Algebra and Combinatorics Seminar

Date: February 5, 2021

Time: 3:00PM - 4:00PM

Location: Zoom

Speaker: Jurij Volcic, TAMU

Title: Freely noncommutative Hilbert's 17th problem

Abstract: One of the problems on Hilbert's 1900 list asked whether every positive rational function can be written as a sum of squares of rational functions. Its affirmative resolution by Artin in 1927 was a breakthrough for real algebraic geometry. The talk addresses the analog of this problem for positive semidefinite noncommutative rational functions. More generally, a rational Positivstellensatz on matricial sets given by linear matrix inequalities will be presented; a crucial intermediate step is an extension theorem on invertible evaluations of linear matrix pencils, which has less to do with positivity and ostensibly more to do with representation theory. One of the consequences of the Positivstellensatz is an algorithm for eigenvalue optimization of noncommutative rational functions. Finally, some contrast between the polynomial and the rational Positivstellensatz in the noncommutative setting will be discussed.