Program Description Hyperbolic and stable polynomials have seen several spectacular applications in combinatorics and optimization in recent years. A hyperbolic polynomial in one variable is just a real polynomial with only real roots, while a hyperbolic polynomial in several variables can be seen as a familiy of such realrooted polynomials in one variable. They appear in several different areas, and a beautiful geometric theory with many surprising features has evolved around their study. Nonnegative polynomials and sums of squares are a classical subject of real algebraic geometry, dating back to Hilbert's 17th problem. There are also rich connections to real analysis via duality and moment problems, as well as to polynomial and combinatorial optimization. The goal of the summer school is to provide an introduction to the latest developments in the theory and practice of hyperbolic polynomials and sums of squares concentrating on the following:
