## Seminars and Colloquia by Series

### An Extension of the Cordoba-Fefferman Theorem on the Equivalence Between the Boundedness Maximal and Multiplier Operators

Series
Analysis Seminar
Time
Wednesday, December 2, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Alexander StokolosGeorgia Southern University
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.

### How likely is Buffon's needle to land near a 1-dimensional Sierspinski gasket? A power estimate via Fourier analysis.

Series
Analysis Seminar
Time
Wednesday, November 18, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Matt BondMichigan State University
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.

### A topological separation condition for attractors of contraction mapping systems

Series
Analysis Seminar
Time
Wednesday, November 11, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Sergiy BorodachovTowson University
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).

### On a Bargmann transform and coherent states for the n-sphere

Series
Analysis Seminar
Time
Wednesday, November 4, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dr Carlos Villegas BlasInstituto de Matematicas UNAM, Unidad. Cuernavaca
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.

### The Extremal Nevanlinna-Pick problem for Riemann Surfaces

Series
Analysis Seminar
Time
Wednesday, October 28, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Mrinal RagupathiVanderbilt University
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.

### From transfinite diameter to order-density to best-packing: the asymptotics of ground state configurations

Series
Analysis Seminar
Time
Friday, October 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doug HardinVanderbilt University
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.

### Inequalities for Derivatives and their Applications

Series
Analysis Seminar
Time
Wednesday, October 21, 2009 - 14:00 for 8 hours (full day)
Location
Skiles 269
Speaker
Yuliya BabenkoSam Houston State University
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.

### Point Perturbation and Asymptotics Orthogonal Polynomials

Series
Analysis Seminar
Time
Thursday, October 15, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 255 **NOTE ROOM CHANGE AND SPECIAL DAY**
Speaker
Lillian WongUniversity of Oklahoma
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.

### Positive Matrices and Applications to Hankel Operators

Series
Analysis Seminar
Time
Wednesday, October 14, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Marcus CarlssonPurdue University
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.

### Embeddings of Modulation Spaces into BMO

Series
Analysis Seminar
Time
Wednesday, October 7, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Ramazan TinaztepeGeorgia Tech
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.