Seminars and Colloquia by Series

The Toeplitz Kernel Approach In Inverse Spectral Theory Of Differential Operators

Series
Analysis Seminar
Time
Wednesday, October 1, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rishika Rupum – Texas A&M
When does the spectrum of an operator determine the operator uniquely?-This question and its many versions have been studied extensively in the field of inverse spectral theory for differential operators. Several notable mathematicians have worked in this area. Among others, there are important contributions by Borg, Levinson, Hochstadt, Liebermann; and more recently by Simon, Gesztezy, del Rio and Horvath, which have further fueled these studies by relating the completeness problems of families of functions to the inverse spectral problems of the Schr ̈odinger operator. In this talk, we will discuss the role played by the Toeplitz kernel approach in answering some of these questions, as described by Makarov and Poltoratski. We will also describe some new results using this approach. This is joint work with Mishko Mitkovski.

Pollaczek multiple orthogonal polynomials ensembles

Series
Analysis Seminar
Time
Monday, June 2, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alexander Aptekarev – Keldysh Institute, Russia
We discuss asymptotics of multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (\sigma_1, \sigma_2) with unbounded supports (supp(\sigma_1) \subset \mathbb{R}_+, supp(\sigma_2) \subset \mathbb{R}_-); moreover, the second measure \sigma_2 is discrete. We focus on deriving the strong and weak asymptotic for a special system of multiple OP from this class with respect to two Pollaczek type weights on \mathbb{R}_+. The weak asymptotic for these polynomials can be obtained by means of solution of an equilibrium problem. For the strong asymptotic we use the matrix Riemann-Hilbert approach.

Bi-parameter singular integrals: recent results and examples

Series
Analysis Seminar
Time
Wednesday, April 16, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Henri Martikainen – Georgia Tech
We discuss bi-parameter Calderon-Zygmund singular integrals from the point of view of modern probabilistic and dyadic techniques. In particular, we discuss their structure and boundedness via dyadic model operators. In connection to this we demonstrate, via new examples, the delicacy of the problem of finding a completely satisfactory product T1 theorem. Time permitting related non-homogeneous bi-parameter results may be mentioned.

Hardy Space Theory on Product Spaces of Homogeneous Type Via Orthonormal Wavelet Bases

Series
Analysis Seminar
Time
Thursday, April 10, 2014 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ji Li – Macquarie University, Sydney, Australia
We study Hardy spaces on spaces X which are the n-fold product of homogeneous spaces. An important tool is the remarkable orthonormal wavelet basis constructed Hytonen. The main tool we develop is the Littlewood-Paley theory on X, which in turn is a consequence of a corresponding theory on each factor space. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving Littlewood-Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing function spaces and the boundedness of singular integrals on spaces of homogeneous type. This is joint work with Yongsheng Han and Lesley Ward.

KP hierarchy for the cyclic quiver

Series
Analysis Seminar
Time
Wednesday, April 9, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Oleg Chalykh – University of Leeds
I will discuss a generalization of the KP hierarchy, which is intimately related to the cyclic quiver and the Calogero-Moser problem for the wreath-product $S_n\wr\mathbb Z/m\mathbb Z$.

Generalizations of Wermer's maximality theorem

Series
Analysis Seminar
Time
Wednesday, March 26, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex Izzo – Bowling Green State University
A classical theorem of John Wermer asserts that the algebra of continuous functions on the circle with holomophic extensions to the disc is a maximal subalgebra of the algebra of all continuous functions on the circle. Wermer's theorem has been extended in numerous directions. These will be discussed with an emphasis on extensions to several complex variables.

Vector-valued inequalities with applications to bi-parameter problems.

Series
Analysis Seminar
Time
Wednesday, March 12, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prabath Silva – Indiana University
In this talk we will discuss applications of a new method of proving vector-valued inequalities discovered by M. Bateman and C. Thiele. We give new proofs of the Fefferman-Stein inequality (without using weighted theory) and vector-valued estimates of the Carleson operator using this method. Also as an application to bi-parameter problems, we give a new proof for bi-parameter multipliers without using product theory. As an application to the bilinear setting, we talk about new vector-valued estimates for the bilinear Hilbert transform, and estimates for the paraproduct tensored with the bilinear Hilbert transform. The first part of this work is joint work with Ciprian Demeter.

A new type of exceptional Laguerre polynomials

Series
Analysis Seminar
Time
Monday, March 10, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 171
Speaker
Conni Liaw – Baylor University
The Bochner Classification Theorem (1929) characterizes the polynomial sequences $\{p_n\}_{n=0}^\infty$, with $\deg p_n=n$ that simultaneously form a complete set of eigenstates for a second order differential operator and are orthogonal with respect to a positive Borel measure having finite moments of all orders: Hermite, Laguerre, Jacobi and Bessel polynomials. In 2009, G\'{o}mez-Ullate, Kamran, and Milson found that for sequences $\{p_n\}_{n=1}^\infty$, with $\deg p_n=n$ (i.e.~without the constant polynomial) the only such sequences are the \emph{exceptional} Laguerre and Jacobi polynomials. They also studied two Types of Laguerre polynomial sequences which omit $m$ polynomials. We show the existence of a new "Type III" family of Laguerre polynomials and focus on its properties.

Carleson and Reverse Carleson measures

Series
Analysis Seminar
Time
Wednesday, February 5, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bill Ross – University of Richmond
This will be a survey talk on the ongoing classification problem for Carleson and reverse Carleson measures for the de Branges-Rovnyak spaces. We will relate these problems to some recent work of Lacey and Wick on the boundedness of the Cauchy transform operator.

Universality in Random Normal Matrices

Series
Analysis Seminar
Time
Wednesday, January 22, 2014 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Roman Riser – ETH, Zurich
In the beginning, the basics about random matrix models and some facts about normal random matrices in relation with conformal map- pings will be explained. In the main part we will show that for Gaussian random normal matrices the eigenvalues will fill an elliptically shaped do- main with constant density when the dimension n of the matrices tends to infinity. We will sketch a proof of universality, which is based on orthogonal polynomials and an identity which plays a similar role as the Christoffel- Darboux formula in Hermitian random matrices. Especially we are interested in the density at the boundary where we scale the coordinates with n^(-1/2). We will also consider the off-diagonal part of the kernel and calculate the correlation function. The result will be illustrated by some graphics.

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