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Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

Series: Analysis Seminar

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.

Series: Analysis Seminar

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

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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).

Series: Analysis Seminar

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.

Series: Analysis Seminar

We use Fourier analysis to establish $L^p$ bounds for Stein's spherical maximal theorem in the setting of compactly supported Borel measures $\mu, \nu$ satisfying natural local size assumptions $\mu(B(x,r)) \leq Cr^{s_{\mu}}, \nu(B(x,r)) \leq Cr^{s_{\nu}}$. As an application, we address the following geometric problem: Suppose that $E\subset \mathbb{R}^d$ is a union of translations of the unit circle, $\{z \in \mathbb{R}^d: |z|=1\}$, by points in a set $U\subset \mathbb{R}^d$. What are the minimal assumptions on the set $U$ which guarantee that the $d-$dimensional Lebesgue measure of $E$ is positive?