## Seminars and Colloquia by Series

Wednesday, January 20, 2016 - 14:05 , Location: Skiles 005 , ChunKit Lai , San Francisco State University , , Organizer: Shahaf Nitzan
We study the construction of exponential bases and exponential frames on general  $L^2$  space with the measures supported on self-affine fractals. This problem dates back to the conjecture of Fuglede. It lies at the interface between analysis, geometry and number theory and it relates to translational tilings.   In this talk, we give an introduction to this topic, and report on some of the recent advances. In particular, the possibility of constructing exponential frames on fractal measures without  exponential bases will be discussed.
Wednesday, December 2, 2015 - 14:05 , Location: Skiles 006 , Walter Van Assche , University of Leuven, Belgium , Organizer: Jeff Geronimo
The Minkowski question mark function is a singular distribution function arising from Number Theory: it maps all quadratic irrationals to rational numbers and rational numbers to dyadic numbers. It generates a singular measure on [0,1]. We are interested in the behavior of the norms and recurrence coefficients of the orthonormal polynomials for this singular measure. Is the Minkowski measure a "regular" measure (in the sense of Ullman, Totik and Stahl), i.e., is the asymptotic zero distribution the equilibrium measure on [0,1] and do the n-th roots of the norm converge to the capacity (which is 1/4)? Do the recurrence coefficients converge (are the orthogonal polynomials in Nevai's class). We provide some numerical results which give some indication but which are not conclusive.
Thursday, November 19, 2015 - 16:35 , Location: Skiles 006 , , Brown University , , Organizer: Galyna Livshyts
[Special time and location] The content of this talk is joint work with Yumeng Ou. We describe a novel framework for the he analysis of multilinear singular integrals acting on Banach-valued functions.Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces, including, in particular, noncommutative Lp spaces. A concrete case of our result is a multilinear generalization of Weis' operator-valued Hormander-Mihlin linear multiplier theorem.Furthermore, we derive from our main result a wide range of mixed Lp-norm estimates for multi-parameter multilinear multiplier operators, as well as for the more singular tensor products of a one-parameter Coifman-Meyer multiplier with a bilinear Hilbert transform. These respectively extend the results of Muscalu et. al. and of Silva to the mixed norm case and provide new mixed norm fractional Leibnitz rules.
Wednesday, November 18, 2015 - 14:00 , Location: Skiles 005 , Betsy Stovall , UW-Madison , Organizer: Michael Lacey
We will discuss the problem of restricting the Fourier transform to manifolds for which the curvature vanishes on some nonempty set.  We will give background and discuss the problem in general terms, and then outline a proof of an essentially optimal (albeit conditional) result for a special class of hypersurfaces.
Wednesday, November 11, 2015 - 14:05 , Location: Skiles 005 , Michael Northington , Vanderbilt University , , Organizer: Shahaf Nitzan
Uncertainty principles are results which restrict the localization of a function and its Fourier transform.  One class of uncertainty principles studies generators of structured systems of functions, such as wavelets or Gabor systems, under the assumption that these systems form a basis or some generalization of a basis.  An example is the Balian-Low Theorem for Gabor systems. In this talk, I will discuss sharp, Balian-Low type, uncertainty principles for finitely generated shift-invariant subspaces of $L^2(\R^d)$.  In particular, we give conditions on the localization of the generators which prevent these spaces from being invariant under any non-integer shifts.
Wednesday, November 4, 2015 - 14:05 , Location: Skiles 005 , B. Chevreau , Univeriste de Bordeaux I , Organizer:
Wednesday, October 28, 2015 - 14:05 , Location: Skiles 005 , Benjamin Jaye , Kent State University , , Organizer: Shahaf Nitzan
We shall describe how the study of certain measures called reflectionless measures can be used to understand the behaviour of oscillatory singular integral operators in terms of non-oscillatory quantities.  The results described are joint work with Fedor Nazarov, Maria Carmen Reguera, and Xavier Tolsa
Wednesday, September 30, 2015 - 14:00 , Location: Skiles 005 , Laura Cladek , University of Wisconsin, Madison , Organizer: Michael Lacey
We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.
Wednesday, September 23, 2015 - 14:05 , Location: Skiles 005 , Alan Sola , University of South Florida , Organizer: Michael Lacey
In my talk, I will discuss coordinate shifts acting on Dirichlet spaces on the bidisk and the problem of finding cyclic vectors for these operators. For polynomials in two complex variables, I will describe a complete characterization given in terms of size and nature of zero sets in the distinguished boundary.
Wednesday, August 26, 2015 - 14:05 , Location: Skiles 005 , Lidia Fernandez , Applied Math Dept, University of Granada , Organizer: Jeff Geronimo
The purpose of this talk is to introduce some recent works on the field of Sobolev orthogonal polynomials. I will mainly focus on our two last works on this topic. The first has to do with orthogonal polynomials on product domains. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions. The second one analyzes a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which involves the outward normal derivatives on the sphere. Using the representation of these polynomials in terms of spherical harmonics, algebraic and analytic properties will be deduced. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of Sobolev orthogonal polynomials. Then explicit expressions for the norms will be obtained, among other properties.