Departmental Colloquium
Michelle Chu & Marcus Michelen
UIC
RAP Colloquium I/II
Abstract: This is the first of two colloquia highlighting our new Research Assistant Professors. Titles/abstracts are as follows:
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(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.
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(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.
Friday September 20, 2019 at 3:00 PM in 636 SEO