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Mathematics & Statistics

Classical Analysis (Leader: Prof. Dmitry Khavinson )

Friday, April 20, 2007

Title
Speaker
Time
Place

General weighted inequalities and their applications
1:00pm-2:00pm
PHY 118

Abstract

We will discuss some weighted inequalities for complex vectors and complex-valued functions. Applications include: integral and coefficient convolutions; Borel-Laplace transform; generalized hypergeometric series, special functions and orthogonal polynomials; binomial coefficients; bi-Hermitian forms; integro-differential inequalities; weighted norm inequalities; conformal mappings; entire functions and formal power series.

Friday, April 13, 2007

Title
Speaker
Time
Place

Nonlinear extremal problems in Bergman spaces, Part II
Catherine Bénéteau
1:00pm-2:00pm
PHY 118

Friday, April 6, 2007

Title
Speaker
Time
Place

Nonlinear extremal problems in Bergman spaces
Catherine Bénéteau
1:00pm-2:00pm
PHY 118

Abstract

In this talk, I will survey a large class of nonlinear extremal problems in Hardy and Bergman spaces. I will discuss the general approach to such problems in Hardy spaces developed by S. Ya. Khavinson in the `60s and will talk about more recent results in Bergman spaces. Finally, I will formulate some “Kryz”-type conjectures for non-vanishing functions in Bergman spaces.

Friday, March 30, 2007

Title
Speaker

Time
Place

Polynomials with prescribed zeros and small norm
Peter Varju
University of Szeged
Szeged, HUNGARY
1:00pm-2:00pm
PHY 118

Abstract

According to a result of Halasz there exist monic polynomials $$P_n$$ of degree $$n$$ such that they vanish at $$1$$, and their supremum norm on the unit circle is $$<1+C/n$$. It immediatly follows that if we are given $$k < n^{1/2}$$ points on the unit circle, then there is a polynomial $$P_n$$ which vanishes at those points and $$|P_n(z)| < 1+O(k_2/n)$$ for $$|z|=1$$. We discuss the question if this estimate can be improved. Such polynomials are used in Turan's power sum method in number theory.

Friday, March 23, 2007

Title
Speaker
Time
Place

Differentially-invariant linear spaces of multivariate polynomials
Wen-Xiu Ma
1:00pm-2:00pm
PHY 118

Abstract

Motivated by a problem of Boris Shekhtman on existence of sub-spaces with special characteristics in two variables, we analyze sub-spaces of differentially-invariant linear spaces of multivariate polynomials and develop ways to extend given differentially-invariant linear spaces by creating new independent polynomials. This is a preliminary report.

Friday, March 9, 2007

Title
Speaker
Time
Place

Analysis in Sub-Riemannian Spaces, Part II
Thomas Bieske
1:00pm-2:00pm
PHY 118

Friday, March 2, 2007

Title
Speaker
Time
Place

Analysis in Sub-Riemannian Spaces
Thomas Bieske
1:00pm-2:00pm
PHY 118

Abstract

The Euclidean space $$R^n$$ and a set of vector linearly independent vector fields $$X_1,X_2,\dotsc,X_m$$ with $$m < n$$ form a sub-Riemannian structure if the vector fields and their Lie brackets span $$R^n$$. Two classic examples of such spaces include the Heisenberg group, which possesses an algebraic group law, and Grushin spaces, which do not. We will examine these spaces in terms of geometry, potential theory and partial differential equations.

Friday, February 9, 2007

Title
Speaker
Time
Place

Smooth Equilibrium Measures and Approximation
Vilmos Totik
1:00pm-2:00pm
PHY 118

Abstract

Approximation by weighted polynomials where the weight changes with the degree has been thoroughly investigated in the last two decades. This talk will present the problem, its history, its relation to potential theory, and a recent breakthrough which solves the problem completely.

Friday, February 2, 2007

Title
Speaker
Time
Place

Parametrization of Ideal Projections, Part II
Boris Shekhtman
1:00pm-2:00pm
PHY 118

Friday, January 26, 2007

Title
Speaker
Time
Place

Parametrization of Ideal Projections
Boris Shekhtman
1:00pm-2:00pm
PHY 118

Abstract

We develop a system of parameters that describe a family of ideal projections in two variables and show that no such system exists in three or more variables. These results are motivated (and are equivalent to) one problem of Carl de Boor.