CCDG Seminar

It is a famous and difficult problem in geometry to find the densest sphere packing in n-dimensional Euclidean space. The answer is known only in dimensions 1 through 3, the proof of the 3-dimensional case by Hales relying on a tremendous amount of computer calculation. Analogous problems of packing among lattices, or on compact spaces such as the sphere or Hamming space, are widely studied in number theory, discrete geometry, coding theory and combinatorics. I will talk about some recent work which studies these problems in the framework of potential energy minimization. This leads to experimental and theoretical techniques, such as gradient descent and linear programming bounds, to approach these optimization problems (and their inverse problems), and to some surprising new results in high dimensions, as well as many natural open questions.