The investigator studies problems in structural and computational number theory, notably the arithmetic of varieties such aselliptic curves and surfaces, and their relations and applicationsto other areas of mathematics such as error-correcting codesand Euclidean lattices. Specific projects include:proof and computational exploitation of new relations between p-adic modular forms of weight 3/2 and the arithmetic of quadratic twists of odd analytic rank; improving and extending results with Henry Cohn and Abhinav Kumar concerning bounds on sphere packings and related questions; refinement of the Shioda-Usui theory relating Mordell-Weil lattices with Weyl groups, and generalization to certain complex reflection groups; continued study of optimal recursive towers and the investigator's modularity conjecture for such towers; extended study of the connection between bilinear (Somos) recurrences, theta sequences, and explicit moduli spaces; and novel applications of lattice reduction to the understanding and efficient computation of solutions of Diophantine equations and inequalities.
The investigator will continue to study problems in number theory concerning the mathematical structure of algebraic solutions of equations and the practical computation 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.
