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Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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 $S_n\wr\mathbb Z/m\mathbb Z$.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

Abstract: In the beginning, the basics about random matrix models andsome facts about normal random matrices in relation with conformal map-pings will be explained. In the main part we will show that for Gaussianrandom 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 orthogonalpolynomials and an identity which plays a similar role as the Christoff el-Darboux formula in Hermitian random matrices.Especially we are interested in the density at the boundary where we scalethe coordinates with n^(-1/2). We will also consider the off -diagonal part of thekernel and calculate the correlation function. The result will be illustratedby some graphics.