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.
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