CAREER: Lattices and Sphere Packings, Arithmetic Geometry and Computational Number Theory

在这个项目中，PI Kumar将研究球体封装、算术几何和计算数论的主题，以及这些研究方向重叠的各个领域。PI和他的合作者利用线性规划界的技术，研究了在8维和24维中找到密度最大的填料的问题。他们研究了势能最小化背景下的球体填充和相关的编码问题。这导致了各种新的问题和技术，如普遍最优性的概念，在包装空间的梯度下降，逆问题可能有应用于分子自组装。本项目将探讨在以前的研究中出现的一些问题，以及其他基本问题，如球填料密度的渐近界的改进。该项目的另一个组成部分是算术几何，特别是对K3表面及其自同构的研究，例如Shioda-Inose结构。PI还提出研究算术应用，如模曲线和曲面的描述、伽罗瓦表示以及模形式和高阶椭圆曲线的计算。该项目还包括计算数论的相关问题。格是提案主题的统一主题，项目的广泛目标之一是明确地理解高维中有趣的格族。在空间中有效填充等尺寸非重叠球体的“蔬菜水果商问题”是一个经典的几何问题。然而，事实证明，它与数学、物理和计算机科学的各个部分有着惊人而美丽的联系。它的理论和实践分支延伸到李代数、例外有限群、二次型、编码理论、密码学和物理中的能量最小化。这个项目将寻求进一步我们对高维球体包装的理解，并探索和利用这些联系。算术几何试图利用代数几何的强大工具来理解自然数的性质。20世纪代数几何最引人注目的成功之一是法尔廷斯定理（以前的莫德尔猜想），它提供了三分法的最后一环，三分法将代数曲线上有理点的数量与其拓扑属联系起来，并大致确定了我们可以期望有多少个解连接两个变量的方程。类似的问题是，对于一个连接三个或更多变量的方程，我们可以期望有多少个解，这仍然是一个谜。PI打算研究的一些问题涉及曲面的算术，特别是在数学和物理中很重要的K3曲面。该项目还希望在计算方面取得进展，比如如何找到这些解决方案。PI还将开发与提案相关主题的课程和研讨会，旨在让研究生和本科生积极参与这项研究。该项目的其他目标将是开发算法和开源代码，以及不同领域有趣代码的目录。

In this project, the PI, Kumar, will investigate the topics of sphere packings, arithmetic geometry and computational number theory, and the various areas of overlap of these directions of research. The PI and his collaborators have worked on the problem of finding the densest packings in 8 and 24 dimensions, using the technique of linear programming bounds. They have studied the sphere packing and associated coding problems in context of potential energy minimization. This has led to various new questions and techniques, such as the concept of universal optimality, gradient descent in the space of packings, and inverse problems which may have applications to molecular self-assembly. This project will explore some of the questions that arose in previous investigations, as well as other fundamental questions such as improvement of asymptotic bounds on the density of sphere packings. Another component of the project is arithmetic geometry, especially the study of K3 surfaces and their automorphisms, such as Shioda-Inose structures. The PI also proposes to study arithmetic applications such as the description of modular curves and surfaces, Galois representations and the computation of modular forms and elliptic curves of high rank. The project also encompasses related questions in computational number theory. Lattices are a unifying theme of the topics of the proposal, and one of the broad goals of the project is to understand explicitly families of interesting lattices in high dimensions.The "greengrocer's problem" of packing equal sized non-overlapping spheres efficiently in space is a classical problem in geometry. Nevertheless, it turns out to have surprising and beautiful connections with various parts of mathematics, physics, and computer science. Its theoretical and practical ramifications extend to Lie algebras, exceptional finite groups, quadratic forms, coding theory, cryptography and energy minimization in physics. This project will seek to further our understanding of sphere packings in high dimensions as well as explore and exploit these connections. Arithmetic geometry seeks to understand the properties of the natural numbers using the powerful tools of algebraic geometry. One of the most spectacular successes of twentieth century algebraic geometry is Faltings' Theorem (erstwhile Mordell's Conjecture), which provides the final link in a trichotomy which links the number of rational points on an algebraic curve to its topological genus, and roughly establishes how many solutions we can expect to an equation linking two variables. The analogous question of how many solutions we can expect to an equation linking three or more variables is still quite a mystery. Some of the questions the PI intends to investigate involve the arithmetic of surfaces, especially K3 surfaces, which are important in mathematics and physics. The project also hopes to make progress on the still murkier computational aspects, such as how to find these solutions. The PI will also develop courses and seminars in topics related to the proposal, and aim to involve graduate and undergraduate students actively in this research. Other aims of the project will be to develop algorithms and open-source code, as well as a catalog of interesting codes in different spaces.