Anabelian methods in arithmetic and algebraic geometry

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This proposal aims to further develop the beautiful, though not yet well-understood, connections between algebraic geometry, number theory, and topology. Algebraic varieties -- the sets of solutions to systems of polynomial equations -- are ubiquitous in mathematics, and their fundamental nature comes in part from the fact that their study lies at the intersection of such varied fields of mathematics. The goal of this proposal is to unwind the connections between these seemingly disparate fields, primarily through the study of the fundamental group of an algebraic variety, an invariant which loosely speaking captures the structure of loops in the variety. Recent discoveries have shown that this invariant is crucial to understanding, for example, rational solutions to systems of polynomial equations -- such so-called "Diophantine" questions have fascinated mathematicians for millenia. Despite their long history, we are only now beginning to understand the connections of such questions to topology, via the section conjecture, the non-abelian Chabauty method, and other extremely recent developments. Broadly speaking, the aspect of algebraic geometry and number theory connected to the fundamental group is called "anabelian geometry," which is the subject of the proposal. Building on my previous work, I plan to better understand anabelian aspects of the topology of algebraic varieties, and in particular the relationship between anabelian geometry and monodromy representations, with the goal of proving two well-known open questions in geometry: the geometric torsion conjecture and the (conjectural) Hard Lefschetz theorem in positive characteristic. Progress on these questions would fundamentally advance our understanding of the topology of algebraic varieties. I also plan to make progress (in joint work with Aaron Landesman) on the Putman-Wieland conjecture, a fundamental question in the topology of surfaces, by bringing to bear algebro-geometric and topological techniques; similarly, my joint work with Li, Salter, and Srinivasan shows that such techniques can yield insight into Grothendieck's section conjecture, perhaps the fundamental (conjectural) connection between anabelian geometry and arithmetic. This work will also yield insight into the topology of moduli spaces, one of the fundamental objects of study in algebraic geometry. Finally, this proposal will build on very recent developments in arithmetic geometry -- in particular, the non-abelian Chabauty method -- to develop practical methods for solving arithmetic questions. In particular, joint work with Eric Katz will yield techniques for running the non-abelian Chabauty method to find rational points on curves of bad reduction, which will be crucial to make the method practical as a way to find solutions to systems of polynomial equations.