# Research

## Analysis

### Friday, November 16, 2007

This week's seminar has been replaced by a colloquium.

### Friday, November 9, 2007

Title: Chebyshev constants and the inheritance problem
Speaker: Vilmos Totik
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

Recently a lower bound was given by K. Schiefemayr for the Chebyshev constants corresponding to a set of finite intervals. In this talk we give a matching upper bound with the aid of the statement of a so-called inheritance problem: To divide an inheritance $$m$$ brothers turn to a judge. Secretly, however, each of them bribes the judge. What a given brother inherits depends continuously and monotonically in the bribes: it is monotone increasing in his own bribe and it is monotone decreasing in everyone else's bribe. Show that if the eldest brother does not give too much to the judge, then the others can choose their bribes so that the decision will be fair, i.e., each of them gets the same share as without bribes.

### Friday, October 26, 2007

Title: Variational identities and their applications to soliton equations
Speaker: Wen-Xiu Ma
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

We will show that associated with matrix spectral problems, there exist variational identities involving solutions to stationary zero curvature equations. The resulting variational identities can be used to generate Hamiltonian structures for the corresponding soliton equations.

### Friday, October 19, 2007

Title: Dirichlet's problem and lightning bolts
Speaker: Erik Lundberg
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

If Dirichlet's problem is posed with any entire data on an ellipse, then the solution is known to be entire. We consider other algebraic curves and present a method for locating singularities developed by analytically continuing a solution with entire data. The technique uses annihilating measures supported on finite sets called “lightning bolts” which go back to Kolmogorov and Arnold's solution of Hilbert's 13th problem.

### Friday, October 12, 2007

This week's seminar has been replaced by a colloquium.

### Friday, October 5, 2007

Title: Approximation on Hausdorff compacts and applications
Speaker: Arthur Danielyan
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

We consider a function defined on a compact Hausdorff space and discuss the problem of pointwise approximation of such function by elements of a closed subspace of continuous (on the compact) complex valued functions. The approximation method is universal in the sense that it gives also uniform approximation on certain subsets whenever such approximation is possible at all.

### Friday, September 28, 2007

Title: Unbounded Subnormal Operators, Part III
Speaker: Sherwin Kouchekian
Time: 4:00pm–5:00pm
Place: PHY 108

### Friday, September 21, 2007

Title: Unbounded Subnormal Operators, Part II
Speaker: Sherwin Kouchekian
Time: 4:00pm–5:00pm
Place: PHY 108

### Friday, September 14, 2007

Title: Unbounded Subnormal Operators, Part I
Speaker: Sherwin Kouchekian
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

We start with a short review of some aspects of the theory of subnormal operators. Then we move on to the concept of subnormality in the unbounded case. It will be shown that the defined notion of subnormality is not only an exercise for the sake of generalization, as there are important concrete examples of unbounded subnormal operators. For instance, the famous Creation Operator is such an example. Finally, we will focus our attention to the class of unbounded multiplication operators and, if time permits, present some of our obtained results in this regard.