Number Theory and Geometry

研究人员研究结构和计算数论中的问题，特别是椭圆曲线和曲面等变种的算术，以及它们与纠错码和欧几里德格等其他数学领域的关系和应用。具体项目包括：将高阶K3曲面的研究扩展到具有正特征的超奇异曲面；继续并推广特征零中高阶K3曲面的显式方程和模的使用，在模曲线和簇以及丢番图记录中的各种应用；进一步研究双线性(Somos)递归、θ序列和显模空间之间的联系；通过李代数的表示来推广Scott Kominers关于调和重量枚举子的结果以处理任意有限域上的线性码；改进和推广Henry Cohn和Abhinav Kumar关于球填充和相关问题的结果；并与Kent Boklan合作，在剩余的剩余类mod18中完成了对7个立方体的Wering问题的求解。研究者将继续研究数论中关于方程代数解的数学结构以及此类解的实际计算和应用的问题。这将使我们更好地理解这些方程的结构和它们的解(椭圆曲线等)，并进一步与计算理论、纠错码和球面填充等其他主题相联系。

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.