Research talk: Ramanujan, partitions, and modular forms

Modular forms, at the interface between number theory, analysis, and representation theory, are one of the central and most beautiful areas of modern mathematical research.  In this talk I will explain a little what they are and how to use them, using Ramanujan and some of his work as a point of departure.  In particular, I will describe his famous congruences modulo 5, 7 and 11 and an extremely simple proof of them, and also tell about his even more famous  paper with Hardy on the asymptotics of the number of partitions of an integer, in which they invented what is now known as the circle method.  I will end by describing some recent research of my own generalizing this last work.  No previous familiarity with the subject will be assumed.