Seminars and Colloquia by Series

Weak limits of optimal discrete measures for Riesz potentials

Series
Analysis Seminar
Time
Wednesday, October 5, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sasha ReznikovVanderbilt
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.

Anisotropic Structures and Sparse Regularization of Inverse Problems

Series
Analysis Seminar
Time
Friday, September 30, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Gitta KutyniokTechnical University of Berlin

Please Note: 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.

Inequalities for eigenvalues of sums of self-adjoint operators and related intersection problems (Part II)

Series
Analysis Seminar
Time
Wednesday, September 28, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wing LiGeorgia Tech
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.

Inequalities for eigenvalues of sums of self-adjoint operators and related intersection problems (Part I)

Series
Analysis Seminar
Time
Wednesday, September 21, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wing LiGeorgia Tech
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.

A weak type estimate for oscillatory singular integrals

Series
Analysis Seminar
Time
Wednesday, September 14, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael LaceyGeorgia Tech
The Ricci-Stein theory of singular integrals concerns operators of the form \int e^{i P(y)} f (x-y) \frac {dy}y.The L^p boundedness was established in the early 1980's, and the weak-type L^1 estimate by Chanillo-Christ in 1987. We establish the weak type estimate for the maximal truncations. This method of proof might well shed much more information about the fine behavior of these transforms. Joint work with Ben Krause.

L^p Estimates for Semi-Degenerate Simplex Multipliers

Series
Analysis Seminar
Time
Wednesday, September 7, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Robert KeslerGeorgia Tech
Multilinear singular integral operators associated to simplexes arise naturally in the dynamics of AKNS systems. One area of research has been to understand how the choice of simplex affects the estimates for the corresponding operator. In particular, C. Muscalu, T. Tao, C. Thiele have observed that degenerate simplexes yield operators satisfying no L^p estimates, while non-degenerate simplex operators, e.g. the trilinear Biest, satisfy a wide range of L^p estimates provable using time-frequency arguments. In this talk, we shall define so-called semi-degenerate simplex multipliers, which as the terminology suggests, lie somewhere between the degenerate and non-degenerate settings and then introduce new L^p estimates for such objects. These results are known to be sharp with respect to target Lebesgue exponents, unlike the best known Biest estimates, and rely on carefully localized interpolation arguments

Two weight estimates with matrix measures for well-localized operators

Series
Analysis Seminar
Time
Wednesday, August 31, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amalia CuliucGeorgia Tech
In this talk we discuss two weight estimates for well-localized operators acting on vector-valued function spaces with matrix weights. We will show that the Sawyer-type testing conditions are necessary and sufficient for the boundedness of this class of operators, which includes Haar shifts and their various generalizations. More explicitly, we will show that it is suficient to check the estimates of the operator and its adjoint only on characteristic functions of cubes. This result generalizes the work of Nazarov-Treil-Volberg in the scalar setting and is joint work with K. Bickel, S. Treil, and B. Wick.

The Z_2^n Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

Series
Analysis Seminar
Time
Wednesday, April 27, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vincent GenestMIT
In this talk, I will discuss the n-dimensional Dirac-Dunkl operator associated with the reflection group Z_2^{n}. I will exhibit the symmetries of this operator, and describe the invariance algebra they generate. The symmetry algebra will be identified as a rank-n generalization of the Bannai-Ito algebra. Moreover, I will explain how a basis for the kernel of this operator can be constructed using a generalization of the Cauchy-Kovalevskaia extension in Clifford analysis, and how these basis functions form a basis for irreducible representations of Bannai-Ito algebra. Finally, I will conjecture on the role played by the multivariate Bannai-Ito polynomials in this framework.

A semidefinite relaxation of k-means clustering

Series
Analysis Seminar
Time
Wednesday, April 20, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dustin MixonOhio state University
Recently, Awasthi et al proved that a semidefinite relaxation of the k-means clustering problem is tight under a particular data model called the stochastic ball model. This result exhibits two shortcomings: (1) naive solvers of the semidefinite program are computationally slow, and (2) the stochastic ball model prevents outliers that occur, for example, in the Gaussian mixture model. This talk will cover recent work that tackles each of these shortcomings. First, I will discuss a new type of algorithm (introduced by Bandeira) that combines fast non-convex solvers with the optimality certificates provided by convex relaxations. Second, I will discuss how to analyze the semidefinite relaxation under the Gaussian mixture model. In this case, outliers in the data obstruct tightness in the relaxation, and so fundamentally different techniques are required. Several open problems will be posed throughout.This is joint work with Takayuki Iguchi and Jesse Peterson (AFIT), as well as Soledad Villar and Rachel Ward (UT Austin).

New results on zeroes of stationary Gaussian functions

Series
Analysis Seminar
Time
Wednesday, April 13, 2016 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Naomi FeldheimStanford University
We consider (complex) Gaussian analytic functions on a horizontal strip, whose distribution is invariant with respect to horizontal shifts (i.e., "stationary"). Let N(T) be the number of zeroes in [0,T] x [a,b]. First, we present an extension of a result by Wiener, concerning the existence and characterization of the limit N(T)/T as T approaches infinity. Secondly, we characterize the growth of the variance of N(T). We will pose to discuss analogues of these results in a few other settings, such as zeroes of real-analytic Gaussian functions and winding of planar Gaussian functions, pointing out interesting similarities and differences. For the last part, we consider the "persistence probability" (i.e., the probability that a function has no zeroes at all in some region). Here we present results in the real setting, as even this case is yet to be understood. Based in part on joint works with Jeremiah Buckley and Ohad Feldheim.

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