A spectrahedron is an affine slice of the cone of positive semidefinite matrices. Spectrahedra form a rich class of convex bodies that are computationally tractable and appear in many areas of mathematics. Examples include polytopes, ellipsoids, and more exotic convex sets, like the convex hull of some curves. I will introduce the theory of spectrahedra with many examples and discuss some applications in distance geometry and combinatorial optimization.
Determinants, Hyperbolicity, and Interlacing
Cynthia Vinzant (North Carolina State University)
MSRI: Simons Auditorium
On the space of real symmetric matrices, the determinant is hyperbolic with respect to the cone of positive definite matrices. As a consequence, the determinant of a matrix of linear forms that is positive definite at some point is a hyperbolic polynomial. A celebrated theorem of Helton and Vinnikov states that any hyperbolic polynomial in three variables has such a determinantal representation. In more variables, this is no longer the case. During this talk, we will examine hyperbolic polynomials that do and do not have definite determinantal representations. Interlacing polynomials will play an interesting role in this story and form a convex cone in their own right.