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

点击翻译按钮获取中文摘要

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.