Analysis and Operator Theory Seminar - Brian C. Hall

Brian
April 17, 2025
12:45 pm - 1:45 pm
Math Tower (MW) 154

Date Range
2025-04-17 12:45:00 2025-04-17 13:45:00 Analysis and Operator Theory Seminar - Brian C. Hall Brian C. HallUniversity of Notre DameTitleEvolution of the zeros of polynomials under repeated differentiationAbstractA basic question in the study of polynomials is how the roots of the derivative of a polynomial are related to the roots of the original polynomial. For high-degree polynomials, various results indicate that the distribution of roots does not change much when taking a single derivative. We will therefore look at how the roots evolve when the number of derivatives is proportional to the degree of the polynomial.In the case of real roots, the evolution of roots under repeated differentiation can be described using a construction from random matrix theory. For complex roots, the situation is more complicated, but still very interesting. I will present recent results of mine with Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko about repeated differentiation of *random* polynomials. In this case, we obtain a very precise description of how the (complex) roots evolve, with connections to PDE and random matrix theory. The talk will be self-contained and have lots of pictures and animations.For More Information About the Seminar Math Tower (MW) 154 America/New_York public

Brian C. Hall
University of Notre Dame

Title
Evolution of the zeros of polynomials under repeated differentiation

Abstract
A basic question in the study of polynomials is how the roots of the derivative of a polynomial are related to the roots of the original polynomial. For high-degree polynomials, various results indicate that the distribution of roots does not change much when taking a single derivative. We will therefore look at how the roots evolve when the number of derivatives is proportional to the degree of the polynomial.

In the case of real roots, the evolution of roots under repeated differentiation can be described using a construction from random matrix theory. For complex roots, the situation is more complicated, but still very interesting. I will present recent results of mine with Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko about repeated differentiation of *random* polynomials. In this case, we obtain a very precise description of how the (complex) roots evolve, with connections to PDE and random matrix theory. The talk will be self-contained and have lots of pictures and animations.

For More Information About the Seminar

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