Mathematics Colloquia and Seminars

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Orthogonally invariant matrix sets – those invariant under left and

right multiplication by orthogonal matrices – appear often in
mathematics. In recent years, an elegant viewpoint has formed: such
sets often inherit algebraic and geometric properties from their
intersections with the subspace of diagonal matrices. Convexity,
smoothness, and algebraicity all follow this paradigm. After surveying
some results of this flavor, I will describe a recent theorem
formalizing the intuition that orthogonally invariant matrix varieties
are in essence no more complicated than their diagonal restrictions.
The two varieties have equal Euclidean distance degrees (an
interesting algebraic complexity measure).

Joint work with Hon-Leung Lee (Washington), Giorgio Ottaviano
(Florence), and Rekha R. Thomas (Washington)