12:00pm - Zoom - Probability and Analysis Webinar
Rocco Servedio - (Columbia University)
Convex influences and a quantitative Gaussian correlation inequality
The Gaussian correlation inequality (GCI), proved by Royen in 2014, states that any two centrally symmetric convex sets (say K and L) in Gaussian space are positively correlated. We establish a new quantitative version of the GCI which gives a lower bound on this correlation based on the "common influential directions" of K and L. This can be seen as a Gaussian space analogue of Talagrand's well known correlation inequality for monotone Boolean functions.
To obtain this inequality, we propose a new approach, based on analysis of Littlewood type polynomials, which gives a recipe for transferring qualitative correlation inequalities into quantitative correlation inequalities. En route, we also give a new notion of influences for symmetric convex symmetric sets over Gaussian space which has many of the properties of influences of Boolean functions over the discrete cube. Much remains to be explored about this new notion of influences for convex sets.
Based on joint works with Anindya De and Shivam Nadimpalli.
1:00pm to 2:00pm - RH 440R - Dynamical Systems
Victor Kleptsyn - (CNRS, University of Rennes 1, France)
Uniform lower bounds on the dimension of Bernoulli convolutions
For any $q\in (0,1)$, one can consider a geometric series
+1 +q +q^2 +…,
and then toss a coin countably many times to decide whether each sign « + » is kept or is replaced by a minus one. The law of this random variable is given by the stationary measure for the random dynamical system, consisting of two affine maps
x\mapsto \pm 1 + qx,
taken with the probability (1/2) each. This stationary measure is called the Bernoulli convolution measure. It is supported on a Cantor set for $q\in (0,1/2)$, and on an interval for $q\in [1/2,1)$. Its properties — and most importantly, whether it is absolutely continuous or signular — have been studied for many years with many famous works and important recent progress in the domain (Erdos, Solomyak, Shmerkin, Varju, …).
My talk will be devoted to our recent work with P. Vytnova and M. Pollicott (https://arxiv.org/abs/2102.07714). I will present a technique for obtaining a lower bound for the Hausdorff dimension for the stationary measure of an affine IFS with similarities (in particular, affine IFS on the real line).
3:00pm to 4:00pm - RH 306 - Number Theory
Yifeng Huang - (UBC)
Matrix enumeration over finite fields (Note the special day!)
I will investigate certain matrix enumeration problems over a finite field, guided by the phenomenon that many such problems tend to have a generating function with a nice factorization. I then give a uniform and geometric explanation of the phenomenon that works in many cases, using the statistics of finite-length modules (or coherent sheaves) studied by Cohen and Lenstra. However, my recent work on counting pairs of matrices of the form AB=BA=0 (arXiv: 2110.15566) and AB=uBA for a root of unity u (arXiv: 2110.15570), through purely combinatorial methods, gives examples where the phenomenon still holds true in the absence of the above explanation. Time permitting, I will talk about a partial progress on the system of equations AB=BA, A^2=B^3 in a joint work with Ruofan Jiang. In particular, it verifies a pattern that I previously conjectured in an attempt to explain the phenomenon in the AB=BA=0 case geometrically.
2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability
Yin-Ting Liao - (UC Irvine)
Large deviations for projections of high-dimensional measures
Random projections of high-dimensional probability measures have gained much attention in asymptotic convex geometry and high-dimensional statistics. While fluctuations at the level of the central limit theorem have been classically studied, only more recently has an inquiry into large deviation principles for such projections been initiated. In this talk, I will review existing work and describe our results on large deviations. I will also talk about sharp large deviation estimates to obtain the prefactor apart from the exponential decay in the spirit of Bahadur and Ranga-Rao. Applications to asymptotic convex geometry and a range of examples including $\ell^p$ balls and Orlicz balls would be given. This talk is based on several joint works with S. S. Kim and K. Ramanan.
9:00am to 9:50am - Zoom - Inverse Problems
Jesse Railo - (University of Cambridge)
Inverse fractional conductivity problem