Number Theory and Geometry

调查研究问题的结构和计算数论，特别是算术的品种，如椭圆曲线和曲面，以及它们的关系和应用程序的其他领域的数学，如纠错码和欧几里德格。 具体项目包括：证明和计算利用权为3/2的p-adic模形式和奇解析秩的二次扭运算之间的新关系;改进和推广亨利科恩和阿比纳夫库马尔关于球面填充上的界和相关问题的结果;改进和推广将Mordell-Weil格与Weyl群联系起来的Shioda-Weil理论，并推广到某些复反射群;继续研究最佳递归塔和调查员的模块性矩阵，这样的塔;扩展研究之间的连接双线性（索莫斯）递归，θ序列，和明确的模空间;和新的应用格约化的理解和有效的计算的解决方案的丢番图方程和不等式。调查员将继续研究问题的数论有关的数学结构的代数解的方程和实际计算这样的解决方案。这将导致双方更好地了解的结构，这些方程及其解（椭圆曲线等），并进一步与其他主题的联系，如计算理论，纠错码和球包装。

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 relationsbetween p-adic modular forms of weight 3/2 and the arithmeticof quadratic twists of odd analytic rank; improving and extendingresults with Henry Cohn and Abhinav Kumar concerning bounds onsphere packings and related questions; refinement of the Shioda-Usuitheory relating Mordell-Weil lattices with Weyl groups, andgeneralization to certain complex reflection groups; continued studyof optimal recursive towers and the investigator's modularity conjecturefor 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 andefficient computation of solutions of Diophantine equationsand inequalities.The investigator will continue to study problems in number theoryconcerning the mathematical structure of algebraic solutionsof equations and the practical computation of such solutions.This should lead both to better understanding of the structureof those equations and their solutions (elliptic curves, etc.),and to further connections with other topics such as the theory ofcomputation, error-correcting codes, and sphere packing.