# Selberg integral and polynomial identities

Fedor Petrov

## Institution:

Steklov Institute of Mathematics at St. Petersburg

## Time:

Monday, February 10, 2020 - 3:00pm to 3:50pm

## Location:

RH 306

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. We discuss a general method based on the multi-dimensional polynomial interpolation identity related to the so called Combinatorial Nullstellensatz of Alon that is powerful enough to establish many such identities, both known before and new, in a simple manner. Based on joint works with R. Karasev, G. Karolyi, Z. Nagy and V. Volkov.

# Phase transition of capacity for the uniform $G_{\delta}$-sets and another counterexample to Nevanlinna's conjecture

## Speaker:

Fernando Quintino

UCI

## Time:

Wednesday, January 15, 2020 - 3:00pm

## Location:

340P

We consider a family of dense $G_{\delta}$ subsets of $[0,1]$, defined as intersections of unions of small uniformly distributed intervals, and study their capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a $G_{\delta}$ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products.

The same mass re-distribution construction that we use to obtain a full capacity statement, allows us to construct another counter-example to a conjecture by Nevanlinna.

# Fourier analysis and discrete structures

## Speaker:

Paata Ivanishvili

UCI

## Time:

Thursday, December 5, 2019 - 11:00am

## Location:

RH 306

Let a_ij be a finite collection of real numbers, and let s_i and s_j be spins (they take only two values +1 or -1). The goal is to choose the spins s_i and s_j so  that to maximize the double sum a_ij s_i s_j, where the summation runs over all indecies from 1 to n such that i does not qual to j.  If a_ij =1 then the sum is maximized if all spins have the same sign which gives the result to be of order n^2. However, if a_ij=-1 then the sum is maximized when half of the spins take value 1 and the other half -1. When a_ij are arbitrary real numbers this becomes a nontrivial problem (also known as Dean's problem). The problem is well understood  in average when a_ij are i.i.d symmetric  +1 or -1 random vairables  (the Sherrington--Kirkpatrick model).  We will speak about possible  lower bounds of  the maximum in terms of arbitrary real numbers a_ij,  its extensions to d-spin case, some conjectures in this area, and their applications in quantum computing.

# Geometry of nodal domains

## Speaker:

Alexander Logunov

## Institution:

Princeton University

## Time:

Thursday, December 19, 2019 - 11:00am

## Location:

RH 306

Courant's theorem states that the k-th eigenfunction of the Laplace operator on a closed Riemannian manifold has at most k nodal domains. Given a ball of radius r, we will discuss how many of nodal domains can intersect a ball (depending on r and k). Based on a joint work (in progress) with S.Chanillo and E.Malinnikova.

# Volume of intersections of convex bodies with their symmetric images and efficient coverings

Tomasz Tkocz

## Institution:

Carnegie Mellon University

## Time:

Thursday, December 12, 2019 - 11:00am

## Location:

RH 306

Let K be a convex body with volume one and barycentre at the origin.
How small is the volume of the intersection of K and -K? I shall
discuss such lower bounds and present applications to the Hadwidger
covering/illumination conjecture. Based on joint work with H. Huang,
B. Slomka and B. Vritsiou.

# The inverse spectral problem for ellipses

Hamid Hezari

UC Irvine

## Time:

Thursday, November 21, 2019 - 11:00am to 12:00pm

## Location:

RH 306

In this talk I will discuss the recent developments in the inverse spectral theory of bounded planner domains with strictly convex smooth boundaries. I will present a joint work with Steve Zelditch in which we prove ellipses of small eccentricity are Laplace spectrally unique among all smooth domains.