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Geometric Analysis Seminar

Convex Algebraic Geometry of Curvature Operators

Renato Ghini Bettiol (Lehman College (CUNY))

Location:  Hill 705
Date & time: Tuesday, 22 October 2019 at 2:50PM - 3:50PM

Abstract:  I will discuss the structure of the set of algebraic curvature operators of n-dimensional Riemannian manifolds satisfying a sectional curvature bound (e.g., nonnegative or nonpositive sectional curvature), under the light of the emerging field of Convex Algebraic Geometry. More precisely, we completely determine in what dimensions n this convex semi-algebraic set is a spectrahedron or a spectrahedral shadow (these are generalizations of polyhedra where linear programing extends to as semidefinite programming, and are of great interest in applied mathematics and optimization). Furthermore, for n=4, we describe the algebraic boundary of this set as the zero set of an explicit irreducible polynomial.

This is based on joint work with M. Kummer (TU Berlin) and R. Mendes (Univ of Oklahoma).