Number Theory and Geometry

研究者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：将高阶K3曲面的研究扩展到正特征的超奇异曲面；继续并扩展了特征零点高阶K3曲面的显式方程和模的使用，并将其应用于模曲线和变体以及丢番图记录；进一步研究双线性（Somos）递归、序列和显模空间之间的联系；利用李代数的表示推广了调和权枚举数在处理任意有限域上线性码的结果与Henry Cohn和Abhinav Kumar一起改进和扩展了关于球体填充及相关问题的结果；并与肯特·博克兰合作，在剩余的剩余类mod 18中完成了韦林问题的七个立方体的解决方案。研究者将继续研究数论中有关方程代数解的数学结构以及这些解的实际计算和应用的问题。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 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.