Sums of Squares and Optimization in Algebraic Varieties

There is a fundamental relation between the optimization of polynomial functions on algebraic varieties and understanding the cone of non-negative polynomials on such varieties. The family of sums of squares is one of the cornerstones for understanding positivity in the setting of algebraic varieties, and even the semi-algebraic setting. The goal of this talk is to illustrate the important role of sums of squares in polynomial optimization and review some important theorems regarding their role in understanding positivity in algebraic varieties.