# Research

## Analysis

### Friday, April 4, 2008

Title: Lubinsky’s approach to Universality for General Measures
Speaker: Marty Findley
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

Recently D. S. Lubinsky discovered a simple relationship between reproducing kernels and Christoffel functions for general measures supported on the interval $$[-1,1]$$. With it, he establishes universality for measures whose weights are positive and continuous. We will discuss his approach and an extension of his result to regular measures with locally Szegö weights as well as an application to the asymptotic spacing of zeros of orthogonal polynomials.

### Friday, March 28, 2008

Title: On equilibrium problems for the logarithmic potential, Part III
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 108

### Friday, March 7, 2008

Title: Geometry of the unit ball of $$L_p$$ and minimal projections
Speaker: Lesław Skrzypek
Time: 3:00pm–4:00pm
Place: PHY 108

### Friday, February 29, 2008

Title: On equilibrium problems for the logarithmic potential, Part II
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 108

### Friday, February 22, 2008

Title: On equilibrium problems for the logarithmic potential, Part I
Speaker: E. A. Rakhmanov
Time: 4:00pm–5:00pm
Place: PHY 108

#### Abstract

We will start with a historic introduction related to two things: (a) Equilibrium (Robin) measure of a compact in complex plane and its applications to investigation of asymptotics for extremal polynomials. (b) Fekete points and their connections (in particular to classical orthogonal polynomials).

The main concept here is an equilibrium measure as minimizing a logarithmic energy functional (typical result: “discrete equilibrium is close to the continuous one”). More recently some approximation problems were investigated by a reduction to a new kind of equilibrium distributions — saddle points of the energy functional (equilibriums are unstable). In turn, there are important connections of these new equilibrium problems and the classical problems and tools of geometric function theory (moduli problems, quadratic differentials and so on).

### Friday, February 1, 2008

This week's seminar has been replaced by a colloquium.

### Friday, January 25, 2008

Title: Limits of Lagrange Projections
Speaker: Boris Shekhtman
Time: 3:00pm–4:00pm
Place: PHY 108

#### Abstract

This is a joint work with Carl de Boor. The fact that every ideal projector in two variables is a limit of Lagrange projectors was proved using non-trivial facts from algebraic geometry. Recently we found a completely elementary proof of this result using nothing more than some elementary facts from linear algebra. I am hoping to present this proof with all the necessary details.