The theory of modular forms, whose infancy goes back to Gauss
and the other great number theorists of the 19th century, was
developed in the 20th century by Hecke and his followers and
is one of the central pillars of modern number theory. In this
lecture series, which does not presuppose any prior knowledge
of modular forms, some less well-known aspects of the theory
which have emerged in recent years will be explored. The contents
of the individual lectures, if all goes according to plan, will
be roughly as follows:
Lecture 1. Tuesday, September 20, 11:15 a.m.
Modular forms and their derivatives.
After reviewing basic definitions, we will discuss the derivatives of modular forms (which are not quite modular but something nearly as good called "quasimodular") and some of their applications, e.g. to a problem concerning periodic functions and to a problem coming from percolation theory.
Lecture 2. Thursday, September 22, 11:15 a.m.
Modular forms and differential equations.
This topic is closely related to the first but with applications in quite different directions, including Beukers's version of Apery's famous proof of the irrationality of &zeta(3).
Lecture 3. Tuesday, September 27, 11:15 a.m.
Quadratic functions and modular forms.
This lecture, perhaps the most elementary of the series (and independent of its predecessors), shows how modular forms arise naturally from - and provide the unexpected answer to - a simple question concerning sums of quadratic polynomials.
Lecture 4. Thursday, September 29, 11:15 a.m.
Period polynomials of modular forms.
It has been known for a long time that one can associate to
modular forms certain polynomials in one variable called "period
which contain all the essential arithmetic information. We will
describe how this works, and also a nice connection with the numbers
(originally discovered by Euler) called "multiple zeta values".
Lecture 5. Tuesday, October 4, 11:15 a.m.
Mock theta functions and theta series of indefinite quadratic forms.
Theta series are one of the most important sources of modular forms and have many applications in pure and applied mathematics (sphere packing, coding theory,...). They are associated to positive definite quadratic forms. Trying to extend the theory to include indefinite forms leads to new arithmetic questions and connections with the mock theta functions of Ramanujan treated in the colloquium lecture.
Lecture 6. Thursday, October 6, 11:15 a.m.
Quantum modular forms.
This topic, related to each of the last three above, has to do with the limiting values of modular forms at roots of unity. At the same time, modular forms (and mock modular forms) have begun to occur in connection with quantum invariants of 3-manifolds, and this promises exciting developments for the future.