University of Notre Dame
Department of ACMS
Network of Minima of The Thomson Problem and Smale's 7th Problem
The Thomson problem is a classic physical/chemical/mathematical problem, and easy to state: find the lowest energy configuration (the global minimum) of N particles interacting via Coulomb interaction and restricted to remain on the surface of a sphere. After briefly presenting the interesting features of the energy landscape of the Thomson problem, I will show how some of the open questions for the model can be addressed by studying the network of minima of the energy landscape. I will discuss our preliminary and promising results. I will also discuss the implications of my results to a mathematical problem called Smale's 7th problem: algorithmically construct a configuration, for a given N, so that it is 'close' to the global minimum of the Thomson problem. I will then spend the remaining part of the talk to discuss interesting future directions we can pursue with the help of networks of minima.