Since Klyashko's major breakthrough in the mid 1990s, there has been great excitements and renewed interests on a wide range of problems related to the Horn's conjecture in representation theory, algebraic geometry, and combinatorics. In particular, significant results were obtained by Balkale, Buch, Fulton, Knutson, Tao, Woodward, and many others. In contrast, the Horn's conjecture has not attracted as much attention in operator theory, even though the conjecture was formulated in terms of self-adjoint matrices. The proof of Horn's conjecture uses highly nontrivial tools from algebraic geometry and combinatorics. For the past ten years, it has been an ongoing project of Bercovici and the PI to find a constructive proof of Horn's conjecture which can be generalized to the von Neumann algebra setting. Recent progress has shown new promises. The propose project will provide new understanding of the eigenvalue problem and the intricate geometry of the eigenflags. The PI and her collaborators also plan to use the machinery that they have developed so far to extend the Horn's conjecture to type II_1 von Neumann factors. A recent result of Collin and Dykema may allow one to use their approach to settle the Connes'embedding problem, i.e., if every type II_1 factor can be embedded in the ultrapower of the hyperfinite II_1 factor, a fundamental question in operator algebra.
The problem of eigenvalues of sums of selfadjoint matrices has intimate connections with algebraic geometry, intersection theory, representation theory, and combinatorics. The proposed project will provide the much needed insight from the operator theory point of view. The generalization to type II_1 factors will bring interesting problems and feedbacks to algebraic geometry and representation theory. The PI is the Georgia Tech ADVANCE professor in the College of Sciences. She is working with others at Georgia Tech to promote the advancement of women in science and engineering in academic. She is also working with AMS on issues that are especially affecting the advancement of women mathematicians in academics. The proposed project will be an essential part of her research program that will help her greatly at this endeavor.
