Abstracts

Week of October 29, 2023

November 1, 2023
Colloquium p-adic hyperbolicity of Shimura varieties 4:00pm -
KT 219

Abstract: A theorem of Borel says that any holomorphic map from a complex algebraic variety to a smooth arithmetic variety is automatically an algebraic map. The key ingredient is to show that any holomorphic map from the (poly) punctured disc to the Baily-Borel compactification of the arithmetic variety has no essential singularity.

I will discuss p-adic analogue of these facts for Shimura varieties of abelian type. Joint with Abhishek Oswal and Ananth Shankar (with an appendix by Anand Patel).
November 2, 2023
Group Actions, Geometry and Dynamics Cancelled 4:00pm -
KT205
Analysis Discrete restriction estimates for manifolds avoiding a line 4:00pm -
KT 219

We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line.
Learning seminar on D-modules 4:00pm -
KT 801

This is the 7th lecture in the series.
November 3, 2023
Friday Morning Seminar Influence in Mixing Measures 10:00am -
KT 801

Abstract: 

The theory of influences in product measures has profound applications in theoretical computer science, combinatorics, and discrete probability.

This deep theory is intimately connected to functional inequalities and to the Fourier analysis of discrete groups.
Originally, influences of functions were motivated by the study of social choice theory, wherein a Boolean function represents a voting scheme, its inputs represent the votes,
and its output represents the outcome of the elections. Thus, product measures represent a scenario in which the votes of the parties are randomly and independently
distributed, which is often far from the truth in real-life scenarios.

We begin to develop the theory of influences for more general measures under mixing or spectral independence conditions.
More specifically, we prove analogues of the KKL and Talagrand influence theorems for Markov Random Fields on bounded degree graphs
when the Glauber dynamics mix rapidly.
We thus resolve a long standing challenge, stated for example by Kalai and Safra (2005).
We show how some of the original applications of the theory of in terms of voting and coalitions extend to these general dependent measures.
Our results thus shed light both on voting with correlated voters and on the behavior of general functions of Markov Random Fields (also called “spin-systems”) where the Glauber dynamics mixes rapidly.