Date : Friday, December 16, 2022
Time: 2pm - 3pm
Location : Goldsmith 300
Abstract: The cotangent bundle of a smooth closed manifold is one of the most basic examples of a symplectic manifold. The smooth topology of the base manifold determines the symplectic topology of its cotangent bundle, but it is a major open question whether the converse is true. In this talk we will focus on the special case where the base manifold is an exotic sphere, reviewing what is known as well as current efforts to go further.
Date: Monday December 19, 2022
Time: 2pm - 3pm
Location : Goldsmith 300
Abstract: One systematic way to construct new manifolds from old is the operation of surgery, removing a simple piece and gluing it back in with a twist. Many manifolds can be obtained from a sphere by repeatedly applying surgery operations, and it is reasonable to ask how many surgeries are needed to produce a given manifold (its "surgery number"). For instance, the surgery number of an oriented surface is precisely its genus. This quantity remains especially mysterious in dimensions 3 and 4, and there has been little headway since the 1990s.
Date: Tuesday, December 20, 2022
Time: 2pm - 3pm
Location: Goldsmith 300
The critical exponent is an important numerical invariant of discrete isometry groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete isometry groups acting on hyperbolic spaces. In particular, I will explain how the numerical invariant is closely related to geometry, dynamics, and representation of the group action on hyperbolic spaces.
Date: Wednesday, December 21, 2022
Time: 2pm - 3pm
Location: Goldsmith 300
Abstract: The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).
Date: Monday, January 9, 2023
Time: 2pm - 3pm
Location: Goldsmith 300
Abstract: A central theme in arithmetic geometry over finite fields is the passage from geometric invariants to
arithmetic information by taking the trace of Frobenius. I will describe a higher version of this procedure with
a particular focus on applications to the Langlands correspondence over function fields. In this case, this
procedure relates the geometric Langlands correspondence with the classical one. Specifically, we obtain that
the space of automorphic functions is the categorical trace (aka Hochschild homology) of Frobenius acting on an
appropriate version of the automorphic category. This leads to a localization of the space of automorphic
functions on a moduli space of Langlands parameters, giving a refinement of V. Lafforgue's spectral
decomposition. This is based on joint works with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.
Date: Thursday, January 12, 2023
Time: 2pm - 3pm
Location: Goldsmith 300
Abstract: The study of group actions gained significant interest in the past several decades, as group actions are a powerful tool when approaching problems from number theory and geometry. We will focus on the dynamical equivalent to vectors with 'infinitely good' diophantine approximation. From this dynamical point of view, Weiss conjectured a complete classification of the relevant trajectories. We will see how tools like representation theory and algebraic geometry can be employed to tackle such questions. This is in part a joint work with Omri Solan, and in part a joint work with Lingmin Liao, Ronggang Shi and Omri Solan.