CAREER: Arithmetic, Algebraic, and Non-Archimedean Geometry

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This is a project to bring abstract and often foundational aspects of number theory, algebra, geometry, topology, and combinatorics to bear on concrete and explicit questions in arithmetic, geometry, and classical number theory. The basic problem is to find or describe all integer solutions of polynomial equations that arise in mathematics, especially cryptography, and the physical sciences. The simplicity and intrinsic beauty of these problems, and the disproportionate depth and effort of their resolution, has inspired their study since ancient Greece. The central questions that arise naturally stratify as follows: Qualitative questions: Does there exist a solution? Do there exist infinitely many solutions? Does the set of solutions have some extra structure (e.g. geometric)? Quantitative questions: How many solutions are there? How large is the smallest solution? How can we explicitly and with certainty find all solutions? Implicit questions: Why do equations have (or fail to have) solutions? Why do some have many and some have none? What underlying mathematical structures control this? The research in this proposal is complemented by educational and outreach activities, including the creation of a summer graduate research workshop on arithmetic geometry and the continued development of the open, collaborative MathOverflow web site. This proposal aims to attack the problems described above with tools from less concrete, often foundational fields (e.g., non-archimedan geometry, algebraic stacks, non-abelian methods, perfectoids), with the goals of proving new results on the generalized Fermat equation; developing techniques to find, with certainty, all solutions to a given equation; developing techniques to bound the number of solutions to a given equation; and building new tools (geometric/cohomological) to study the structure of solutions of equations.