Abstracts

Week of May 1, 2022

May 2, 2022
Group Actions and Dynamics Uniform spectral gaps and orthogeodesic counting for strong convergence of geometrically finite Kleinian groups 4:00pm -
LOM206

Spectral gaps of the Laplacian are interesting geometric quantities that have applications in dynamics. In this talk we will study the convergence of small eigenvalues for geometrically finite hyperbolic $n$-manifolds under strong limits. We will apply this together with the exponentially mixing properties of geodesic flow to count uniformly along the sequence the number of orthogeodesics between converging Margulis tubes, which can give uniform control of the size of regions of small injectivity radius. This is joint work with Beibei Liu.
May 3, 2022
Geometry & Topology Dimers, webs, and local systems 4:15pm -
LOM 214

For a planar bipartite graph G equipped with a SLn-local system, we show that the determinant of the associated Kasteleyn matrix counts “n-multiwebs” (generalizations of n-webs) in G, weighted by their web-traces. We use this fact to study random n-multiwebs in graphs on some simple surfaces. This is joint work with Daniel Douglas and Richard Kenyon.
Algebra and Number Theory Seminar p-adic aspects of L-functions, with a view toward unitary groups 4:30pm -
p-adic aspects of L-functions, with a view toward unitary groups

The study of p-adic properties of values of L-functions dates back to Kummer's study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After Kummer's ideas lay mostly dormant for a century, Iwasawa’s conjectures about the meaning of p-adic L-functions led to renewed interest, and Serre’s discovery of p-adic modular forms opened up a new approach to studying congruences between values of L-functions, forming the foundation for continued developments today.

With a viewpoint that encompasses several settings, including modular forms (on GL_2) and automorphic forms on unitary groups, I will introduce p-adic L-functions and a recipe for constructing them, which relies partly on properties of Fourier coefficients of modular (and automorphic) forms. Along the way, I will introduce several recent developments (including my joint work with Harris, Li, and Skinner for unitary groups) and put them in the context of constructions of Serre, Katz, and Hida.

Zoom join link: https://yale.zoom.us/j/94467913606
Passcode: 1729
May 4, 2022
Colloquium Arithmetic patterns in dense sets 4:15pm -
LOM 206

Abstract: Some of the most important problems in combinatorial number theory ask for the size of the largest subset of the integers in an interval lacking points in a fixed arithmetically defined pattern. One example of such a question is, “What is the largest subset of {1,…,N} with no nontrivial k-term arithmetic progression x,x+y,…,x+(k-1)y?”. Gowers initiated the study of higher order Fourier analysis while seeking to answer this question, and used it to give the first good upper bounds for arbitrary k. In this talk, I will discuss recent progress on polynomial and multidimensional variants of this question and on related problems in additive combinatorics, harmonic analysis, and ergodic theory, and explain what higher order Fourier analysis is and why it is relevant to the study of certain arithmetic patterns.
May 6, 2022
Friday Mornings Friday Morning Seminar 9:00am -

We discuss topics of common interest in the areas of geometry, probability, and combinatorics.
Geometric Analysis and Application Kähler-Einstein metrics on complex hyperbolic cusps 2:00pm -

Abstract:

A complex hyperbolic cusp is an end of a finite-volume quotient of complex hyperbolic space. Up to a finite cover, any such cusp can be realized as the punctured unit disk bundle of a negative line bundle over an abelian variety. The Dirichlet problem for complete Kähler-Einstein metrics on this space with boundary data prescribed on the unit circle bundle is well-posed. We determine the precise asymptotics of its solutions towards the zero section. Time permitting I will also mention an application to gluing constructions for Kähler-Einstein metrics on surfaces of general type. This is joint work with Xin Fu and Xumin Jiang.