Period Domains and Number Theory

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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.