CAREER: Randomness in Number Theory and Beyond

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

This project will lead to a deeper understanding of the statistics of fundamental behaviors of numbers, including how numbers factor and how many solutions there are to given equations. The approach is to use random models to study these seemingly deterministic questions. This research will investigate how mysterious microlevel numerical structures aggregate into universal macrolevel patterns. A fundamental example of this micro-to-macro phenomenon is seen in the ubiquity of the bell curve, which describes many distributions seen in nature, even when they come from different sources. The project will help uncover the analog of the bell curve for fundamental behaviors of numbers. Specifically, the distribution of class groups of number fields and function fields will be studied using tools from probability, random groups, arithmetic geometry, algebraic geometry, topology, and number theory. The project will develop models of random groups that are good approximations for class groups and their non-abelian analogs, and determine the basic probabilistic structure of these models. Tools from algebraic geometry and topology will be used to prove that distributions of class groups of function fields have behavior that agrees with what is predicted by the models. This will exhibit new structure in the class groups of number fields and their non-abelian analogs.