Research Seminars @ Illinois

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Tailored for undergraduate researchers, this calendar is a curated list of research seminars at the University of Illinois. Explore the diverse world of research and expand your knowledge through engaging sessions designed to inspire and enlighten.

To have your events added or removed from this calendar, please contact OUR at ugresearch@illinois.edu

Probability Seminar

Event Type
Seminar/Symposium
Sponsor
Department of Mathematics
Location
149 Henry Administration Bldg
Date
Nov 14, 2024   2:00 - 2:50 pm  
Speaker
Promit Ghosal (UChicago)
Contact
Xuan Wu
E-Mail
xuanw@illinois.edu
Views
35
Originating Calendar
Probability Research Area Calendar

Title:    Universality of Persistence of Random Polynomials.

Abstract:    Consider random polynomials of the form $\sum_{i=0}^n a_ix^i$ where the coefficients $a_i$ are independent with mean zero and finite second moments. We discuss how the probability that such a polynomial (when $n$ is even) has no zeros decays as $n^{-b+o(1)}$, where $b$ is a universal constant independent of the distributions of $a_i$. This resolves a conjecture by Poonen and Stoll. Notable progress on this conjecture was made by Dembo, Poonen, Shao, and Zeitouni, who demonstrated the same result when coefficients are identically and independently distributed (i.i.d.) with all higher moments being finite. Extending their results to the finite moment case remained an open problem until now. Our approach, in contrast to previous works in this area, relies on the combinatorics of multiscale analysis of random polynomials. This work is based on joint collaboration with Sumit Mukherjee.

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