# Research

## Classical Analysis

### Friday, December 5, 2014

Title: Two-dimensional orthogonal polynomials: subleading correction in $$1/N$$-expansion
Speaker: Seung-Yeop Lee
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

Two-dimensional orthogonal polynomials appear naturally in two-dimensional Coulomb gas. We conjecture a general form of the subleading term in the $$1/N$$-expansion of the polynomials. The subleading term is obtained by solving some scalar Riemann-Hilbert problems, assuming certain statistical properties of Coulomb gas. This is a joint work with Roman Riser.

### Friday, November 21, 2014

Title: Zero sets of polynomials on the torus and cyclicity in Dirichlet type spaces
Speaker: Catherine Bénéteau
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

I will continue my talk about cyclicity of polynomials in Dirichlet type spaces. I will define a Riesz capacity on the torus and examine the Riesz capacities of the zero curves of polynomials. I will also investigate a geometric characteristic of curves called “type”, and will discuss the relationship between type, Riesz capacity, and cyclicity.

### Friday, November 14, 2014

Title: Cyclicity of Polynomials of Two Variables in the Dirichlet Spaces, II
Speaker: Catherine Bénéteau
Time:4:00pm–5:00pm
Place: CMC 130

### Friday, October 24, 2014

Title: Cyclicity of Polynomials of Two Variables in the Dirichlet Spaces
Speaker: Catherine Bénéteau
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk, I will give a characterization of polynomials in two complex variables that are cyclic with respect to the coordinate shifts acting on Dirichlet-type spaces in the bidisk, which include the Hardy space and the Dirichlet space of the bidisk. The cyclicity of a polynomial depends on both the size and nature of the zero set of the polynomial on the distinguished boundary. The techniques in the proof come from real analytic function theory, determinantal representations for stable polynomials, and harmonic analysis on curves.

### Friday, October 10, 2014

Title: Deformation theory for algebraic curves and function spaces
Speaker: Razvan Teodorescu
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

When applied to a class of algebraic curves, deformation theory provides an analytic tool to describe points in moduli spaces. It can also be interpreted as a smooth flow on the parameter space of the curve, and therefore extended to function spaces on the same manifold. We will discuss applications of this method ranging from (weak) resolution of singularities, to stability of a class of curves under special deformations. Two specific examples are integrable deformation of conic singularities in two complex variables, and the lack of stability of polynomial lemniscate families.

### Friday, October 3, 2014

Title: On one class of Hermite projections, II
Speaker: Boris Shekhtman
Time:4:00pm–5:00pm
Place: CMC 130

### Friday, September 26, 2014

Title: On one class of Hermite projections
Speaker: Boris Shekhtman
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

A finite-dimensional projection $$P$$ is called ideal if kerP is an ideal. In particular every Lagrange interpolation projection is ideal. An ideal projection is Hermite if it is a limit of Lagrange projection. Approximation of ideal projections by Lagrange projections is a special form of a resolution of singularities of a variety defined by the ker$$P$$.

In three or more dimension not every ideal projection is a limit of Lagrange projections, i.e., not every singularity of a zero-dimensional variety can be resolved this way. Starting with a very elementary introduction to the subject I will describe a special class of ideal projections that are Hermite.

### Friday, September 19, 2014

Title: On some extremal problems in geometric function theory and its connections in approximation theory, II
Speaker: E. A. Rakhmanov
Time:4:00pm–5:00pm
Place: CMC 130

### Friday, September 12, 2014

Title: On some extremal problems in geometric function theory and its connections in approximation theory
Speaker: E. A. Rakhmanov
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

We start with reformulating of a particular class of moduli problems in electrostatic terms. The latter has a discrete version which may be, in turn, reformulated in terms of polynomial solutions of second order linear DE with polynomial coefficients (a.k.a Heine–Stieltjes polynomials; in particular, Padé polynomials). We plan to discuss those connections in some details.

If time allow then the perspectives will be briefly discussed of generalizations to higher order DE, Hermite–Padé polynomials and associated electrostatics.

### Friday, September 5, 2014

Title: Extremal functions in modules of systems of measures, II
Speaker: Alexander Vasiliev, University of Bergen
Time:4:00pm–5:00pm
Place: CMC 130

### Friday, August 29, 2014

Title: Extremal functions in modules of systems of measures
Speaker: Alexander Vasiliev, University of Bergen
Time:4:00pm–5:00pm
Place: CMC 130

#### Abstract

We study Fuglede's $$p$$-modules of systems of measures in condensers in the Euclidean spaces.

First, we generalize the result by Rodin that provides a way to compute the extremal function and the 2-module of a family of curves in the plane to a variety of other settings. More specifically, in the Euclidean space we compute the $$p$$-module of images of families of connecting curves and families of separating sets with respect to the plates of a condenser under homeomorphisms with some assumed regularity. Then we calculate the module and find the extremal measures for the spherical ring domain on polarizable Carnot groups and extend Rodin's theorem to the spherical ring domain on the Heisenberg group. Applications to special functions and examples will be provided.