Period Domains and Number Theory

数论是数学的一个分支，有两千多年的历史，但它今天仍然是一个充满活力的学科，连接着几个不同的数学领域和物理学。这门学科最近的许多进展都来自于对数论与几何和代数之间联系的探索。本研究计画将研究允许退化的算术与分析对象的延伸分类空间，并将结果应用于数论。 本课题涉及五名研究生，主要研究数域上的混合Hodge结构、混合p-adic Hodge结构、Drinfeld模和动机。 研究者打算将扩展分类空间上的先前结果应用于研究岩泽理论、Sharifi拓扑、高度的渐近行为、调节器、数域上动机的高度配对以及Hodge度量及其p-adic Hodge版本的退化。 该研究项目将开发数学和科学的不同领域之间的新的相互作用，包括算术，物理，谐波分析和表示理论。 Drinfeld模的广义分类空间的研究将应用于函数域的Langlands对应，而Hodge结构的退化与物理学有关。

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.