# Research

## Analysis

### Friday, December 6, 2013

Title: On minimal Haar covering of $$d$$-dimensional spaces, Part II
Speaker: Boris Shekhtman
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, November 29, 2013

Thanksgiving holiday — no seminar this week.

### Friday, November 22, 2013

Title: On minimal Haar covering of $$d$$-dimensional spaces
Speaker: Boris Shekhtman
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

An $$n$$-dimensional subspace $$H$$ of $$C(K)$$ is called Haar if every non-zero function in $$H$$ has at most $$(n-1)$$ zeroes. Space of polynomials of degree $$n$$ is the prime example of a Haar space.

It is well-known that Haar spaces of functions of several (more than 1) variables do not exist. To compensate for this “loss of Haar” we introduce a notion of “Haar covering” and discuss some questions associated with this notion. In addition to approximation theory these questions are relevant to algebraic geometry (in the complex case) and algebraic topology (in the real case).

### Friday, November 15, 2013

Title: Radial interpolation in the unit disc and related questions
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The talk will present a new approach for the peak-interpolation theorem of E. Bishop. We also discuss further interpolation problems and present new results in particular on the radial interpolation in the unit disc which are related to the Rudin-Carleson theorem.

### Friday, November 8, 2013

Title: Stahl's theorem on convergence of Padé approximants and various attempts to generalize it for Hermite-Padé approximants, Part III
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, November 1, 2013

Title: Groups $$\mathrm{SL}(2,R)$$ and its symmetric spaces
Speaker: Igor Chitikov
Time: 4:00pm–5:00pm
Place: CMC 130

TBA

### Friday, October 25, 2013

Title: Stahl's theorem on convergence of Padé approximants and various attempts to generalize it for Hermite-Padé approximants, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 18, 2013

Title: Two dimensional orthogonal polynomials $$=$$ orthogonal polynomials on complex contours
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Motivated by the normal matrix ensemble with the external potential given by $$Q(z)=|z|^2+(\text{harmonic})$$, we study the special case where the corresponding orthogonal polynomials are equivalent to the orthogonal polynomials that are orthogonal with respect to certain contour integrals.

### Friday, October 11, 2013

Title: Stahl's theorem on convergence of Padé approximants and various attempts to generalize it for Hermite-Padé approximants
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Generalizations did not go too far yet. I am going to discuss some very recent progress in spirit of classical orthogonal polynomials based on DE for first kind Hermite-Padé polynomials for a special class of analytic function.

### Friday, October 4, 2013

Title: Sharpness of connectivity bounds for quadrature domains, Part III
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 27, 2013

Title: Sharpness of connectivity bounds for quadrature domains, Part II
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 20, 2013

Title: Sharpness of connectivity bounds for quadrature domains, Part I
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Given the degree of a rational function, the “possible” topology of the associated quadrature domain can be identified (http://arxiv.org/pdf/1307.0487.pdf). We further claim that there exists a quadrature domain for each possible topology. This is a joint work with Nikolai Makarov.