Wednesday, January 25, 2017 - 14:05 , Location: Skiles 005 , Ishwari Kunwar , Georgia Tech , Organizer: Shahaf Nitzan
We show that multilinear dyadic paraproducts and Haar multipliers, as well as their commutators with locally integrable functions, can be pointwise dominated by multilinear sparse operators. These results lead to various quantitative weighted norm inequalities for these operators. In particular, we introduce multilinear analog of Bloom's inequality, and prove it for the commutators of the multilinear Haar multipliers.
Wednesday, January 18, 2017 - 14:05 , Location: Skiles 005 , Chris Heil , Georgia Tech , Organizer: Shahaf Nitzan
The Linear Independence of Time-Frequency Translates Conjecture, also known as the HRT conjecture, states that any finite set of time-frequency translates of a given $L^2$ function must be linearly independent. This conjecture, which was first stated in print in 1996, remains open today. We will discuss this conjecture, its relation to the Zero Divisor Conjecture in abstract algebra, and the (frustratingly few) partial results that are currently available.
Wednesday, January 11, 2017 - 13:05 , Location: Skiles 005 , Walter Van Assche , Katholieke University Lueven , Organizer: Shahaf Nitzan
We show that the multiwavelets, introduced by Alpert in 1993, are related to type I Legendre-Angelesco multiple orthogonal polynomials. We give explicit formulas for these Legendre-Angelesco polynomials and for the Alpert multiwavelets. The multiresolution analysis can be done entirely using Legendre polynomials, and we give an algorithm, using Cholesky factorization, to compute the multiwavelets and a method, using the Jacobi matrix for Legendre polynomials, to compute the matrices in the scaling relation for any size of the multiplicity of the multiwavelets.Based on joint work with J.S. Geronimo and P. Iliev
Friday, December 16, 2016 - 12:00 , Location: Skiles 005 , Prof. Jorge Arvesu Carballo , Universida Carlos III de Madrid , email@example.com , Organizer: Doron Lubinsky
I will present a discrete family of multiple orthogonal polynomials defined by a set of orthogonality conditions over a non-uniform lattice with respect to different q-analogues of Pascal distributions. I will obtain some algebraic properties for these polynomials (q-difference equation and recurrence relation, among others) aimed to discuss a connection with an infinite Lie algebra realized in terms of the creation and annihilation operators for a collection of independent ascillators. Moreover, if time allows, some vector equilibrium problem with constraint for the nth root asymptotics of these multiple orthogonal polynomials will be discussed.
Wednesday, November 9, 2016 - 14:05 , Location: Skiles 005 , John Jasper , University of Cincinnati , firstname.lastname@example.org , Organizer: Shahaf Nitzan
An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound. Though they arise in many applications, there are only a few known methods for constructing ETFs. One of the most popular classes of ETFs, called harmonic ETFs, is constructed using the structure of finite abelian groups. In this talk we will discuss a broad generalization of harmonic ETFs. This generalization allows us to construct ETFs using many different structures in the place of abelian groups, including nonabelian groups, Gelfand pairs of finite groups, and more. We apply this theory to construct an infinite family of ETFs using the group schemes associated with certain Suzuki 2-groups. Notably, this is the first known infinite family of equiangular lines arising from nonabelian groups.
Wednesday, November 2, 2016 - 14:05 , Location: Skiles 005 , Beatrice-Helen Vritsiou , University of Michigen , email@example.com , Organizer: Shahaf Nitzan
The thin-shell or variance conjecture asks whether the variance of the Euclidean norm, with respect to the uniform measure on an isotropic convex body, can be bounded from above by an absolute constant times the mean of the Euclidean norm (if the answer to this is affirmative, then we have as a consequence that most of the mass of the isotropic convex body is concentrated in an annulus with very small width, a "thin shell''). So far all the general bounds we know depend on the dimension of the bodies, however for a few special families of convex bodies, like the $\ell_p$ balls, the conjecture has been resolved optimally. In this talk, I will talk about another family of convex bodies, the unit balls of the Schatten classes (by this we mean spaces of square matrices with real, complex or quaternion entries equipped with the $\ell_p$-norm of their singular values, as well as their subspaces of self-adjoint matrices).In a joint work with Jordan Radke, we verified the conjecture for the operator norm (case of $p = \infty$) on all three general spaces of square matrices, as well as for complex self-adjoint matrices, and we also came up with a necessary condition for the conjecture to be true for any of the other p-Schatten norms on these spaces. I will discuss how one can obtain these results: an essential step in the proofs is reducing the question to corresponding variance estimates with respect to the joint probability density of the singular values of the matrices.Time permitting, I will also talk about a different method to obtain such variance estimates that allows to verify the variance conjecture for the operator norm on the remaining spaces as well.
Wednesday, October 26, 2016 - 14:05 , Location: Skiles 005 , Irina Mitrea , Temple University , Organizer: Michael Lacey
The Integration by Parts Formula, which is equivalent withthe DivergenceTheorem, is one of the most basic tools in Analysis. Originating in theworks of Gauss, Ostrogradsky, and Stokes, the search for an optimalversion of this fundamental result continues through this day and theseefforts have been the driving force in shaping up entiresubbranches of mathematics, like Geometric Measure Theory.In this talk I will review some of these developments (starting from elementaryconsiderations to more sophisticated versions) and I will discuss recentsresult regarding a sharp divergence theorem with non-tangential traces.This is joint work withDorina Mitrea and Marius Mitrea from University of Missouri, Columbia.
Wednesday, October 19, 2016 - 14:05 , Location: Skiles 005 , Joel Rosenfeld , University of Florida , firstname.lastname@example.org , Organizer: Shahaf Nitzan
I will present results on numerical methods for fractional order operators, including the Caputo Fractional Derivative and the Fractional Laplacian. Fractional order systems have been of growing interest over the past ten years, with applications to hydrology, geophysics, physics, and engineering. Despite the large interest in fractional order systems, there are few results utilizing collocation methods. The numerical methods I will present rely heavily on reproducing kernel Hilbert spaces (RKHSs) as a means of discretizing fractional order operators. For the estimation of a function's Caputo fractional derivative we utilize a new RKHS, which can be seen as a generalization of the Fock space, called the Mittag-Leffler RKHS. For the fractional Laplacian, the Wendland radial basis functions are utilized.
Wednesday, October 5, 2016 - 14:05 , Location: Skiles 005 , Sasha Reznikov , Vanderbilt , email@example.com , Organizer: Shahaf Nitzan
The problem in the talk is motivated by the following problem. Suppose we need to place sprinklers on a field to ensure that every point of the field gets certain minimal amount of water. We would like to find optimal places for these sprinklers, if we know which amount of water a point $y$ receives from a sprinkler placed at a point $x$; i.e., we know the potential $K(x,y)$. This problem is also known as finding the $N$-th Chebyshev constant of a compact set $A$. We study how the distribution of $N$ optimal points (sprinklers) looks when $N$ is large. Solving such a problem also provides an algorithm to approximate certain given distributions with discrete ones. We discuss connections of this problem to minimal discrete energy and to potential theory.