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Inequalities for eigenvalues of sums of self-adjoint operators and related intersection problems

Consider Hermitian matrices A, B, C : H → H on an n-dimensional Hilbert space such that C = A + B. Let α = (α1 ≥ α2 ≥ . . . ≥ αn ≥ 0), β = (β1 ≥ β2 ≥ . . . ≥ βn ≥ 0), and γ = (γ1 ≥ γ2 ≥ . . . ≥ γn ≥ 0) be sequences of eigenvalues of A, B, and C counting multiplicity, arranged in decreasing order, respectively. It was conjectured by A. Horn in 1960 that α, β, γ must satisfy a collection of inequalities (later named as Horn inequalities) and the collection is also sufficient to characterize the relation A + B = C. This conjecture was proved by the work of Klyachko and Knutson-Tao in the late 1990s, using highly sophisticated methods from algebraic geometry and intricate combinatorics.

In this talk I will start by discussing some of the history of the Horn’s conjecture and then move on to its more recent developments. We will show that these inequalities are also valid for selfadjoint elements in a finite factor, for types of torsion modules over division rings, and for singular values for products of matrices. If time permits, some more recent work and open questions will also be discussed.

This is a survey talk and will be accessible to math graduate students and general audience.


Biography:

Dr. Wing Suet Li has been with Georgia Tech since 1992. Her research specialties are functional analysis and operator theory, with a focus on understanding the fundamental structures of linear operators on Hilbert spaces. Her research has been continuously supported by the National Sciences Foundation since 1993. She has published over 30 scholarly articles in refereed journals, and is an established expert on problems related to intersection theory in algebraic geometry and operator theory, and operator algebra from analysis. Dr. Li has attended many national and international conferences and held several prestigious short-term visiting positions in Europe and Asia. She has been invited to numerous mathematics departments to give colloquium talks nationally and internationally. A passionate teacher, Dr. Li has taught a wide range of courses at Georgia Tech, including Analysis, Algebra, Statistics, and Numerical Analysis. She has taught three times at the Georgia Tech Lorraine campus and currently is teaching at Georgia Tech Shenzhen and she is very committed to increasing Georgia Tech's global influence.