Abstracts

Proof of the Geometric Langlands Conjecture

Abstract:

In the talk we will explain the various contexts for the geometric Langlands conjecture (de Rham, Betti, l-adic), and the ideas that go into its proof.

This is joint work with: Dima Arinkin,  Dario Beraldo, Justin Campbell, Lin Chen, Joakim Faergeman, Kevin Lin and Nick Rozenblyu,

Area-Minimizing Currents Mod an Integer

Abstract:

Currents mod p are a suitable generalization of classical chains mod p, i.e. of finite combinations of smooth submanifolds with coefficients in the cyclic group Zp. By the pioneering work of Federer and Fleming it is possible to minimize the area in this context and, for instance, represent mod p homology classes with area minimizers. For p > 2 typically (i.e. away from a small set of exceptional points) one would expect such minimizers to be a union of smooth minimal surfaces joining together (“in multiples of p’s”) at some common boundary. This is however surprisingly challenging to prove, especially for even p’s, and up until recently only known for p = 3 and 4 in codimension 1.

In this talk I will explain the difficulties and outline the outcome of a series of more recent works (some joint of the speaker with Hirsch, Marchese, Stuvard and Spolaor, some by Minter and Wickramasekera, and some joint of the speaker with Minter and Skorobogatova) which confirms this picture, with varying degrees of precision, for all p’s, every dimension and codimension, and general ambient Riemannian manifolds.

Higher Derivatives of Zeta Functions and Volume of Moduli Shtukas

Abstract:

It is well-known that the volume of a locally symmetric space (quotient of a symmetric space by an arithmetic subgroup, with respect to a suitably normalized invariant measure) is essentially a product of special values of zeta functions. We give an extension of this result in the function field case, so far only for general linear groups and unitary groups. In the new result, locally symmetric spaces will be replaced by the moduli space of Drinfeld Shtukas with multiple legs, and special values of zeta functions will be replaced by their higher derivatives. The result is already nontrivial (but not hard) in the case of GL(1).

This is joint work with Tony Feng and Wei Zhang.