## Oliver Club

Abstract: Suppose that you are given self-adjoint matrices A and B with known eigenvalues and unknown eigenvectors. What can you say about eigenvalues of C=A+B? It took the entire 20th century to obtain deterministic characterizations of the eigenvalues in the work of Weyl, Horn, Klyachko, and Knutson-Tao. In the talk we will discuss the probabilistic version of the problem, in which A and B are random and an important role is played by the random matrix parameter Beta, that takes values 1, 2, or 4, depending on whether we deal with real, complex, or quaternionic matrices. I will explain how this parameter can be taken to be an arbitrary positive real number (identified with the inverse temperature in the terminology of the statistical mechanics) and outline a rich asymptotic theory as Beta tends to zero and to infinity.