Topics in Geometry and Dynamics

Brown University, Providence RI

Investigators

Abstract

This project concerns research in geometry and dynamical systems, some of which uses computer experimentation to study simply-stated questions about which little is known. The idea is that new and powerful computing tools will provide insight that was not available to earlier generations of mathematicians. One such question, known as Thomson's problem, asks how some number of electrons will arrange themselves on the sphere so as to minimize their total potential energy. Such a problem has wide-ranging applications because in many scientific fields it is quite useful to know how to evenly distribute points or sensors or energy sources in a space. The principal investigator recently resolved a large part of the story for the case of five points on the sphere and plans to continue to develop this theory. Another problem asks about the stability of the solar system in a highly simplified model of celestial mechanics called outer billiards. It is known that there exist initial conditions for which the corresponding orbits are unbounded; this project aims to further develop a theory that provides detailed theoretical pictures of these orbits in a special case. Another problem under investigation in the project is the famous square peg problem, which asks if every loop in the plane has four points on it that make the corners of a square. In technical terms, the project focuses on the following areas. First, it is planned to extend results related to a phase-transition conjecture for the 5-electron problem. The principal investigator proved that there exists a constant c such that the triangular bi-pyramid is the energy minimizer for the Riesz s-potential if and only if s is not greater than c. It is conjectured that some pyramid with square base is the energy minimizer for the Riesz s-potential if and only if s is not less than c. Second, the project aims to continue developing the plaid mode, a combinatorial construction that assigns to each rational number a collection of polyhedral surfaces contained in a cube. When the model is sliced in one coordinate direction, it gives loops which accurately model certain orbits of outer billiards with respect to the kite with the same parameter. When the model is sliced in the other two coordinate directions it gives loops that are combinatorially isomorphic to those found in Truchet tilings and in corner percolation. Third, the principal investigator plans to continue an ongoing study of the space of inscribed rectangles in a Jordan loop. He has established that every Jordan loop has associated to it a connected set of rectangles such that every point of the loop, with at most 4 exceptional points, is the vertex of one of the rectangles in the set. This result leads to a question of how a rectangle can slide continuously along a pair of arcs, a problem akin to studying the solutions of a differential equation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.