Eigenvalue Inequalities, Intersection of Schubert Varieties, and Related Problems

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This project will continue the study on various problems naturally arise in operator theory and operator algebra related to the Horn's Conjecture, and their relationships to representation theory, combinatorics, and algebraic geometry. The Horn's Conjecture was originally formulated for the eigenvalues of sums of self-adjoint matrices and had shown many important connections with algebraic geometry and representation theory. So far, there were successful generalizations to the infinite dimensional settings, including compact operators on infinite dimensional Hilbert spaces and elements in finite von Neumann algebras. This work also revealed some intricate structure properties on the lattices of invariant subspaces of certain operators and properties on the algebra. This project will help to gain insights in the structure of finite von Neumann algebras and may bring new directions to representation theory and algebraic geometry. Horn's Conjecture was formulated as a problem in linear algebra and it has natural analogues in the infinite dimensional setting and in the von Neumann algebra setting. The tools involved in solving the original Horn's Conjecture were coming from algebraic geometry, representation theory, and combinatorics. The recent advances make a compelling case to continue the investigator's collaborative effort. It is also hopeful that the insights developed here may result in new discoveries, directions and problems for algebraic geometry and representation theory. The investigator is also the ADVANCE Professor of College of Sciences at Georgia Tech, working towards a more family friendly working environment for all faculty, in particular for women and as a role model and advocate for junior women faculty in the college. She plans to continue her work providing networking opportunities and encouraging discussions among women mathematicians, especially junior women mathematicians.