Modular and quasimodular forms, partitions, and representations of symmetric groups

In 1995, Robbert Dijkgraaf found a "mirror symmetry in dimension one"
statement saying that the generating function counting genus g ramified
coverings of a torus was what is now called a quasimodular form on the
full modular group (i.e., a polynomial in the Eisenstein series E_2,
E_4 and E_6) of weight 2g-2. A mathematical proof of this was given
by Kaneko and myself in a small paper in which we also gave the
definition and basic properties of quasimodular forms. A few years
later, Bloch and Okounkov gave a vast generalization showing that
the generating series associated to a wide class of functions on
partitions are also quasimodular forms. In the course I will explain
this theorem and give a very short proof of it, as well as reviewing
the necessary background material on modular and quasimodular forms
and on the representation theory of finite groups and in particular
of symmetric groups. If time permits, I will also discribe some
extensions and applications of the theory coming from my recent
work with Martin Möller, which was in turn motivated by questions
coming from the theory of flat surfaces and their moduli spaces.