixed Hodge structure on the integral cohomology groups of the fibers of the complex analytification of the scheme over points of a punctured disc.
Chat Notes
Roman Bezrukavnikov
Title: Unipotent characters and l-adic sheaves.
Abstract: The talk will be based on a joint project in progress with David Kazhdan and Yakov Varshavsky where we relate characters of unipotent representations of a p-adic group on compact elements to l-adic sheaves. This is partly inspired by conjectures by Lusztig and by the conjectural endoscopic decomposition for invariant distributions."
Slides Chat
Dennis Gaistgory
Title: Geometric 2nd adjointness via nearby cycles, a report on the work of Lin Chen.
Abstract: we will explain a theorem of Lin Chen that establishes an analog of Bernstein's 2nd adjointness theorem for the semi-infinite category of sheaves on the affine Grassmannian. The key construction providing the unit of the duality is given by a nearby cycles procedure along the Vinberg (a.k.a. DeConcini-Procesi) degeneration of G to B\underset{T}\times B^-.
Slides Chat
Michael Finkelberg
Title: Elliptic zastava.
Abstract: For a semisimple group G and a smooth curve C, open zastava space Z(G,C) is a smooth variety, affine over a configuration space of C. In case C is the additive or multiplicative group, Z(G,C) carries a natural symplectic form, and the projection to the configuration space is an integrable system (open Toda lattice for G=SL(2)). I will explain what happens when C is an elliptic curve. This is a joint work with Alexander Polishchuk.
Slides Chat
David Kazhdan
Title: Hecke operators and the Langlands correspondence for curves over local fields.
Erez Lapid
Title: Results and conjectures about parabolic induction for the general linear group over a p-adic field.
Abstract: Representations of p-adic groups, especially the general linear group, was one of the fields dramatically transformed by Bernstein, through his joint work with Zelevinsky and beyond. Some basic questions in the theory remain outstanding. I will discuss some of these questions and their connection to other themes in representation theory. Joint with Alberto Mínguez.
Slides Chat
Yiannis Sakellaridis
Title: Periods and L-functions.
Abstract: This will be a survey talk on the relationship between periods of automorphic forms and L-functions, starting from the fall of 2004, when Joseph Bernstein encouraged my investigations into spherical varieties and L-functions, and ending with ongoing joint work with Jonathan Wang and David Ben Zvi– Akshay Venkatesh.
Slides Chat
Peter Sarnak
Title : Gap Sets for the Spectra of Cubic Graphs
Abstract :The Alon-Bopanna/Ramanujan Graph interval (2sqrt2,3) ,is a maximal spectral gap interval for the eigenvalues of the adjacency matrix of cubic graphs and it renders optimal expanders. Other applications seek gaps at different locations of the spectrum. We construct such gap intervals and sets, including some maximal ones. Among our tools to do so is a triangle adding transformation on cubic graphs, abelian covers and Fekete's theorem on the transfinite diameter of sets containing algebraic integers and their conjugates.
Joint work with Alicia Kollar.
Slides Chat
Akshay Venkatesh
Title: Infinite sums of L-functions.
Abstract: There exist surprising identities relating sums of different families of L-functions. Following work of Kuznetsov, Motohashi and Reznikov, I will describe some of these identities and how to view them in terms of representation theory. Then I will speculate on the role that such infinite sums of L-functions may play in the "relative Langlands duality" proposed by Ben-Zvi, Sakellaridis and myself.
Slides Chat
Sasha Yom Din
Title: A Paley-Wiener theorem for spherical p-adic spaces and Bernstein morphisms.
Abstract: I will try to discuss a Paley-Wiener type theorem for spherical homogeneous p-adic spaces (which contains much less information than the Paley-Wiener theorem of Delorme, Harinck and Sakellaridis but is more elementary and encompasses more spaces), and to discuss the construction of Bernstein morphisms via this approach.
Slides Notes Chat