## Seminars and Colloquia by Series

Wednesday, January 20, 2010 - 14:00 , Location: Skiles 269 , Úlfar Stefánsson , Georgia Tech , Organizer: Plamen Iliev
Müntz polynomials arise from consideration of Müntz's Theorem, which is a beautiful generalization of Weierstrass's Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials on the interval of orthogonality, and in particular obtain new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. This is the first time that such asymptotics have been obtained for general Müntz exponents. We also look at the asymptotic behavior outside the interval, and the asymptotic properties of the associated Christoffel functions.
Wednesday, January 13, 2010 - 14:00 , Location: Skiles 269 , Gerald Beer , California State University, Los Angeles , Organizer: Plamen Iliev
Sandro Levi and I have investigated variational strengthenings of uniform continuity and uniform  convergence of nets or sequences of functions with respect to a family of subsets of the domain. Out of our theory comes  an answer to this basic question: what is the weakest topology  stronger than the topology of pointwise convergence in which continuity is preserved under taking limits?   We argue that the classical theory constitues a misunderstanding of what is fundamentally a variational phenomenon.
Tuesday, December 8, 2009 - 16:00 , Location: Skiles 269 , Xuan Duong , Macquarie University , Organizer: Michael Lacey
In this talk,we study  weighted L^p-norm inequalities  for general spectralmultipliersfor self-adjoint positive definite operators on L^2(X), where X is a space of homogeneous type. We show that the sharp weighted Hormander-type spectral multiplier theorems  follow  from the appropriate estimatesof the L^2 norm of the kernel of spectral multipliers and the Gaussian boundsfor the corresponding heat kernels. These results are applicable to spectral multipliersfor group invariant  Laplace operators acting on Lie groups of polynomialgrowth and   elliptic operators on compact manifolds. This is joint work with Adam Sikora and Lixin Yan.
Wednesday, December 2, 2009 - 14:00 , Location: Skiles 269 , Alexander Stokolos , Georgia Southern University , Organizer: Brett Wick
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.
Wednesday, November 18, 2009 - 14:00 , Location: Skiles 269 , Matt Bond , Michigan State University , Organizer: Brett Wick
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.
Wednesday, November 11, 2009 - 14:00 , Location: Skiles 269 , Sergiy Borodachov , Towson University , Organizer: Plamen Iliev
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).
Wednesday, November 4, 2009 - 14:00 , Location: Skiles 269 , , Instituto de Matematicas UNAM, Unidad. Cuernavaca , , Organizer: Jean Bellissard
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.
Wednesday, October 28, 2009 - 14:00 , Location: Skiles 269 , Mrinal Ragupathi , Vanderbilt University , Organizer: Brett Wick
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.
Friday, October 23, 2009 - 14:00 , Location: Skiles 269 , Doug Hardin , Vanderbilt University , Organizer: Jeff Geronimo
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.
Wednesday, October 21, 2009 - 14:00 , Location: Skiles 269 , Yuliya Babenko , Sam Houston State University , Organizer: Brett Wick
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.