Symmetry and Simulation: How Geometry Affects Scientific Computing from the Solar System to your Microwave Oven

Ari Stern, ACM

How do we model change in physical systems? Ever since Newton proposed his laws of motion, the answer has been: differential equations. Solving these equations lets us make predictions about the future, or even the past- but for most complex systems in science and engineering, it is impossible or impractical to obtain exact solutions. Therefore, we must rely on numerical simulation to compute approximate answers. With computing power cheap and ubiquitous, these simulations can now be carried out better and faster than ever before. However, even some of the "best" numerical methods have a serious problem: in trying to simulate the laws of motion accurately, they end up breaking other physical laws, such as the conservation laws for momentum and energy. For many problems- from long-time simulations of the solar system, to molecular dynamics, to computing the resonant frequencies of a microwave oven- failure to preserve these sorts of features can result in a major loss of predictive power. These properties can best be understood in terms of geometric mechanics, a powerful branch of classical mechanics that incorporates mathematical tools from differential geometry and calculus of variations, but which has until recently seen very little use among applied mathematicians. In this talk, I hope to bring the audience on a tour of recent advances at this rich intersection of geometry, physics, and computation.