Mathematics 697SS - ST- Sums of Squares

The theory of sums of squares (SOS) blends exciting ideas from optimization, real algebraic geometry and convex geometry. Indeed, Hilbert's famous characterization of nonnegative polynomials that are SOS in 1888, and Artin's affirmative answer to Hilbert's 17th problem on whether all nonnegative polynomials are SOS of rational functions are at the origins of this topic. Over the last two decades, interest in the theory and application of SOS polynomials has exploded because of the work of Shor, Nesterov, Lasserre and Parrilo that connects SOS polynomials to modern optimization via semidefinite programming. Since then, there has been many thrilling applications in combinatorics, theoretical computer science, and engineering. This course will cover both the theory and some applications.

Pre-requisites: Math 235 and Math 455, and ideally either a course on polytopes or linear programming, though a motivated student will be able to pick up the notions needed from these topics as we go along.