The Schur-Horn theorem characterizes eigenvalues and diagonal entries of Hermitian matrices. The Horn problem characterizes eigenvalues of sums of Hermitian matrices. There are similar results for singular values of sums and products of arbitrary matrices. After summarizing recent work on these problems, we will discuss some variations, including the problem of characterizing invariants of off-diagonal blocks in terms of invariants of the whole matrix. The solutions involve the geometry of Schubert calculus as well as some combinatorics involving Littlewood-Richardson coefficients.

Large part of the talk will be based on the recent paper by S.F., W.F., Chi-Kwong Li, and Yiu-Tung Poon.

(The speakers will speak sequentially rather than simultaneously.)