## Algebra and Combinatorics Seminar

**Date: ** November 15, 2019

**Time: ** 3:00PM - 3:50PM

**Location: ** BLOC 628

**Speaker: **Jurij Volcic, Texas A&M University

**Title: ***The Procesi-Schacher conjecture and positive trace polynomials*

**Abstract: **Hilbert’s 17th problem asked whether every positive polynomial can be written as a sum of squares of rational functions. An affirmative answer by Artin is one of the cornerstones of real algebraic geometry. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a H17 problem for a universal central simple $*$-algebra of degree $n$. It has a positive answer for $n=2$. Recently we proved that the answer for $n=3$ is negative. Nevertheless, we obtained several positivity certificates (Positivstellensätze) for trace polynomials on semialgebraic sets of $n\times n$ matrices. The talk will be a gentle introduction to this mix of central simple algebras, invariant theory and real algebraic geometry.