Seminars and Colloquia by Series

Friday, December 16, 2016 - 12:00 , Location: Skiles 005 , Prof. Jorge Arvesu Carballo , Universida Carlos III de Madrid , jarvesu@math.uc3m.es , 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 30, 2016 - 14:05 , Location: Skiles 005 , Scott Spencer , Georgia Tech , Organizer: Shahaf Nitzan
Wednesday, November 9, 2016 - 14:05 , Location: Skiles 005 , John Jasper , University of Cincinnati , john.jasper@uc.edu , 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 , vritsiou@umich.edu , 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 , joelar@ufl.edu , 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 , aleksandr.b.reznikov@vanderbilt.edu , 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.
Friday, September 30, 2016 - 14:05 , Location: Skiles 005 , Gitta Kutyniok , Technical University of Berlin , kutyniok@math.tu-berlin.de , Organizer: Shahaf Nitzan

Note the unusual time.

Many important problem classes are governed by anisotropic structures such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic features for regularization of inverse problems is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified  treatment of the continuum and digital realm, leading to faithful implementations. In this talk, we will first provide an introduction to sparse regularization of inverse problems, followed by an introduction to the anisotropic representation system of shearlets and presenting the main theoretical results. We will then analyze the effectiveness of using shearlets for sparse regularization of exemplary inverse problems such as recovery of missing data and magnetic resonance imaging (MRI) both theoretically and numerically.
Wednesday, September 28, 2016 - 14:05 , Location: Skiles 005 , Wing Li , Georgia Tech , Organizer: Shahaf Nitzan
Consider Hermitian matrices A, B, C  on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order.  Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor,  for types of torsion modules over division rings,  and for  singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.
Wednesday, September 21, 2016 - 14:05 , Location: Skiles 005 , Wing Li , Georgia Tech , Organizer: Shahaf Nitzan
Consider Hermitian matrices A, B, C  on an n-dimensional Hilbert space such that C=A+B. Let a={a_1,a_2,...,a_n}, b={b_1, b_2,...,b_n}, and c={c_1, c_2,...,c_n} be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order.  Such a triple of real numbers (a,b,c) that satisfies the so-called Horn inequalities, describes the eigenvalues of the sum of n by n Hermitian matrices. The Horn inequalities is a set of inequalities conjectured by A. Horn in 1960 and later proved by the work of Klyachko and Knutson-Tao. In these two talks, I will start by discussing some of the history of Horn's conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor,  for types of torsion modules over division rings,  and for  singular values for products of matrices, and how additional information can be obtained whenever a Horn inequality saturates. The major difficulty in our argument is the proof that certain generalized Schubert cells have nonempty intersection. In the finite dimensional case, it follows from the classical intersection theory. However, there is no readily available intersection theory for von Neumann algebras. Our argument requires a good understanding of the combinatorial structure of honeycombs, and produces an actual element in the intersection algorithmically, and it seems to be new even in finite dimensions. If time permits, we will also discuss some of the intricate combinatorics involved here. In addition, some recent work and open questions will also be presented.

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