Math 205: Topics in Number Theory (Winter 2018)

Arithmetic Statistics

Arithmetic statistics is the study of the distribution of invariants of objects of interest in number theory. Some typical questions in arithmetic statistics are the following. What is the probability that a random integer is squarefree, or is prime? How many number fields of degree d are there with discriminant of absolute value at most X? What does the class group of a random quadratic field look like? 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 course, we will discuss some of these topics, but we will also concentrate on those questions with a special emphasis on what the statistics looks like when we start from the field of rational functions over a finite field (or, more generally, the function field of a curve over a finite field) instead of the field of rational numbers. Then the questions about number fields become questions about covers of the projective line, the questions about class groups become questions about groups of rational points on Jacobians, and so on.

Instructor:	Alina Bucur (alina at math dot ucsd dot edu), APM 7151
Lectures:	MWF 1-2pm, APM 2402
Office hours:	by appointment only (email!)

Books:	There are no textbooks on the subject!
Other sources:	Will be posted as we go along.

Prerequisites:	MATH 204ABC.
Work:	There will be some assignments, no exams, and a project. The goal of the assignments is to supplement what is said in class. The goal of the project (which I promise to read!) is for you to elaborate on some topic in arithmetic statistics beyond what is covered in class. Your topic will be chosen in consultation with me; I will have some suggestions as the term goes on.

First lecture: Monday, January 8.
Last lecture: Friday, March 16.
UCSD holidays this term and therefore no lecture: Monday, January 15 and Monday, February 19.
Other dates with no lectures: Monday March 5 and Wednesday March 7. I will schedule makeup lectures to replace these.