Schur polynomials in analysis: from matrix positivity preservers to majorization
The PDE & Analysis seminar covers topics in PDE and analysis.
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Description
Abstract:
We explain recent results involving Schur polynomials – viewed as functions on the positive orthant – which have led to progress along several classical directions of study:
(1) Understanding functions that preserve positive semidefiniteness when applied entrywise to matrices. This journey takes us from work of Schur, Schoenberg, Rudin, and Loewner (1900s) to modern-day results.
(2) The proofs of these recent results lead to characterizing (weak) majorization of real tuples using Schur polynomials; this has since been generalized to all Weyl groups. Majorization inequalities have been studied since Maclaurin and Newton (1700s).
(Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar; and with Terence Tao.)
Location
Seminar Room 1.33
Hanna Neumann Building 145
Science Road
Action 2601