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Mathematics & Statistics

(Leader: Prof. Dmitry Khavinson )

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General weighted inequalities and their applications

Arcadii Grinshpan

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.

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Nonlinear extremal problems in Bergman spaces, Part II

Catherine Bénéteau

1:00pm-2:00pm

PHY 118

**Title**

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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.

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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.

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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.

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Analysis in Sub-Riemannian Spaces, Part II

Thomas Bieske

1:00pm-2:00pm

PHY 118

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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.

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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.

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Parametrization of Ideal Projections, Part II

Boris Shekhtman

1:00pm-2:00pm

PHY 118

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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.