Stability Phenomena in Number Theory, Algebraic Geometry, and Topology

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Number theory is one of the oldest and purest areas of mathematics, unchanged in many ways since the time of Euclid, but in recent years it has incorporated ideas and techniques from a wide range of other mathematical areas. This research project stands at the interface between classical questions about whole numbers and ideas from other subjects. In one main project, the PI and his collaborators show how results in algebraic topology, the study of high-dimensional shapes and the relations between them, translate into statements about the arithmetic of various number systems. In another, he and another group develop a new form of representation theory (the study of symmetries of linear spaces), which sheds light on phenomena of stabilization in number theory, topology, and algebra. To a first approximation, the work answers the question: when can an infinite object be described by a finite amount of data? The PI will also continue his work in mathematical outreach, including a general-audience book to be released in 2014. This project investigates the Cohen-Lenstra conjectures concerning the variation of the p-part of the class group of number fields, and, more generally, distributional questions about the discriminants of G-extensions for G an arbitrary finite group. The methods used are novel -- the PI and collaborators show that the Cohen-Lenstra conjectures follow from assertions about the cohomology of certain moduli spaces of branched covers of the complex projective line, known as Hurwitz spaces. These spaces can be defined purely topologically, and in fact the thrust of the work has been to show that new theorems in algebraic topology imply many popular conjectures about arithmetic statistics over function fields. What's more, the topological results serve as a kind of machine for generating conjectures, or at least heuristics, about questions concerning the distribution of G-extensions over Q which have not yet been investigated. For instance, the results suggest that if N is a random squarefree integer chosen uniformly from a large range, and X is the number of totally real quintic extensions with discriminant N, then X has the Poisson distribution with mean 1/120. A new aspect of the project is the theory of FI-modules, developed by the PI in collaboration with Tom Church and Benson Farb. This theory represents a new approach to homological stability, whose natural domain of application is not sequences of unadorned vector spaces but rather sequences of vector spaces whose nth term is a representation of the symmetric group on n letters. It turns out that there is a natural abelian category, called the category of FI-modules, which captures a broad spectrum of phenomena ranging from cohomology of moduli spaces to the coinvariant algebras arising in algebraic combinatorics to the statistics of squarefree polynomials and tori in Lie groups over finite fields. In this research project, besides continuing investigation of the inherent structure of the category of FI-modules, it is planned to bring this work into contact with other work with Venkatesh and Westerland. A typical question to be investigated is: are there infinitely many cubic extensions of a rational function field over a finite field (or, better: what is the expected number of cubic extensions, asymptotically) whose discriminant is prime (i.e. an irreducible polynomial over the finite field)? The corresponding question over Q is a well-known open problem. The project will also address a suite of other problems; geometric analogues of and approaches to the Kakeya problem in harmonic analysis, random matrices and the proportion of ordinary curves over finite fields, and, on the applied side, some questions about the application of geometry to problems in data science.