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Series: Analysis Seminar

I will speak about an extension of Cordoba-Feﬀerman Theorem on the equivalence between boundedness properties of certain classes of maximal and multiplier operators. This extension utilizes the recent work of Mike Bateman on directional maximal operators as well as my work with Paul Hagelstein on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions.

Series: Analysis Seminar

It is well known that a needle thrown at random has zero
probability of intersecting any given irregular planar set of finite
1-dimensional Hausdorff measure. Sharp quantitative estimates for fine open
coverings of such sets are still not known, even for such sets as the
Sierpinski gasket and the 4-corner Cantor set (with self-similarities 1/4
and 1/3). In 2008, Nazarov, Peres, and Volberg provided the sharpest known
upper bound for the 4-corner Cantor set. Volberg and I have recently used
the same ideas to get a similar estimate for the Sierpinski gasket. Namely,
the probability that Buffon's needle will land in a 3^{-n}-neighborhood of
the Sierpinski gasket is no more than C_p/n^p, where p is any small enough
positive number.

Series: Analysis Seminar

We consider finite systems of contractive homeomorphisms of a complete metric space, which are non-redundanton every level. In general, this condition is weaker than the strong open set condition and is not equivalent to the weak separation property. We show that the set of N-tuples of contractive homeomorphisms, which satisfy this separation condition is a G_delta set in the topology of pointwise convergence of every component mapping with an additional requirement that the supremum of contraction coefficients of mappings in the sequence be strictly less than one.We also give several sufficient conditions for this separation property. For every fixed N-tuple of dXd invertible contraction matrices from a certain class, we obtain density results for vectors of fixed points, which defineN-tuples of affine contraction mappings having this separation property. Joint work with Tim Bedford (University of Strathclyde) and Jeff Geronimo (Georgia Tech).

Series: Analysis Seminar

We will introduce a Bargmann transform from the space of square integrable functions on the n-sphere onto a suitable Hilbert space of holomorphic functions on a null quadric. On base of our Bargmann transform, we will introduce a set of coherent states and study their semiclassical properties. For the particular cases n=2,3,5, we will show the relation with two known regularizations of the Kepler problem: the Kustaanheimo-Stiefel and Moser regularizations.

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.