Seminars and Colloquia by Series

Results on two variable orthogonal polynomials associated with Bernstein-Szego measures on the circle and square.

Series
Analysis Seminar
Time
Wednesday, February 1, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeff GeronimoGeorgia Tech
The theory of two variable orthogonal polynomials is not very well developed. I will discuss some recent results on two variable orthogonal polynomials on the bicircle and time permitting on the square associate with orthogonality measures that are one over a trigonometric polynomial. Such measures have come to be called Bernstein-Szego measures. This is joint work with Plamen Iliev and Greg Knese.

Sparse Domination of Multilinear Dyadic Operators and Their Commutators

Series
Analysis Seminar
Time
Wednesday, January 25, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ishwari KunwarGeorgia Tech
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.

The HRT Conjecture

Series
Analysis Seminar
Time
Wednesday, January 18, 2017 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris HeilGeorgia Tech
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.

Alpert multiwavelets and Legendre-Angelesco multiple orthogonal polynomials

Series
Analysis Seminar
Time
Wednesday, January 11, 2017 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Walter Van AsscheKatholieke University Lueven
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

Multiple q-Meixner polynomials of the first kind

Series
Analysis Seminar
Time
Friday, December 16, 2016 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Prof. Jorge Arvesu Carballo Universida Carlos III de Madrid
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.

Equiangular tight frames from association schemes

Series
Analysis Seminar
Time
Wednesday, November 9, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John JasperUniversity of Cincinnati
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.

On the thin-shell conjecture for the Schatten classes

Series
Analysis Seminar
Time
Wednesday, November 2, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Beatrice-Helen VritsiouUniversity of Michigen
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.

The art of integration by parts

Series
Analysis Seminar
Time
Wednesday, October 26, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Irina Mitrea Temple University
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.

Fractional Calculus, Reproducing Kernel Hilbert Spaces, and Approximation Theory

Series
Analysis Seminar
Time
Wednesday, October 19, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joel RosenfeldUniversity of Florida
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.

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