Principal Investigator: Lajos Molnár

The research will bring new useful results related to the analytic, algebraic, geometric and probabilistic aspects of structures of matrices and linear operators, especially, positive definite cones and their generalizations. The geometrical properties of that cone are especially important in applications. We investigate various metrics and divergences and study corresponding transformations of positive definite cones. We extend our recently developed theory of several fine variable operator monotone functions as free functions on positive definite cones to the infinite variable case. We plan to solve the Champagne problem of McCarthy. We develop a theory for CAT(k) spaces providing Central Limit Theorems and apply it to the study of means on positive definite cones by generalizing the gradient flow theory. Finally, we make contributions to the theory of integrable systems, our main goal is to understand the scattering and the duality properties of the hyperbolic van Diejen systems.