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Mathematics & Statistics

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**Title**

**Speaker**

**Time**

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Two-dimensional orthogonal polynomials: subleading correction in \(1/N\)-expansion

Seung-Yeop Lee

4:00pm-5:00pm

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.

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Zero sets of polynomials on the torus and cyclicity in Dirichlet type spaces

Catherine Bénéteau

4:00pm-5:00pm

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.

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Cyclicity of Polynomials of Two Variables in the Dirichlet Spaces, II

Catherine Bénéteau

4:00pm-5:00pm

CMC 130

**Title**

**Speaker**

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Cyclicity of Polynomials of Two Variables in the Dirichlet Spaces

Catherine Bénéteau

4:00pm-5:00pm

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.

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Deformation theory for algebraic curves and function spaces

Razvan Teodorescu

4:00pm-5:00pm

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.

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On one class of Hermite projections, II

Boris Shekhtman

4:00pm-5:00pm

CMC 130

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On one class of Hermite projections

Boris Shekhtman

4:00pm-5:00pm

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.

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On some extremal problems in geometric function theory and its connections in approximation theory, II

E. A. Rakhmanov

4:00pm-5:00pm

CMC 130

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**Speaker**

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On some extremal problems in geometric function theory and its connections in approximation theory

E. A. Rakhmanov

4:00pm-5:00pm

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.

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Extremal functions in modules of systems of measures, II

Alexander Vasiliev

University of Bergen

4:00pm-5:00pm

CMC 130

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Extremal functions in modules of systems of measures

Alexander Vasiliev

University of Bergen

4:00pm-5:00pm

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.