The investigator studies problems in structural and computational number theory, notably the arithmetic of varieties such as elliptic curves and surfaces, and their relations and applications to other areas of mathematics such as error-correcting codes and Euclidean lattices. Specific projects include: Extending the investigation of high-rank K3 surfaces to supersingular surfaces in positive characteristic; continuing and extending the use of explicit equations and moduli of high-rank K3 surfaces in characteristic zero, with various applications to modular curves and varieties and to Diophantine records; further study of the connection between bilinear (Somos) recurrences, theta sequences, and explicit moduli spaces; generalizing results with Scott Kominers on harmonic weight enumerators via representations of Lie algebras to treat linear codes over arbitrary finite fields; improving and extending results with Henry Cohn and Abhinav Kumar on sphere packings and related questions; and working with Kent Boklan on completing the solution of Waring's problem for seven cubes in the remaining residue classes mod 18.
The investigator will continue to study problems in number theory concerning the mathematical structure of algebraic solutions of equations and the practical computation and applications of such solutions. This should lead both to better understanding of the structure of those equations and their solutions (elliptic curves, etc.), and to further connections with other topics such as the theory of computation, error-correcting codes, and sphere packing.
