Title: Inequalities for eigenvalues of sums of self-adjoint operators, II

Abstract: (This is a continuation of my last talk. However it will be self-contained.) Consider two self-adjoint operators $A,B:{\cal H}\to{\cal H}$ on a finite-dimensional Hilbert space. Let $\{{\lambda}_j(A)\}$, $\{{\lambda}_j(B)\}$, and $\{{\lambda}_j(A+B)\}$ be sequences of eigenvalues of A,B, and A+B counting multiplicity, arranged in decreasing order. In 1912, while working on problems in PDE, H. Weyl raised the following question: what are all the inequalities that $\{{\lambda}_j(A)\}$, $\{{\lambda}_j(B)\}$, and $\{{\lambda}_j(A+B)\}$ must satisfy? This problem was only solved completely by A. Klyachko in 1998. The proof uses the moduli space of stable bundles on the projective plane. Since Klyachko's break through, a great deal of activities have been generated, led by Buch, Fulton, Klyachko, Knotson, Tao, Woodward, and others. (Two excellent articles on the subject appeared recently in the AMS Bulletin and the AMS Notices: W. Fulton, Eigenvalues, invariant factors, hightest weights, and Schuber calculus, AMS Bulletin 37 (2000), pp. 209-249, and A. Knutson and T. Tao, Honeycombs and Sums of Hermitian Matrices, AMS Notices 48 (2001), pp. 175 - 186.) We will see how to generalize the finite dimensional setting to the infinite dimensional setting and obtain similar theorems for self-adjoint compact operators. Also I will talk about analogus of Weyl's question for self-adjoint elements in von Neumann algebras with finite trace. This has the potential to lead to some new theories on symmetric functions, infinite dimensional representation theory, and infinite dimensional algebraic geometry.