The 2019-2020 Colloquium will take place every Wednesday from 4:30 to 5:30 at CMSA, 20 Garden Street, Room G10. This year’s colloquium will be organized by Aghil Alaee, Ryan Thorngren and Sergiy Verstyuk. The schedule below will be updated as speakers are confirmed. Information on previous colloquia can be found here.
|9/18/2019||Bill Helton (UC San Diego)||
Abstract: The last decade has seen the development of a substantial noncommutative (in a free algebra) real and complex algebraic geometry. The aim of the subject is to develop a systematic theory of equations and inequalities for (noncommutative) polynomials or rational functions of matrix variables. Such issues occur in linear systems engineering problems, in free probability (random matrices), and in quantum information theory. In many ways the noncommutative (NC) theory is much cleaner than classical (real) algebraic geometry. For example,
◦ A NC polynomial, whose value is positive semidefinite whenever you plug matrices into it, is a sum of squares of NC polynomials.
◦ A convex NC semialgebraic set has a linear matrix inequality representation.
◦ The natural Nullstellensatz are falling into place.
The goal of the talk is to give a taste of a few basic results and some idea of how these noncommutative problems occur in engineering. The subject is just beginning and so is accessible without much background. Much of the work is joint with Igor Klep who is also visiting CMSA for the Fall of 2019.