Convex Algebraic Geometry of Curvature Operators

We study the structure of set algebraic curvature operators satisfying a sectional bound under light emerging field Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic is spectrahedron or spectrahedral shadow; particular, for $n\geq5$, these give new counter-examples to Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ test membership such set. For using semidefinite programming obtained from hierarchies inner approximations by shadows and outer relaxations spectrahedra.