Ken Ono, Pennsylvania State University and University of Wisconsin, Madison, U.S.A.
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Lecture 1.
The history of the partition function

In this lecture I will describe some of the most classical results about p(n), the ordinary partition function. This will include simple consequences of the form of its generating function, its asymptotic formula, and the Ramanujan congruences. This will essentially be a review of works of Euler, Hardy, Ramanujan, Rademacher, Watson, Atkin, Swinnerton-Dyer, Dyson, Andrews, Garvan, Kim and Stanton.

Lecture 2.
Ramanujan's congruences and modern number theory

In the second lecture I will revisit Ramanujan's congruences for p(n). I will illustrate how advances in the theory of modular forms allows us to understand p(n) from a deeper perspective. For example, I will give a brief overview of some works of Deligne, Serre and Shimura which allows us to prove that every positive integer M coprime to 6 has many pairs of positive integers A and B for which

        p(An+B)=0 mod M

for every n. I will also describe how this perspective clearly suggests that partitions label elements in Tate-Shafarevich groups of certain modular motives. These new connections suggest that generalizations of the ranks and cranks reveal deeper structures in arithmetic geometry.

View Lectures Text in: pdf format, ps