Research

Mathematically, I study point configurations and energy minimization, using tools from potential theory, discrete geometry, harmonic analysis, and approximation theory. I have also worked in extremal graph theory. I am particularly interested in the conjectured universal optimality of the hexagonal lattice (also referred to as A_2 or the equitriangular lattice). The conjecture is a major open problem and is discussed at the end of the following article: Out of a Magic Math Function, One Solution to Rule Them All | Quanta Magazine 

It is conjectured (first by Cohn and Kumar) that an approach using so-called "linear programming bounds" can prove the conjecture. Such bounds have already been used in groundbreaking work by Viazovska and others to prove the analogous universal optimality result in dimensions 8 and 24, generalizing their earlier solutions of the corresponding sphere packing problems. Viazovska received the 2022 Fields Medal in large part due to these results. 

 In my thesis and related papers, we used a variant of the linear programming bounds to show the optimality of certain periodic configurations for a wide class of potentials. In certain cases, optimal configurations arise from the hexagonal lattice. By degrees of freedom, these results (as of April 2024) constitute the broadest classes of configurations for which the hexagonal lattice is known to be universally optimal.