# Research

## Analysis

### Friday, December 4, 2015

Title: On the Maximal Relative Projection Constants
Speaker: Lesɫaw Skrzypek
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We will investigate the maximal $$m$$-dimensional spaces with regard to the relative projection constant and demonstrate how this notion translates to the problem of maximizing the sum of the m largest eigenvalues. Surprisingly (or not) this problem exhibits interesting connections to other fields like graph theory (Seidel matrices, Taylor graphs and the properties of their adjacency matrix) and equiangular tight frames. Based on our approach we will reprove the famous Kadec–Snobar inequality and its refinements. We will also apply Random Matrix techniques (semicircle law) to partially reverse the Kadec–Snobar inequality.

### Friday, November 20, 2015

Title: Spectral properties of single layer operators
Speaker: Seyed Zoalroshd
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk, we discuss the spectral properties of single layer operators. We give some criteria for Schatten class membership and also also injectivity of layer potentials. Regarding eigenfunctions, we show that polynomial eigenfunctions of layer potentials can, to some extent, determine the underlying curve. If time permits, some open problems will be discussed.

### Friday, November 13, 2015

Title: Approximating $$z$$-bar in the Bergman Space
Speaker: Matt Fleeman
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We consider the problem of approximating $$z$$-bar in the Bergman Space. We characterize the best approximation in terms of the solution of the Dirichlet problem with data $$|z|^2$$ and give some examples. Finally we obtain an “isoperimetric sandwich” for the distance from $$z$$-bar to the Bergman space which yields the celebrated St. Venant inequality.

### Friday, November 6, 2015

Title: Valence of harmonic polynomials and lemniscates
Speaker: Seung-Yeop Lee
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

The maximal valence of harmonic polynomials is an open problem. In a (failed) attempt to solve the problem, we consider the critical lemniscates of the harmonic polynomials. It is an elementary fact (by Sheil-Small) that a convex component of the lemniscate can have only one root of the harmonic polynomial. We explain the necessary and sufficient condition that a lemniscate can contain more than one root. This is a work with Khavinson and Saez.

### Friday, October 30, 2015

Title: Polar Coordinates in spaces of Controlled Geometry
Speaker: Tom Bieske
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We examine when polar coordinates can be constructed for various geometric versions of $$R^n$$. We explore the consequences of having these polar coordinates and discuss the open problem of finding the largest class of spaces where polar coordinates can be constructed.

### Friday, October 23, 2015

Title: Fatou's interpolation theorem implies the Rudin-Carleson theorem
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In this talk it will be shown that the classical Rudin-Carleson interpolation theorem is a direct corollary of Fatou's much older interpolation theorem (of 1906). Some related questions will also be considered.

### Friday, October 16, 2015

Title: Ergodic decomposition of group actions on rooted trees, Part II
Speaker: Dima Savchuk
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, October 9, 2015

Title: Ergodic decomposition of group actions on rooted trees
Speaker: Dima Savchuk
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, will be considered in order to demonstrate applications of the theorem.

### Friday, October 2, 2015

Title: Questions related to ideal interpolation, Part II
Speaker: Boris Shekhtman
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 25, 2015

Title: Questions related to ideal interpolation
Speaker: Boris Shekhtman
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In an effort to solicit comments from the audience, I will start with Jenya’s remark that the Newton series converges to the Taylor series, and discuss its generalization to polynomials of several variables. In particular I will try to outline some open and solved problems related to this subject and their connections to irreducibility of certain affine varieties, numerical approximation of differential operators, properties of sequences of commuting matrices and resolution of singularities.

### Friday, September 18, 2015

Title: Stahl's theorem on convergence of Padé Approximants and beyond, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, September 11, 2015

Title: Stahl's theorem on convergence of Padé Approximants and beyond
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

To state the theorem one needs to start with a discussion of several key concepts of complex analysis and approximation theory, such as Pade Approximations, analytic (multivalued) functions with “small” set of branch points, minimal capacity sets, quadratic differential, etc.

Generalizations to, say, Hermite-Padé Approximants are often discussed, but nothing really general is known. It would require going deep into Riemann surfaces and/or equilibrium problems and/or boundary value problems to formulate even a conjecture on poles distribution in rather simple situations.

### Friday, September 4, 2015

Title: Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants, Part II
Speaker: Alan Sola
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, August 28, 2015

Title: Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants
Speaker: Alan Sola
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

In joint work with Bénéteau, Khavinson, Liaw, and Seco, we consider certain polynomial subspaces of Hilbert function spaces over the disk generated by a fixed function, and study the properties of certain families of polynomials associated with such subspaces. The location of zeros, and how it depends on both the given function and the Hilbert space, is the main focus of our work.