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, if n=4, we describe this set as an algebraic interior with respect to an explicit irreducible polynomial.

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

### Geometry-Topology Seminar

Thursday, November 7, 2019 - 4:30pm

#### Renato Ghini Bettiol

CUNY