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 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.

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.

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.