Non-commutative Optimization - Where Algebra, Analysis and Computational Complexity Meet

We briefly describe a flurry of recent activity in the interaction between the theory of computation and several mathematical areas, that has led to many applications on both sides. The core results are mainly new algorithms for basic problems in invariant theory, arising from computational questions in algebraic complexity theory. However, as understanding evolved, connections were revealed to many other mathematical disciplines, as well as to optimization theory. In particular, the most basic tools of convex optimization in Euclidean space extend to a far more general geodesic setting of Riemannian manifolds that arise from the symmetries of non-commutative groups. This paper extends a section in my book, Mathematics and Computation [54] devoted to an accessible exposition of the theory of computation. Besides covering many of the different parts of this theory, the book discusses its connections with many different areas of mathematics, and many of the sciences.