Spring 2019

Problems in Matrix Variables: A Taste of Noncommutative (Convex) Algebraic Geometry *starts at 10:15 a.m. sharp*

Apr 24, 2019 10:15 am – 11:15 am 

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Parent Program: 

Bill Helton (UC San Diego) 


Room 116

Over the last decade a substantial noncommutative (free) real and complex algebraic geometry has developed within the  mathematics community. 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. In many  ways the theory is incomplete, certainly with regard to handing engineering problems. The goal of the talk is to give a taste of some basic results and an introduction to a technique which permeates the subject.