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Seminars

Seminar on Matrix Theory

Limiting Horn inequalities in infinite dimensions

2024/12/09 (Mon.) 15:00~17:00
  • Date : 2024/12/09 (Mon.) 15:00~17:00
  • Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
  • Speaker : Colin McSwiggen (AS)
  • Abstract : Horn's problem is a fundamental question in linear algebra that asks about the spectra of sums of matrices: if A and B are two n-by-n Hermitian matrices with known eigenvalues, what are the possible eigenvalues of A+B? Following decades of work by many scholars, this problem was finally solved in 1998. The solution takes the form of a recursive procedure for generating a list of linear inequalities depending on the eigenvalues of A and B; then the possible spectra of A+B are precisely the vectors that satisfy all of these inequalities. Moreover, this same system of inequalities turns out to play a fundamental role in the algebraic geometry of Grassmannians and in the representation theory of the general linear group. In this talk, we will study the Horn inequalities in the limit as n (the size of the matrices) goes to infinity. This limit tells us about the spectra of sums of operators on a Hilbert space. I will first give a brief, elementary introduction to Horn problems in infinite dimensions, without assuming any background on operator algebras. Then I will present some new results on the Horn inequalities that apply in this setting, based on recent joint work with Samuel Johnston (https://arxiv.org/abs/2410.08907). In particular, we will find that in infinite dimensions, the Horn inequalities have a strange "self-characterization" property, whereby the system of inequalities encodes an exact description of itself.
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