- You are here:
- GT Home
- Home
- News & Events

Series: Analysis Seminar

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.

Series: Analysis Seminar

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.

Series: Analysis Seminar

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

Series: Analysis Seminar

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.

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.