Anabelian Methods in Arithmetic and Algebraic Geometry

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Algebraic geometry and number theory are both concerned with systems of polynomial equations. The former seeks to understand the shapes they define, and the latter seeks to understand the set of (say) integer or rational solutions to the system. One of the most important themes of 21st century mathematics is the interplay between these two viewpoints: the properties of the shape in question influence, and are in turn influenced by, the arithmetic of the equations. This project seeks to understand this fundamental interplay through a study of the arithmetic of fundamental groups of algebraic varieties.Specifically, this project is concerned with (1) an analysis of the representation theory of arithmetic fundamental groups, taking as input the geometric Langlands program, tools from p-adic analysis, and p-adic Hodge theory, and (2) an application of these results to concrete problems in arithmetic and algebraic geometry: for example, the geometric torsion conjecture, the non-abelian Chabauty-Kim method, and the section conjecture. The project aims to expand on new techniques developed by the author and collaborators to investigate these difficult open questions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.