# Research

## Analysis

### Friday, April 23, 2010

Title: The density problem for self-commutators of Toeplitz operators, Part II
Speaker: Sherwin Kouchekian
Time: 4:00pm–5:00pm
Place: PHY 108

### Friday, April 9, 2010

Title: The density problem for self-commutators of Toeplitz operators
Speaker: Sherwin Kouchekian
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

We will recall the Berger-Shaw Theorem in the bounded case and its obtained extension to unbounded case. Then we will discuss the obtained result for the density problem of the self-commutator of an unbounded Toeplitz operator. (Some of the results is joint work with J. Thomson.)

### Friday, March 26, 2010

Title: Malmheden’s theorem revisited
Speaker: Dmitry Khavinson
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

In 1934 Harry Malmheden discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin 23 years later, it still does not seem to be widely known. In this talk we return to Malmheden’s theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in $$R^n$$.

Joint work with M. Agranovsky and H. S. Shapiro.

### Friday, February 26, 2010

Title: On a polynomial approximation problem of L. Zalcman
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

The purpose of this talk is the solution of a problem on the uniform polynomial approximation of a continuous function on an arbitrary closed subset of the unit circle (in the complex plane) such that the approximating polynomials are uniformly bounded on the unit circle. In a particular case the problem has been solved by L. Zalcman in 1982. As a further application of the main result of this talk we also present a new proof for the classical interpolation theorem of W. Rudin and L. Carleson.

### Friday, February 12, 2010

Title: Univalent Harmonic Mappings
Speaker: Allen Weitsman, Purdue University
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

In my first talk I gave a general outline of problems and approaches to problems involving minimal graphs. One approach was via univalent harmonic mappings.

In the second talk I showed how the conformal invariant extremal length can be used to study minimal graphs. Again, the univalent harmonic mappings were a part of this.

In the third talk, I will focus more on the univalent harmonic mappings. I will show how function theoretic properties of these mappings deliver results in the theory of minimal surfaces, and conversely how minimal surfaces could potentially be used to study univalent harmonic mappings.

### Friday, February 5, 2010

Title: Viscosity solutions to the Minimal Surface Equation
Speaker: Tom Bieske
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

In this talk, we will compare viscosity solutions, weak solutions, to classical solutions of the minimal surface equation.

### Friday, January 29, 2010

Title: Spiraling Minimal Graphs
Speaker: Allen Weitsman, Purdue University
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

A graph given by $$z=u(x,y)$$ over an unbounded domain $$D$$ will be called “spiraling” if a boundary component of $$D$$ is an arc on which $$\arg z$$ tends to infinity. I will show that spiraling minimal graphs exist, and prove that the rate of spiraling must be limited.

Although the problem is somewhat specialized, the methods may be of general interest.

The construction is achieved through quasiconformal approximation and the proof is carried out by the method of extremal length.

### Friday, January 22, 2010

Title: Minimal graphs in $$R^3$$
Speaker: Allen Weitsman, Purdue University
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

There are many ways to study the minimal surface equation. I will discuss a number of problems and the different approaches. Of special interest are growth rates of solutions to the minimal surface equation, and shapes of the resulting surfaces.

The techniques include function theoretic, potential theoretic, differential geometric methods as well as nonlinear pde methods.

This will be a general survey and is intended for a general audience.