Richard Schwartz plans to continue his work in geometry and dynamical systems. The first area of his focus will be the dynamics of particles in the complement of planar polygonal regions. Building on his solution of the 1960 Moser-Neumann question, Schwartz plans to continue developing the theory, finding connections to such subjects as Diophantine approximtion, self-similar tilings, and polytope exchange transformations. The second ares of his focus will be on optimization problems, such as the question of how a finite number of electrons optimally distribute themselves on the sphere. Specifically, Schwartz would like to extend his solution of Thomson's 5-electron problem to the case of other power laws. Finally, Schwartz plans to study a variety of geometric iterations, such as high dimensional iterated barycentric subdivision.
For the most part, Schwartz's research involves studying what happens when one makes a simple geometric construction over and over again. For instance, one could follow a ball as it bounces endlessly inside a triangular table, or follow the motion of traffic as it moves along an infinite grid of one-way streets according to certain rules, or study the shapes one gets when one cracks a tetrahedron into smaller and smaller tetrahedra. Often these simple constructions, when done repeatedly, produce very beautiful and unexpected patterns. Schwartz likes to study such problems, at first, by doing computer experiments and then, later, by trying to construct rigorous proofs that the phenomena uncovered by the experiments actually exist.
