Seminars and Colloquia by Series

Interpretation of some integrable systems via multiple orthogonal polynomials

Series
Analysis Seminar
Time
Wednesday, April 21, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dolores BarriosPolytechnical University of Madrid
Some discrete dynamical systems defined by a Lax pair are considered. The method of investigation is based on the analysis of the matrical moments for the main operator of the pair. The solutions of these systems are studied in terms of properties of this operator, giving, under some conditions, explicit expressions for the resolvent function.

Tangent cones and regularity of real hypersurfaces

Series
Analysis Seminar
Time
Wednesday, April 14, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mohammad GhomiGeorgia Tech
The tangent cone of a set X in R^n at a point p of X is the limit of all rays which emanate from p and pass through sequences of points p_i of X as p_i converges to p. In this talk we discuss how C^1 regular hypersurfaces of R^n may be characterized in terms of their tangent cones. Further using the real nullstellensatz we prove that convex real analytic hypersurfaces are C^1, and will also discuss some applications to real algebraic geometry.

Extrapolation of Carleson measures

Series
Analysis Seminar
Time
Monday, April 5, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Steven HofmannUniversity of Missouri
We discuss joint work with J.-M. Martell, in which werevisit the ``extrapolation method" for Carleson measures, originallyintroduced by John Lewis to proveA_\infty estimates for certain caloric measures, and we present a purely real variable version of the method. Our main result is a general criterion fordeducing that a weight satisfies a ReverseHolder estimate, given appropriate control by a Carleson measure.To illustrate the useof this technique,we reprove a well known theorem of R. Fefferman, Kenig and Pipherconcerning the solvability of the Dirichlet problem with data in some L^p space.

Interpolation in the Drury-Arveson Space

Series
Analysis Seminar
Time
Wednesday, March 17, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Brett WickGeorgia Tech
The Drury-Arveson space of functions on the unit ball in C^n has recently been intensively studied from the point of view function theory and operator theory. While much is known about this space of functions, a characterization of the interpolating sequences for the space has still remained elusive. In this talk, we will discuss the relevant background of the problem, and then I will discuss some work in progress and discuss a variant of the question for which we know the answer completely.

For compactly supported measures, universality holds in measure

Series
Analysis Seminar
Time
Wednesday, March 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doron LubinskyGeorgia Tech
Let mu be a measure with compact support, with orthonormal polynomials {p_{n}} and associated reproducing kernels {K_{n}}. We show that bulk universality holds in measure in {x:mu'(x)>0}. The novelty is that there are no local or global conditions on the measure. Previous results have required regularity as a global condition, and a Szego condition as a local condition.As a consequence, for a subsequence of integers, universality holds for a.e. x. Under additional conditions on the measure, we show universality holds in an L_{p} sense for all finite p>0.

Hardy Inequalities for Fractional Integrals on Convex Domains

Series
Analysis Seminar
Time
Wednesday, February 24, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Craig SloaneGeorgia Tech
We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a convex domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda. Further, the Hardy term in this inequality is stronger than the one in the classical case. The result can be extended as well to more general domains

Gasper's identity and the Markov sequence problem

Series
Analysis Seminar
Time
Wednesday, February 10, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Jeff GeronimoGeorgia Tech
Gasper in his 1971 Annals of Math paper proved that the Jacobi polynomials satisfy a product formula which generalized the product formula of Gegenbauer for ultraspherical polynomials. Gasper proved this by showing that certains sums of triple products of Jacobi polynomials are positive generalizing results of Bochner who earlier proved a similar results for ultraspherical polynomials. These results allow a convolution structure for Jacobi polynomials. We will give a simple proof of Gasper's and Bochner's results using a Markov operator found by Carlen, Carvahlo, and Loss in their study of the Kac model in kinetic theory. This is joint work with Eric Carlen and Michael Loss.

Orthogonal Polynomials on the Unit Circle. Spectral transformations and their applications to integrable systems

Series
Analysis Seminar
Time
Wednesday, February 3, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Francisco MarcellánUniversidad Carlos III de Madrid
In this talk we will present some recent results about the matrix representation of the multiplication operator in terms of a basis of either orthogonal polynomials (OPUC) or orthogonal Laurent polynomials (OLPUC) with respect to a nontrivial probability measure supported on the unit circle. These are the so called GGT and CMV matrices.When spectral linear transformations of the measure are introduced, we will find the GGT and CMV matrices associated with the new sequences of OPUC and OLPUC, respectively. A connection with the QR factorization of such matrices will be stated. A conjecture about the generator system of such spectral transformations will be discussed.Finally, the Lax pair for the GGT and CMV matrices associated with some special time-depending deformations of the measure will be analyzed. In particular, we will study the Schur flow, which is characterized by a complex semidiscrete modified KdV equation and where a discrete analogue of the Miura transformation appears. Some open problems for time-depending deformations related to spectral linear transformations will be stated.This is a joint work with K. Castillo (Universidad Carlos III de Madrid) and L. Garza (Universidad Autonoma de Tamaulipas, Mexico).

On the independence of complex exponentials

Series
Analysis Seminar
Time
Thursday, January 28, 2010 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Mishko MitkovskiTexas A&M
Given a set of complex exponential e^{i \lambda_n x} how large do you have to take r so that the sequence is independent in L^2[-r,r] ? The answer is given in terms of the Beurling-Mallivan density.

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