# Research

## Classical Analysis

### Friday, April 28, 2017

Title: Fatou's interpolation theorem and related questions
Speaker: Arthur Danielyan
Time:4:00pm–5:00pm
Place: CMC 204

#### Abstract

Fatou's interpolation theorem has many important applications in complex analysis. In this talk a review of its some new applications will be given. A new extension of Fatou's theorem will be presented as well. We also consider several interpolation problems in the unit disc, when the interpolation of a continuous function on a closed subset of the unit circumference is required to be done by the limiting values of a bounded analytic function.

### Friday, April 21, 2017

Title: On polynomials with vanishing Hessian and some density problems
Speaker: Boris Shekhtman
Time:4:00pm–5:00pm
Place: CMC 204

#### Abstract

In the talk I will introduce an intriguing conjecture regarding properties of partial derivatives of homogeneous polynomials.

The conjecture is accessible to anyone with the knowledge of Calculus III.

I will explain the relationship between the conjecture and the subjects mentioned in the title and provide some partial results. The rest of the time, if any, I will beg for help with the conjecture.

This is a joint work with Tom McKinley.

### Friday, April 14, 2017

Title: Zero distribution for Angelesco Hermite-Padé polynomials, III
Speaker: E. A. Rakhmanov
Time:4:00pm–5:00pm
Place: CMC 204

### Friday, April 7, 2017

Title: Zero distribution for Angelesco Hermite-Padé polynomials, II
Speaker: E. A. Rakhmanov
Time:4:00pm–5:00pm
Place: CMC 204

### Friday, March 31, 2017

Title: Zero distribution for Angelesco Hermite-Padé polynomials
Speaker: E. A. Rakhmanov
Time:4:00pm–5:00pm
Place: CMC 204

#### Abstract

We consider the problem of zero distribution of Hermite-Padé polynomials associated with a vector function $$\vec f=\left(f_1,\dotsc,f_s\right)$$ whose each component $$f_k$$ is defined as element at infinity which has unlimited analytic continuation in plane with a deleted finite set (so, each component has a finite number of branch points). We assume that branch sets $$e_k$$ of component functions are well enough separated (which constitute the Angelesco case). Under this condition polynomials have limit zero distribution.

The limit measures may be defined in terms of a vector equilibrium problem determined by $$\vec f$$ (or, rather, by sets $$e_k$$). There is an equivalent characterization in terms of a third kind Abelian integral on an algebraic Riemann surface determined by branch sets too. We will explain definitions and go into some details.

### Friday, March 24, 2017

Title: Extremal hypergeometric functions and nonlinear damping on the real line
Speaker: Razvan Teodorescu
Time:4:00pm–5:00pm
Place: CMC 204

#### Abstract

I will describe an (open) extremal problem on the space of measures induced by hypergeometric functions, and associated best approximations. I will then show how this problem leads to a curious “selection” theorem and an even curiouser property of classical orthogonal polynomials.

The (open) problem is joint work with Ar. Abanov (Texas A&M). The property was last rediscovered in 2006.

### Friday, March 17, 2017

Spring Break — no seminar this week.

### Friday, March 3, 2017

Title: Bounded Point Derivations on Certain Function Algebras
Speaker: James Brennan, University of Kentucky
Time:4:00pm–5:00pm
Place: CMC 204

<4>Abstract

Let $$X$$ be a compact nowhere dense subset of the complex plane $$\mathbb{C}$$, let $$C(X)$$ be the linear space of all continuous functions on $$X$$ endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in $$\mathbb{C}$$. Denote by $$R(X)$$ the closure in $$C(X)$$ of the set of all rational functions having no poles on $$X$$. It is well-known that if $$X$$ is suficiently massive, then the functions in $$R(X)$$ can inherit many of the properties usually associated with functions smooth in a neighborhood of $$X$$ such as various degrees of differentiability, and even the uniqueness property the analytic functions itself. In this talk I shall examine the extent to which some of those properties are inherited by the larger algebra $$H^\infty(X)$$, which by definition is the weak-$$*$$ closure of $$R(X)$$ in $$L^\infty(X)=L^\infty(X,dA)$$.

### Friday, February 24, 2017

Title: Distribution of critical points of polynomials
Speaker: Vilmos Totik
Time:4:00pm–5:00pm
Place: CMC 204

#### Abstract

The distribution of the critical points (zeros of the derivatives) of polynomials is considered. Some history related to Springer's conjecture will be reviewed, and then results will be given on the asymptotic distribution of critical points when the distribution (say $$m$$) of the zeros is given. One result is that if the support of $$m$$ has connected complement, then the distribution of the critical points is again $$m$$ (this need not be the case when the complement is disconnected, as the example $$z^n-1$$, $$n=1,2,\dotsc$$ shows).

### Friday, February 17, 2017

Title: Weighted planar orthogonal polynomials with logarithmic potential as multiple orthogonal polynomials of Type II, Part II
Speaker: Ming Yang
Time:4:00pm-5:00pm
Place: CMC 204

### Friday, February 10, 2017

Title: Weighted planar orthogonal polynomials with logarithmic potential as multiple orthogonal polynomials of Type II
Speaker: Ming Yang
Time:4:00pm-5:00pm
Place: CMC 204

#### Abstract

When the potential is given by $$Q(z)=|z|^2+\sum\alpha_j\log\left|z-a_j\right|$$, we will write down the Riemann-Hilbert problem satisfied by the planar orthogonal polynomials when there are two log-singularities.

### Friday, February 3, 2017

Title: Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed, III
Speaker: Catherine Bénéteau
Time:4:00pm-5:00pm
Place: CMC 204

### Friday, January 27, 2017

Title: Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed, II
Speaker: Catherine Bénéteau
Time:4:00pm-5:00pm
Place: CMC 204

### Friday, January 20, 2017

Title: Optimal polynomial approximants, connections to Jacobi matrices and orthogonal polynomials: Mysteries Revealed
Speaker: Catherine Bénéteau
Time:4:00pm-5:00pm
Place: CMC 204

#### Abstract

In this talk, I will discuss the zeros of optimal polynomial approximants in Dirichlet-type spaces. I will explain how the smallest modulus of any zero of an optimal approximant is connected to the norm of a special Jacobi matrix constructed from a three-term recursion relation. I will discuss how the solution of this extremal problem differs for spaces such as the Bergman space compared to a spaces such as the Hardy space or the Dirichlet space. Finally, I will discuss results about the accumulation of the zeros of these approximants that are similar to Jentzsch's theorem about the accumulation of the zeros of Taylor polynomials. This talk could also be called: what Daniel did not say.

### Friday, January 13, 2017

Title: Derivatives of rational inner functions
Speaker: Alan Sola, Stockholm University
Time:4:00pm-5:00pm
Place: CMC 204

#### Abstract

Rational inner functions are important examples of bounded analytic functions in polydisks. I will present several results concerning boundary behavior of rational inner functions, with a focus on integrability properties of their derivatives.

This reports on joint work with K. Bickel and J. Pascoe.