# Research

## Analysis

### Friday, April 12, 2013

Title: On the Putnam Inequality for Toeplitz Operators on the Bergman Space, Part II
Speaker: Matt Fleeman
Time: 4:00pm–5:00pm
Place: CMC 130

### Friday, April 5, 2013

Title: On the Putnam Inequality for Toeplitz Operators on the Bergman Space
Speaker: Matt Fleeman
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

I will show that in a Bergman Space setting the Putnam Inequality can not be sharp for Toeplitz Operators with analytic symbol. This is in contrast to the Hardy Space setting, which I will also briefly discuss.

### Friday, March 29, 2013

Title: On regularity of multivariate Birkhoff interpolation
Speaker: Boris Shekhtman
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

I will present a solution to a conjecture of Ron-Qing Jia and A. Sharma regarding regularity of certain multivariate Birkhoff interpolation schemes. While originally conjectured for polynomials over reals, the conjecture turns out to be true over the complex field and false over the real field. Time permitted, I will discuss some related issues regarding multivariate interpolation and subspace arrangements.

### Friday, March 22, 2013

Title: Cyclicity in Dirichlet-type spaces and extremal polynomials
Speaker: Catherine Bénéteau
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

For functions $$f$$ in the Dirichlet-type spaces $$D$$, I will study how to determine optimal polynomials $$p$$ that minimize the norm of $$pf-1$$ among all polynomials $$p$$ of degree at most $$n$$. I will then give upper and lower bounds for the rate of decay of this “best approximation” as the degree of the polynomials approach infinity. Further, I will examine a generalization of a weak version of the Brown-Shields conjecture and, if time, will discuss some computational phenomena about the zeros of optimal polynomials for some special cases.

This talk is a continuation of a talk I gave in the analysis seminar last semester.

### Friday, March 8, 2013

Title: Variants of Bernstein's inequality
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

101 years ago S. N. Bernstein proved his famous inequality on the derivative of trigonometric polynomials. The talk (just missing the centenary) will discuss some history and some generalizations which turn out to be ...

### Friday, March 1, 2013

Title: A solution to Sheil-Small's harmonic mapping problem for Jordan polygons
Speaker: Allen Weitsman, Purdue University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Simple as they are, Poisson integrals of (complex) step functions play a prominent role in the theory of univalent harmonic mappings, and also in the study of the minimal surface equation. In both cases, the important consideration is whether or not the mappings so defined are univalent.

I will discuss the problem first in the setting of minimal graphs, then explain the general problem. The solution to this problem involves a new elementary univalence criterion.

### Friday, February 1, 2013

Title: Approximation by meromorphic matrix-valued functions
Speaker: Alberto Condori, Florida Gulf Coast University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

Given an “admissible” bounded matrix-valued function $$\Phi$$ defined on the unit circle, it is well-known that there exists a unique “superoptimal” meromorphic approximant $$Q$$ to $$\Phi$$ with a fixed number of poles in the unit disc. In this talk, I will give an introduction to this area of research and discuss the role of Hankel and Toeplitz operators on this topic. In particular, I will prove that the Toeplitz operator with symbol $$\Phi-Q$$ is Fredholm and compute its index with the help of Hankel operators.

### Friday, January 25, 2013

Title: AOn complex Banach manifolds similar to Stein manifolds
Speaker: Imre Patyi, East Carolina University
Time: 4:00pm–5:00pm
Place: CMC 130

#### Abstract

We give an abstract definition, similar to the axioms of a Stein manifold, of a class of complex Banach manifolds in such a way that a manifold belongs to the class if and only if it is biholomorphic to a closed split complex Banach submanifold of a separable Banach space.