Linear Analysis Seminar

Date: October 4, 2019

Time: 4:00PM - 5:00PM

Location: BLOC 220

Speaker: Jurij Volcic, TAMU

Title: Noncommutative polynomials describing convex sets

Abstract: The free semialgebraic set D determined by a hermitian noncommutative polynomial f is the set of tuples of hermitian matrices X such that f(X) is positive semidefinite. When f is a hermitian monic linear pencil, D is called a free spectrahedron. Since it is the feasible set of a linear matrix inequality (LMI), it is evidently convex. Conversely, it is well-known that every convex free semialgebraic set is a free spectrahedron. This talk presents a solution to the basic problem of determining those noncommutative polynomials f for which D is convex. A consequence is an effective probabilistic algorithm that not only determines if D is convex, but also produces its LMI representation. Further results address 1x1 noncommutative polynomials describing convex sets.