Period Domains and Number Theory

数论是数学的一个分支，有两千多年的历史，但今天它仍然是一个充满活力的学科，连接了几个不同的数学领域和物理学。这一学科最近的许多进展都来自于探索数论与几何和代数的联系。本研究项目将研究允许退化的算术和解析对象的扩展分类空间，并将结果应用于数论。这个项目有五名研究生参与研究。本课题研究混合Hodge结构、混合p进Hodge结构、Drinfeld模块和数字域上的动机。研究者打算将先前在扩展分类空间上的成果应用于Iwasawa理论、Sharifi猜想、数域上动机的高度、调节和高度配对的渐近行为、Hodge度量及其p进Hodge版本的退化等方面的研究。这个研究项目将发展数学和科学不同领域之间的新互动，包括算术、物理、谐波分析和表示理论。对Drinfeld模的扩展分类空间的研究将应用于函数场的Langlands对应，而Hodge结构的退化则与物理有关。

Number theory is a branch of mathematics with a history of over two thousand years, but it remains a vibrant subject today, linking several different mathematical fields and physics. Much recent progress in the subject has come from exploring links of number theory with geometry and algebra. This research project will study extended classifying spaces of arithmetic and analytic objects that allow degenerations, and apply the results to number theory. The project involves five graduate students in the research.This project studies mixed Hodge structures, mixed p-adic Hodge structures, Drinfeld modules, and motives over number fields. The investigator intends to apply prior results on extended classifying spaces to the study of Iwasawa theory, the Sharifi conjectures, asymptotic behaviors of heights, regulators, and height pairings of motives over number fields, and degeneration of Hodge metrics and their p-adic Hodge versions. This research project will develop new interactions between different areas of mathematics and science, including arithmetic, physics, harmonic analysis, and representation theory. The study of extended classifying spaces of Drinfeld modules will have applications to Langlands correspondences for function fields, and degenerations of Hodge structures are related to physics.