Number Theory and Geometry

研究结构和计算数论中的问题，特别是椭圆曲线和椭圆曲面的算法，以及它们在其他数学领域的关系和应用，如纠错码和欧几里得格。具体项目包括：权值3/2的p进模形式与奇解析秩的二次扭转的新关系的证明和计算开发；改进和推广了Henry Cohn和Abhinav Kumar关于球界填充及相关问题的结果；改进了有关Mordell-Weil格与Weyl群的Shioda-Usuitheory，并将其推广到某些复反射群继续研究最优递归塔及其研究者的模块化猜想扩展了双线性（Somos）递归、序列和显模空间之间联系的研究；以及晶格约简在丢芬图方程和不等式解的理解和有效计算中的新应用。研究者将继续研究数论中有关方程代数解的数学结构和这些解的实际计算的问题。这将有助于更好地理解这些方程的结构及其解（椭圆曲线等），并进一步与其他主题（如计算理论、纠错码和球体填充）建立联系。

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.