Topics in geometry and dynamics

Brown University, Providence RI

Investigators

Abstract

Richard Schwartz plans to continue his research in geometric dynamics. A central focus of his research is the study of outer billiards. In 2006, Schwartz resolved the central problem in this 50-year-old subject, the Moser-Neumann problem concerning the stability of outer billiards orbits. Schwartz will continue to develop the theory, exploring connections to self-similar tilings, limit sets of discrete groups, and renormalization. Schwartz also plans to continue studying irrational polygonal billiards, with a focus on the Triangular Billiards Problem, which asks if every triangular shaped billiard table has a periodic billiard path. Finally, Schwartz plans to study basic geometric iterations such as those defined by iterated barycentric subdivision. The common theme in Schwartz's proposed research is the analysis of what happens when a simple geometric construction is repeated over and over again. For example, in outer billiards, a toy model for planetary motion, a point (the satellite) moves around the outside of a convex planar shape (the planet) according to a simple geometric rule. Resolving a 50 year old question about this system, Schwartz found examples of convex shapes, and starting positions for the point, such that the point wanders unboundedly far away from the shape while following the rules of the game. One might say that these shapes are examples of planets whose satellites can wander off into space. Iterated geometric constructions such as outer billiards are intellectually appealing and scientifically important because their simple definitions sometimes lead to extremely rich and mysterious behavior, as do analogous systems in the natural sciences.