Cohomological periods and high rank lattices

正如笛卡尔所认识到的那样，代数方程的解可以用几何方法来实现。这一发现是代数和几何之间丰富互动的开始。本专题将研究数论中的两个主题。第一个问题涉及算术流形的形状，即，由数论对称性定义的几何PI已经证实了新结构的存在，这些新结构控制着它们的形状（从数学上讲，它们的拓扑结构），他将对此进行更详细的研究。第二个主题涉及到高维格。这些都是现代密码学中的一个有趣的话题;另一方面，他们还没有一个令人满意的数学理论，这个项目的目的是发展这样一个理论。更具体地说，PI提出了一个猜想，该猜想指定了算术局部对称空间的“周期”的值，即，将同调类与标准化微分形式配对得到的数。这个猜想很有趣，因为它暗示了这些同调群和某些动机上同调群之间的关系。PI将研究这个猜想，并试图证明它。关于格，PI将特别研究以下问题：什么是n维格空间的直径，在一个典型的n维格中的短向量如何表现，为什么LLL格约化算法表现得这么好？一个基本的工具将是分析GL（n）上的自守形式，对于大的n，PI也将研究高维极限下自守形式的相关问题。

As was realized by Descartes, the solution of algebraic equations can be realized geometrically. This observation was the start of a rich interaction between algebra and geometry. This project will study two topics in number theory. The first concerns the shape of arithmetic manifolds -- i.e., certain geometries defined by their number theoretic symmetries. The PI has conjectured the existence of new structures that govern their shape (mathematically speaking their topology), which he will investigate in more detail. The second topic relates to lattices of high dimension. These are a topic of interest in modern cryptography; on the other hand, a satisfactory mathematical theory of them is not yet available, and this project aims to develop such a theory. More specifically, the PI has formulated a conjecture that specifies the values of "periods" of arithmetic locally symmetric spaces -- i.e., the numbers obtained by pairing homology classes with normalized differential forms. This conjecture is interesting because it suggests a relationship between these homology groups, and certain motivic cohomology groups. The PI will study this conjecture and attempt to give evidence for it. Concerning lattices, the PI will study in particular the following questions: What is the diameter of the space of n-dimensional lattices, how do the short vectors in a typical n-dimensional lattice behave, and why does the LLL lattice reduction algorithm behave so well? A basic tool will be the analysis of automorphic forms on GL(n) for large n, and the PI will also study related questions about automorphic forms in the high-dimensional limit.