# Research

## Analysis

### Friday, November 16, 2018

Title: Connections between tau functions and the Kontsevich matrix integral, Part II
Speaker: Nathan Hayford
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, November 9, 2018

Title: Connections between tau functions and the Kontsevich matrix integral
Speaker: Nathan Hayford
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

It has been known since the 90’s that the tau function for the KdV hierarchy is equivalent to the partition function for a certain matrix model. Because of the broad scope of problems random matrix theory is capable of addressing (2D quantum gravity, combinatorics, quantum computing, etc.), results of this form are particularly relevant, in that they provide an analytic tool for computing matrix integrals.

By way of example, I will attempt to explain what a tau function is and why knowledge of it is useful; the example shall come from the theory of linear ordinary differential equations with Fuchsian singularities. I will also attempt to establish (or at least sketch) the connection of such a tau function to the Kontsevich matrix integral, first introduced in [1]. Finally, if time permits, I’d like to try and sell the utility of this result by providing a few explicit examples of where this matrix integral has made an impact.

### Friday, October 28, 2018

Title: Regularity results for weak solutions of $$p$$-Laplacian-type equations, Part II
Speaker: Diego Ricciotti
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 19, 2018

Title: Regularity results for weak solutions of $$p$$-Laplacian-type equations
Speaker: Diego Ricciotti
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Minimizing integral functionals of the calculus of variations with $$p$$-growth in Sobolev spaces leads to consider weak solutions to equations of $$p$$-Laplacian type. A focal point of the theory is then to investigate additional regularity properties possessed by such solutions. In this regard, after presenting an overview of results and techniques developed for the $$p$$-Laplace equation, I will discuss some recent advances pertaining ‘anisotropic’ $$p$$-Laplacian-type equations.

### Friday, October 12, 2018

Title: Some theorem(s) from the spectral theory of Sturm–Lioulille operator, Part III
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 5, 2018

Title: Some theorem(s) from the spectral theory of Sturm–Lioulille operator, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 21, 2018

Title: Some theorem(s) from the spectral theory of Sturm–Lioulille operator
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We will discuss (mainly discrete) Sturm-Lioulille operators, however most part of basic theorems are similar. One of the most interesting recent advances is a C. Remling theorem a.k.a Breimesser–Pearson theorem, a.k.a Denisov–Rakhmanov–Remling theorem. Loosely speaking, the theorem says that if the potential of an operator is obtained by certain limiting process, then its spectral measure is reflectionless. The “mess” in the names shows that there are, at least, several connections (versions).

I will try to explain some of them.

### Friday, September 14, 2018

Title: Some thoughts on uniform approximation by polyanalytic functions, Part II
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 7, 2018

Title: Some thoughts on uniform approximation by polyanalytic functions, Part I
Speaker: Dima Khavinson
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We shall discuss the problem of uniform approximation of continuous functions on a compact subset $$K$$ of the plane by the so-called rational modules, consisting of functions $$f$$ satisfying $$(d/dz*)^n f=0$$, $$n\geq 1$$, near $$K$$. (For $$n=1$$, this is a well-known old problem of approximation by rational functions with poles off $$K$$. For $$n=2$$, the approximants already do not form an algebra, but a module of dimension 2 over $$R(K)$$ with the basis $$1,z*$$, where $$*$$ denotes complex conjugation.) The main idea is to extend the concept of analytic content introduced by DK in '82 for rational approximation to this new situation and reduce the problem to approximating just one particular function.

### Friday, August 31, 2018

Title: Optimal polynomial approximants and simultaneous zero-free approximation
Speaker: Myrto Manolaki
Time: 4:00pm–5:00pm
Place: CMC 108

#### Abstract

Given a Hilbert space $$H$$ of analytic functions on the unit disc and a function $$f \in H$$, a polynomial $$p_n$$ is called an optimal polynomial approximant of degree $$n$$ of $$1/f$$ if $$p_n$$ minimizes the norm of $$pf-1$$ over all polynomials $$p$$ of degree at most $$n$$. Optimal polynomial approximants arise naturally in the study of cyclicity in Dirichlet-type spaces. In this talk, we will focus on the boundary behavior of such approximants in the Hardy space $$H^2$$, and we will discuss an auxiliary result on simultaneous zero-free approximation, which is of independent interest. (Joint work with Catherine Bénéteau, Oleg Ivrii and Daniel Seco.)

### Friday, August 24, 2018

Title: Theorems and problem in harmonic mappings
Speaker: Daoud Bshouty, Technion — Israel Institute of Technology
Haifa, Israel
Time: 4:00pm–5:00pm
Place: CMC 108

#### Abstract

Since de Branges’s proof of the Bieberbach conjecture many complex analysts were driven towards the study of univalent harmonic mappings due to the analogy of this field to univalent analytic functions as presented by Clunie and Sheil-Small in their 1984 paper. For the past 30 years several directions were explored via complex analytic methods. I shall present a few of these techniques, theorems and problems in four topics: zeroes of harmonic polynomials, existence and uniqueness of the Riemann mapping theorem, univalent harmonic mappings with finite Blaschke dilatations, and boundary behavior of univalent harmonic mappings.