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Spectrahedra and their ShadowsAlgebra & Discrete Mathematics
This lecture concerns the algebraic geometry that underlies semidefinite programming. After a brief discussion of quartic spectrahedra in 3-space, we focus on spectrahedral shadows, that is, convex sets representable by linear matrix inequalities. We characterize the polynomials that vanish on the boundary of a spectrahedral shadow when the defining matrices are generic. The colorful pictures shown in this lecture can be enjoyed by everyone.