Georgia Institute of Technology College of Sciences

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

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

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)

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

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.

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 $N$

A fundamental result in Harmonic Analysis states that many functions defined over the interval [-\pi,\pi] can be decomposed into a Fourier series, that is, decomposed as sums of sines and cosines with integer frequencies. This allows one to describe very complicated functions in a simple way, and therefore provides with a strong tool to study the properties of different families of functions.However, the above decomposition does not hold -- or holds but is not efficient enough-- if the functions are no longer defined over an interval,( e.g.

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.

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

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