Investigations in the areas of Sphere Packing and the Arithmetic of K3 surfaces

本课题涉及K3曲面的球面填充和算术几何等领域。Kumar将继续他在高维球面填充及相关问题上的研究。球面填充的中心问题是求出欧几里得空间中最密集的非重叠球面填充。答案只有在3维以内才知道，一般来说，这是一个很难但又很自然的问题。Cohn和Kumar在8维和24维上取得了进展，表明来自E_8和Leech晶格的球体填料非常接近于这些维度上最密集的填料。他们正在研究相关的问题，如球体表面的球体填充（球面代码），双曲球体填充，以及如何利用势能最小化来实现这些问题。Kumar还将研究K3曲面的算法：具有平凡规范束和零不规则性的射影代数光滑曲面。它们的几何和模已经被代数几何学者详尽地研究过了，但是关于它们的算术问题，如有理点、与模形式的联系等，还没有被很好地理解。Kumar研究了一些K3曲面族与2属曲线之间的联系，并应用了算术应用，如寻找高秩椭圆曲线和计算希尔伯特模曲面的方程。这个项目将集中在球体填充和算术几何领域。球体填充问题要求用大小相等的非重叠球体来填充空间，以覆盖尽可能大的体积。作为一个几何问题，它与数学的其他部分有许多联系，如数论、群论和李论。它在数学之外也有许多重要的应用，例如，在编码和通信理论以及物理学中。例如，它在连接cd上使用的纠错码、手机信号中使用的球形码以及无线电或更一般的模拟信号时很有用。许多相关的问题，如势能的最小化，是物理学家和材料科学家感兴趣的，例如，纳米粒子的自组装。PI， Kumar和他的合作者在理解高维的球体填充和相关问题上取得了进展，这个项目将继续这项研究。算术几何领域将代数和代数几何的技术应用于数论领域，代数几何是对多项式方程解集的几何研究。例如，人们可能希望知道某个多项式方程组（如费马方程，x^n + y^n = z^n）的整数或有理数解。Kumar的研究重点是K3曲面的算法研究。这些美丽的物体与数学和物理的许多部分都有联系，比如晶格理论和镜像对称。弦理论试图理解宇宙的物理本质，K3表面和相关的几何物体是弦理论的基本兴趣。这个项目将进一步研究这类曲面的数论性质。

This project involves the fields of sphere packing and arithmetic geometry of K3 surfaces. Kumar will continue his research on sphere packing in high dimensions and related problems. The central question in sphere packing asks for the densest packing of Euclidean space by congruent non-overlapping spheres. The answer is known only in dimensions up to 3, and in general this is a very difficult though natural question. Cohn and Kumar have made progress in 8 and 24 dimensions, showing that the sphere packings coming from the E_8 and Leech lattices are very close to being the densest packings in these dimensions. They are investigating related problems, such as that of sphere packing on the surface of a sphere (spherical codes), hyperbolic sphere packing, and how to make these using potential energy minimization. Kumar will also work on the arithmetic of K3 surfaces: projective algebraic smooth surfaces with trivial canonical bundle and zero irregularity. Their geometry and moduli have been studied exhaustively by algebraic geometers, but questions about their arithmetic, such as rational points, connections with modular forms, etc., are not so well understood. Kumar has investigated the connection between some families of K3 surfaces and curves of genus two, with arithmetic applications such as finding elliptic curves of high rank and computing equations of Hilbert modular surfaces. This project will focus on the areas of sphere packing and arithmetic geometry. The sphere packing question asks for a packing of space by equal sized non-overlapping spheres, so as to cover the maximum possible fraction of volume. Stated as a problem in geometry, it nevertheless has many connections to other parts of mathematics, such as number theory, group theory and Lie theory. It also has many important applications outside of mathematics, for instance, in coding and communication theory, and in physics. For instance, it is useful in connection with error correcting codes used on CDs, spherical codes used in cellphone signals, and radio or more general analog signals. Many related problems, such as minimization of potential energy, are of interest to physicists and materials scientists, for instance, in the self-assembly of nanoparticles. The PI, Kumar, and his collaborators have made progress in understanding sphere packing and related problems in high dimensions, and this project will continue that research. The field of arithmetic geometry applies techniques from algebra and algebraic geometry, which is the study of the geometry of sets of solutions to polynomial equations, to the field of number theory. For instance, one might wish to know the solutions in integers or rational number to a certain system of polynomial equations (such as Fermat's equation, x^n + y^n = z^n). Kumar's research focuses on the study of the arithmetic of K3 surfaces. These beautiful objects have connections to many parts of mathematics and physics, such as lattice theory and mirror symmetry. K3 surfaces and related geometric objects are of fundamental interest in string theory, which tries to understand the physical nature of the universe. This project will further the research into the number-theoretic properties of such surfaces.