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Series: Analysis Seminar

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

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.