Ruijie Yang (Kansas) Archimedean zeta function and Hodge theory Sponsored by the Meyer Fund
Thu, Jan. 16 3:35pm (MATH 3…
Probability
Jonas Jalowy (University of Münster)
X
How do zeros of polynomials evolve under the action of differential operators? Consider a random polynomial with i.i.d. (rescaled) coefficients and look at the empirical distribution of their complex roots. We investigate the evolution of the roots when the polynomial undergoes the action of differential operators, such as the heat flow and repeated differentiation. In one prominent example of Weyl polynomials undergoing the heat flow, the limiting zero distribution evolves from the circular law into the elliptic law until it collapses to the Wigner semicircle law.
In this talk, I present an overview over results of such type with a focus on complex zeros. We determine the general limiting zero distribution and describe the dynamics of the roots from various points of view such as (optimal) transport, differential equations and free probability. The connection to the latter is particularly fruitful for real rooted polynomials and will be the focus of a second part. This talk will include illustrative simulations leading to intriguing open questions.
This is based on joint works with Brian Hall, Ching Wei Ho, Zakhar Kabluchko and Antonia Höfert.
Evolution of complex zeros of random polynomials