Vinberg's theory of reflection groups has wide applications. We discuss some of these, to monodromy groups of hypergeometric and Painlevé equations. The nonlinear case is intimately connected to affine Markoff surfaces and it is a central ingredient in the Diophantine analysis of these surfaces.

This talk aims to summarize the status and goals of a broad research project. The main messages of this project are summarized below; I plan to describe, through examples, many of the concepts they refer to, and the evolution of ideas leading to them. No special background is assumed.

- We extend some basic algorithms of convex optimization from Euclidean space to the far more general setting of Riemannian manifolds, capturing the symmetries of non-commutative group actions. The main tools for analyzing these algorithms combine central results from invariant and representation theory.
- One main motivation for studying these problems and algorithms comes from algebraic complexity theory, especially attempts to separate Valiant’s algebraic analogs of the P and NP. Symmetries and group actions play a key role in these attempts.
- The new algorithms give exponential (or better) improvements in run-time for solving algorithmic many specific problems across CS, Math and Physics. In particular, in algebra (testing rational identities in non-commutative variables), in analysis (testing the feasibility and tightness of Brascamp-Lieb inequalities), in quantum information theory (to the quantum marginals problem), optimization (testing membership in “moment polytopes”), and others. This work exposes old and new connections between these diverse areas.