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**Lecture 1.**
**The history of the partition function**

**Abstract**

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**

**Abstract**

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**