Abstract: We are familiar with the prime numbers as those integers which cannot be factored into smaller integers, but if we consider systems of numbers larger than the integers, the primes may indeed factor in those larger systems. We discuss various questions mathematicians ask how the primes may react in larger number systems. We will talk about both classical results and current research on the topic, and give a sense of the kind of tools needed to tackle these questions.
NB: PDF version of this announcement (suitable for posting).
Abstract: There are certain finite abelian groups that arise from objects in number theory that are quite mysterious and of great interest, for example the class group arising from a finite extension of the rational numbers, or the Tate-Shafarevich group of an elliptic curve y^2=x^3+ax+b (for some rational numbers a,b). We discuss the question of what a class group of a random extension, or the Tate-Shafarevich group of a random elliptic curve, looks like, and explain heuristics, including the Cohen-Lenstra heuristics, for how these random groups are distributed. In particular, we will explain a new theorem that the conjectured distribution is "universal" (in the sense that the Central Limit Theorem shows that the normal distribution is universal).
Abstract: The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois group of the maximal unramified extension as a non-abelian generalization of the class group. We will explain non-abelian analogs of the Cohen-Lenstra heuristics due to Boston, Bush, and Hajir and joint work with Boston proving cases of the non-abelian conjectures in the function field analog.