Title: Geometry and Algebra in Optimization
Abstract: While optimization is a well established field with myriad tools, the use of algebraic methods to tackle optimization problems, beyond those from linear algebra, is relatively new. Algebraic models inherently bring in geometry via the language of algebraic geometry, which studies solutions to polynomial equations and inequalities. Convexity enters naturally as well since optimizing a linear function over a set is the same as optimizing the function over its convex hull.
In this talk I will illustrate a sequence of ideas and results that use polynomials to tackle a variety of optimization problems. On the theoretical side, this will touch on theta bodies, lifts of convex sets, tightness of relaxations, and the use of symmetries. Applications include problems from computer vision and combinatorics.