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Determinants, Hyperbolicity, and Interlacing

Hot Topics: Kadison-Singer, Interlacing Polynomials, and Beyond March 09, 2015 - March 13, 2015

March 09, 2015 (02:00 PM PDT - 03:00 PM PDT)
Speaker(s): Cynthia Vinzant (University of Washington)
Location: MSRI: Simons Auditorium
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC



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.

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