The question of energy minimization is widely studied in physics and mathematics to find a configuration of points which minimize a given form of potential energy. I will describe some progress on the inverse problem of contructing a potential function which has a given target configuration as its (provable) global minimum.
For simplicity, consider a finite collection of points on a sphere in dimensions, though some of the techniques apply in broader generality. I will first describe a necessary and sufficient condition for the inverse question to have a solution, and then an algorithm which attempts to construct a solution as a linear combination of a finite set of specificied potential functions.
Finally, I will illustrate the technique in the case of several symmetrical examples, such as the cube, the dodecahedron and the 120-cell. In these cases one can in fact display solutions which are decreasing and convex as a function of distance, and use linear programmming bounds and spherical design properties to give simpleproofs of the global minimality.
This is joint work with Henry Cohn from Microsoft Research.
Abhinav Kumar is an associate professor of Mathematics at MIT. He attended MIT as an undergraduate, despite obtaining the first rank in the entrance examination for the Indian Institute of Technology (out of a few hundred thousand applicants). He received S.B. degrees in EECS, mathematics and physics from MIT in 2002. A two-time Putnam Fellow, Kumar received the Jon A. Bucsela Prize for the strongest undergraduate in mathematics at MIT. He also received the Harvard Putnam and GSAS fellowships for graduate school at Harvard, where he completed a Ph.D. mathematics in 2006 under the supervision of Noam Elkies and Barry Mazur. Kumar then took a postdoctoral research appointment at Microsoft, 2006-07, before joining the MIT mathematics faculty as assistant professor in 2007. He was promoted to associate professor in 2012. Professor Kumar works in the field of arithmetic algebraic geometry, lattices and sphere packings, and combinatorics and discrete geometry.
He is also interested in related questions in physics and computer science. In 2007, he was selected by the MIT School of Science for support from the Solomon Buchsbaum AT&T research fund. In 2010 he received a Faculty Early Career Development (CAREER) award from the NSF.
Host: Ravi Sundaram