Kateryna Tatarko, University of Alberta
"Geometric methods in isoperimetric problems and random matrix theory"
High-dimensional convex geometry is concerned with geometric parameters of convex bodies such as volume, surface area, or mean width. These parameters are at the core of both classical extremal problems and the study of probabilistic geometric constructions, for example, random polytopes. The latter type of problems appears in other areas such as random matrix theory or randomized algorithms. Although these two directions lead to questions of very different nature, convex geometric methods are very important in both of them.
In this talk, we will discuss such methods in relation to estimates for the smallest singular value of certain random matrices as well as an isoperimetric problem.
More specifically, we will explain how the smallest singular value controls the invertibility of a matrix and why estimating it is a delicate task. We will also provide an upper bound on the smallest singular value of matrices with i.i.d. entries of zero mean and unit variance.
In the second part of the talk, we will discuss the question of reversing the classical isoperimetric inequality for convex bodies. We present the solution of the reverse isoperimetric problem in the class of $\lambda$-concave bodies (this part is based on the joint work with R. Chernov and K. Drach).
M3 3103