### Athens-Atlanta number theory seminar 2 - Arithmetic statistics over function fields

- Series
- Algebra Seminar
- Time
- Tuesday, April 16, 2013 - 17:15 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jordan Ellenberg – University of Wisconsin

What is the probability that a random integer is squarefree? Prime? How
many number fields of degree d are there with discriminant at most X?
What does the class group of a random quadratic field look like? These
questions, and many more like them, are part of the very active subject
of arithmetic statistics. Many aspects of the subject are well-understood,
but many more remain the subject of conjectures, by Cohen-Lenstra,
Malle, Bhargava, Batyrev-Manin, and others.
In this talk, I explain what arithmetic statistics looks like when we
start from
the field Fq(x) of rational functions over a finite field instead of
the field Q
of rational numbers. The analogy between function fields and number
fields
has been a rich source of insights throughout the modern history of number
theory. In this setting, the analogy reveals a surprising relationship
between
conjectures in number theory and conjectures in topology about stable
cohomology of moduli spaces, especially spaces related to Artin's braid
group. I will discuss some recent work in this area, in which new theorems
about the topology of moduli spaces lead to proofs of arithmetic
conjectures
over function fields, and to new, topologically motivated questions
about
counting arithmetic objects.