Michelle Chu & Marcus Michelen : RAP Colloquium I/II

Posted by Ian Tobasco , part of the Departmental Colloquium.

At

Sept. 20, 2019, 3 p.m.

In

636 SEO

Abstract

This is the first of two colloquia highlighting our new Research Assistant Professors. Titles/abstracts are as follows: ------ (Michelle) Title: Arithmetic hyperbolic 3-manifolds Abstract: The study of virtual properties of 3-manifolds groups has played a key role in major recent developments in 3-manifold topology. In this talk I will motivate and introduce the study of arithmetic hyperbolic manifolds and discuss some recent results on quantifying their virtual properties. ------ (Marcus) Title: Zeros of Polynomials and Central Limit Theorems Abstract: Let $f$ be a polynomial with non-negative real coefficients. Pemantle conjectured that if $f$ has no roots close to $1 \in \mathbb{C}$, then the coefficients of $f$ roughly trace out a Gaussian bell curve. In the language of probability, this says that the random variable $X$ defined by $$\frac{f(z)}{f(1)} = \sum_k \mathbb{P}(X = k)z^k $$ is close to a normal variable provided the variance of $X$ is large and $f$ has no roots near $1$. I will discuss a complete resolution of this conjecture in a strong quantitative form. Additionally, if $f$ has no roots with small argument, then $X$ must be approximately normal, again in a sharp quantitative form. Time permitting, I will discuss an application of these results to probability and combinatorics. This talk is based on joint work with Julian Sahasrabudhe.