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

职业：格子和球堆积、算术几何和计算数论

• 批准号: 0952486
• 负责人: Abhinav Kumar
• 金额: $ 40万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-03-01 至 2016-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0952486&HistoricalAwards=false
• 关键词: CAREER; Lattices; Sphere; Packings; Arithmetic

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.

在这个项目中，PI，Kumar，将研究球面填充，算术几何和计算数论的主题，以及这些研究方向的各个重叠领域。PI和他的合作者使用线性规划界限的技术，致力于寻找8维和24维最密集的填充的问题。他们在势能最小化的背景下研究了球体堆积和相关的编码问题。这导致了各种新的问题和技术，如泛最优性的概念，填充空间中的梯度下降，以及可能应用于分子自组装的反问题。这个项目将探索在以前的研究中出现的一些问题，以及其他基本问题，如改进球形填充密度的渐近界。该项目的另一个组成部分是算术几何，特别是对K3曲面及其自同构的研究，例如Shioda-Inor结构。PI还建议研究算术应用，如模曲线曲面的描述、伽罗瓦表示以及高阶模形式和椭圆曲线的计算。该项目还包括计算数论中的相关问题。格子是该提案主题的统一主题，该项目的广泛目标之一是明确地理解高维上有趣的格子族。在空间中有效地包装大小相等的非重叠球体的“蔬菜水果问题”是几何中的经典问题。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令人惊讶和美丽的联系。它的理论和实践分支扩展到李代数、例外有限群、二次型、编码理论、密码学和物理学中的能量最小化。这个项目将寻求加深我们对高维球体填充的理解，以及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。二十世纪代数几何最引人注目的成功之一是福林斯定理(以前的莫德尔猜想)，它提供了三分法中的最后一个环节，该三分法将代数曲线上的有理点的数量与其拓扑亏格联系起来，并大致确定了连接两个变量的方程可以有多少解。一个类似的问题是，我们可以期待一个连接三个或更多变量的方程有多少解，这仍然是一个相当神秘的问题。PI打算调查的一些问题涉及曲面的算法，特别是K3曲面，这在数学和物理中很重要。该项目还希望在更模糊的计算方面取得进展，比如如何找到这些解决方案。PI还将开发与该提案相关的主题的课程和研讨会，并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放源代码，以及不同领域的有趣代码目录。