# Research

## Analysis

### Friday, April 18, 2014

Title: On the peak points
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We consider the well-known uniform algebra of all complex functions which are continuous on a given compact set and analytic in the interior of the set. It will be shown that some classical results of E. Bishop directly imply that each boundary point of a compact set with connected complement is a peak point for the mentioned uniform algebra. An application of a result of Bishop for a known problem (related to Mergelyan’s theorem) on the so-called zero free polynomial approximation will be presented as well.

### Friday, April 11, 2014

Title: Three problems of A. Kroo on multiple Chebyshev polynomials
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Chebyshev polynomials are fundamental in many questions of mathematics. In the case when there are several weight functions present, their so called multiple version have been recently introduced in analogy with multiple orthogonality. We shall review these concepts and show how to solve three problems of A. Kroo about the existence and unicity of multiple Chebyshev polynomials.

### Friday, March 28, 2014

Title: A Remarkably Successful Mathematical Model of a Semiconductor Diagnostic Tool
Speaker: A. David Snider, Professor Emeritus
Electrical Engineering, USF
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The corona-Kelvin technique involves electrostatically focusing a beam of ions onto a small patch of surface of a silicon dioxide wafer and tracing the voltage behavior as the ions diffuse away from the target. The voltage-time graph has defied description by any of the known patterns. We devised a physical-mathematical model of the process and successfully shepherded the complicated analytics into revealing the nature of the graph, including a novel reparametrization that renders the measurements along a straight line whose slope and intercept enable the extraction of the diffusion coefficient and the layer thickness.

### Friday, February 28, 2014

Title: Cyclic Functions and Extremal Polynomials in the Dirichlet Spaces of the Bidisk, Part II
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, February 21, 2014

Title: Cyclic Functions and Extremal Polynomials in the Dirichlet Spaces of the Bidisk
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk, I will discuss Dirichlet-type spaces of analytic functions in the unit bidisk and their cyclic elements. These are the functions $$f$$ whose polynomial multiples generate the whole space. I will build on results discussed previously for one variable, and show how to use comparisons between norms in one and two variables to get some information about cyclic functions and “rates of best approximation” for certain good functions in the bidisk. In particular, a somewhat surprising and interesting phenomenon occurs in the bidisk that doesn't occur in one variable: there are examples of polynomials with no zeros on the bidisk that are not cyclic in the (classical) Dirichlet space of the bidisk. If time permits, I will point out the necessity of a capacity zero condition on zero sets (in an appropriate sense) for cyclicity in the setting of the bidisk.

### Friday, February 14, 2014

Title: Two-dimensional Coulomb gas by orthogonal polynomials
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

I will talk about some approaches to study “two-dimensional Coulomb gas on the plane” using orthogonal polynomials. The corresponding orthogonal polynomials have weights that varies over the plane. Strong asymptotics for such orthogonal polynomials are not known in general.

However there are a few cases where the strong asymptotics can be found and, in such cases, the behavior of Coulomb gas can be studied in details. I will mention some results with Roman Riser about the case with Hermite polynomials. Then I will mention a few open problems, and a general (yet unsuccessful) approach using Dbar problem.

### Friday, January 31, 2014

Title: Deformation of equilibrium measures as a dynamical process
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The classical problem for the equilibrium measure on arcs in the complex plane, in the presence of an external potential, has been linked to several other types of spectral problems, from approximation theory to nonlinear hyperbolic equations. When the support $$S$$ is not restricted to segments on the real line, the dependence of the boundary of $$S$$ on the total mass (and parameters of the external potential) can be regarded as a dynamical process. We will discuss several recent results in this direction and their applications.

### Friday, January 24, 2014

Title: TBA
Speaker: Razvan Teodorescu
Time: 4:00pm–5:00pm
Place: CMC 130

TBA

### Friday, January 17, 2014

Title: A convexity property of discrete random walks
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We discuss the discrete analogue of a recent result about the convexity of the densities of certain harmonic measures. The discrete statement is actually stronger than the continuous theorem. The higher dimensional case is also considered, when the statement is that the densities of certain harmonic measures are subharmonic. (Joint work with G. V. Nagy.)