Algebraic Geometry of Curvature and Matrices with Partitioned Eigenvalues

Abstract: This talk is a combined discussion of an upcoming paper with Paul Breiding and Kristian Ranestad on the enumerative geometry of the curvature of algebraic varieties and a past paper called Real Symmetric Matrices with Partitioned Eigenvalues. Curvature is an important concept in differential geometry. We approach curvature from the perspective of algebraic geometry, studying the critical curvature locus of an algebraic variety. A curvature feature known as an umbilical point occurs when the eigenvalues of the second fundamental form coincide. This leads us to a discussion of the real algebraic variety of matrices with eigenvalue multiplicities determined by a partition.