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Series: Analysis Seminar

Given points $z_1,\ldots,z_n$ on a finite open Riemann
surface $R$ and complex scalars $w_1,\ldots,w_n$, the Nevanlinna-Pick
problem is to determine conditions for the existence of a holomorphic
map $f:R\to \mathbb{D}$ such that $f(z_i) = w_i$.
In this talk I will provide some background on the problem, and then
discuss the extremal case. We will try to discuss how a method of
McCullough can be used to provide more qualitative information about
the solution. In particular, we will show that extremal cases are
precisely the ones for which the solution is unique.

Series: Analysis Seminar

I will review recent and classical results concerning the
asymptotic properties (as N --> \infty) of 'ground state' configurations
of N particles restricted to a d-dimensional compact set A\subset {\bf R}^p
that minimize the Riesz s-energy functional
\sum_{i\neq j}\frac{1}{|x_{i}-x_{j}|^{s}}
for s>0.
Specifically, we will discuss the following
(1) For s < d, the ground state configurations have limit distribution as
N --> \infty given by the equilibrium measure \mu_s, while the first
order asymptotic growth of the energy of these configurations is given by
the 'transfinite diameter' of A.
(2) We study the behavior of \mu_s as s approaches the critical
value d (for s\ge d, there is no equilibrium measure). In the case that
A is a fractal, the notion of 'order two density' introduced by Bedford
and Fisher naturally arises. This is joint work with M. Calef.
(3) As s --> \infty, ground state configurations approach best-packing
configurations on A. In work with S. Borodachov and E. Saff we show that
such configurations are asymptotically uniformly distributed on A.

Series: Analysis Seminar

In this talk we will discuss Kolmogorov and Landau type inequalities for the derivatives. These are the inequalities which estimate the norm of the intermediate
derivative of a function (defined on an interval, R_+, R, or
their multivariate analogs) from some class in terms of the norm of the
function itself and norm of its highest derivative.
We shall present several new results on sharp inequalities of this type
for special classes of functions (multiply monotone and absolutely
monotone) and sequences. We will also highlight some of the techniques
involved in the proofs (comparison theorems) and discuss several
applications.

Series: Analysis Seminar

In this talk, I will discuss some results obtained in my Ph.D. thesis.
First, the point mass formula will be introduced. Using the formula, we
shall see how the asymptotics of orthogonal polynomials relate to the
perturbed Verblunsky coefficients. Then I will discuss two classes of
measures on the unit circle -- one with Verblunsky coefficients \alpha_n -->
0 and the other one with \alpha_n --> L (non-zero) -- and explain the
methods I used to tackle the point mass problem involving these measures.
Finally, I will discuss the point mass problem on the real line. For a long
time it was believed that point mass perturbation will generate
exponentially small perturbation on the recursion coefficients. I will
demonstrate that indeed there is a large class of measures such that that
proposition is false.

Series: Analysis Seminar

Given an "infinite symmetric matrix" W we give a simple condition, related
to the shift operator being expansive on a certain sequence space, under
which W is positive. We apply this result to AAK-type theorems for
generalized Hankel operators, providing new insights related to previous
work by S. Treil and A. Volberg. We also discuss applications and open
problems.

Series: Analysis Seminar

Modulation spaces are a class of Banach spaces which provide a quantitative time-frequency analysis of functions via the Short-Time Fourier Transform. The modulation spaces are the "right" spaces for time-frequency analysis andthey occur in many problems in the same way that Besov Spaces are attached to wavelet theory and issues of smoothness. In this seminar, I will talk about embeddings of modulation Spaces into BMO or VMO (the space of functions of bounded or vanishing mean oscillation, respectively ). Membership in VMO is central to the Balian-Low Theorem, which is a cornerstone of time-frequency analysis.

Series: Analysis Seminar

The trigonometric Grassmannian parametrizes specific solutions of the KP hierarchy which correspond to rank one solutions of a differential-difference bispectral problem. It can be considered as a completion of the phase spaces of the trigonometric Calogero-Moser particle system or the rational Ruijsenaars-Schneider system.
I will describe the characterization of this Grassmannian in terms of representation theory of a suitable difference W-algebra. Based on joint work with L. Haine and E. Horozov.

Series: Analysis Seminar

We consider multipoint Padé approximation to Cauchy transforms of
complex measures. First, we recap that if the support of a measure is
an analytic Jordan arc and if the measure itself is absolutely
continuous with respect to the equilibrium distribution of that arc
with Dini-continuous non-vanishing density, then the diagonal
multipoint Padé approximants associated with appropriate interpolation
schemes converge locally uniformly to the approximated Cauchy
transform in the complement of the arc. Second, we show that this
convergence holds also for measures whose Radon–Nikodym derivative is
a Jacobi weight modified by a Hölder continuous function. The
asymptotics behavior of Padé approximants is deduced from the analysis
of underlying non–Hermitian orthogonal polynomials, for which the
Riemann–Hilbert–∂ method is used.

Series: Analysis Seminar

We describe how time-frequency analysis is used to analyze boundedness
and Schatten class properties of pseudodifferential operators and
Fourier integral operators.

Series: Analysis Seminar

We will survey recent developments in the area of two weight inequalities, especially those relevant for singular integrals. In the second lecture, we will go into some details of recent characterizations of maximal singular integrals of the speaker, Eric Sawyer, and Ignacio Uriate-Tuero.