Georgia Institute of Technology College of Sciences

In the study of perturbation theories in Dynamical systems one is often interested in solving differential equations involving frequencies satisfying number theoretic properties. We will present some estimates ofsums involving Diophantine frequencies leading to sharp estimates on the differential equations.

The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. We develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, in the case of Pauli measurements) with explicit dependence of the bounds on the rank and other complexity parameters.Such bounds are established for several statistically relevant distances, including quantum versions of Kullback-Leibler divergence (relative entropy distance) and of Hellinger distance (so called Bures distance), and Schatten p-norm distances. Sharp upper bounds and oracle inequalities for least squares estimator with von Neumann entropy penalization are obtained showing that minimax lower bounds are attained (up to logarithmic factors) for these distances.

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, generalizing those of toric varieties and their moment maps. Another special class, including Gaussian graphical models, are varieties of inverses of symmetric matrices satisfying linear constraints. We develop a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. Joint work with Mateusz Michałek, Bernd Sturmfels, and Piotr Zwiernik.

The theory of combinatorial limits provides analytic ways of representing large discrete objects. The theory has opened new links between analysis, combinatorics, computer science, group theory and probability theory. In this talk, we will focus on limits of dense graphs and their applications in extremal combinatorics. We will present a general framework for constructing graph limits corresponding to solutions of extremal graph theory problems, which led to constructing counterexamples to several conjectures concerning graph limits. At the end, we will discuss limits of sparse graphs and possible directions to unify the existing approaches related to dense and sparse graphs.