Automorphic Forms, Sphere Packing, and Energy Minimization in Euclidean Space

经典的球体填充问题（由开普勒于1611年提出）问的是，大小相等、不重叠的球体在一个空间区域内的密度如何。这个抽象的数学问题在理解化学、物理和材料科学中的晶体结构方面起着基本的作用。这个问题可以在数学上推广到任何数量的维度，其中它与其他主题相关，例如构建健壮的、错误证明的通信代码。这个问题的完整解决方案现在在维度1、2、3、8和24中是已知的。在这个项目中，首席研究员和合作者的目标是应用解析数论和模形式的新技术来解决一个更普遍的问题，即什么点的配置使它们之间的能量最小。这个问题的另一个相关版本是“为什么某些固体中的原子会形成重复的晶体？”这个问题的答案将为开普勒球体堆积猜想提供新的证明，并解决几何中的其他悬而未决的问题。在技术方面，该项目的工作涉及谐波分析和自同构形式。首席研究员和合作者的目标是使用准模形式来解决Cohn和Kumar提到的“普遍最优性猜想”，特别是在第8维和第24维。其他涉及自同构形式的项目对表示理论中的酉对偶问题也有潜在的应用，特别是对分裂实约群的单能Arthur包的残差爱森斯坦级数和非朗兰兹元的分析。最后，该项目旨在使用自同构分布在解析数论中创建新的voronoi式求和公式。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The classical sphere packing problem (stated by Kepler in 1611) asks how densely equal-sized, non-overlapping spheres can be fit into a region of space. This abstract mathematical question plays a fundamental role in understanding crystalline structures in chemistry, physics, and materials science. The question can be mathematically generalized to any number of dimensions, where it is related to other topics such as the construction of robust, error proof communication codes. The full solution to the problem is now known in dimensions 1, 2, 3, 8, and 24. In this project, the principal investigator and collaborators aim to apply new techniques from analytic number theory and modular forms to settle a more general question about what configuration of points minimizes energy between them. A related version of this question would be "why do atoms in certain solids form repeating crystals?" An answer to this question would provide a new proof of the Kepler's sphere packing conjecture, as well as resolve other open questions in geometry. On the technical side, the work of the project involves harmonic analysis and automorphic forms. The principal investigator and collaborators aim to use quasi-modular forms to solve the "universal optimality conjecture" of Cohn and Kumar alluded to above, particularly in dimensions 8 and 24. Other projects involving automorphic forms have potential applications to the unitary dual problem in representation theory, particularly an analysis of residual Eisenstein series and non-Langlands elements of unipotent Arthur packets for split real reductive groups. Finally, the project aims to create new Voronoi-style summation formulas in analytic number theory using automorphic distributions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.