The subject of modular forms is central to several branches of mathematics, ranging from number theory, algebraic geometry, representation theory, and complex analysis. It also has intimate connections to theoretical physics. The celebrated Fermat's last theorem could not have been solved without this theory. The new techniques and tools introduced in the last thirty years make a basic understanding of the theory of modular forms indispensable for any graduate student of pure mathematics. This intense short course is motivated by these developments.
The course is primarily aimed at Master's and Ph.D. students with appropriate background and interest. The length of the course will be two weeks. The format will be as follows: each morning, there will be a lecture on the topics listed below for about 1.5 hours. In the afternoon, there will be a tutorial of about 2 hours long, indicating how one applies the theoretical knowledge of the morning lecture to concrete problems.
- Lecture 1: q-series (Aug 24)
- Lecture 2: The modular group (Aug 25)
- Lecture 3: The upper half-plane (Aug 26)
- Lecture 4: Modular forms of level one (Aug 27)
- Lecture 5: The Ramanujan 𝝉-function (Aug 29)
- Lecture 6: Modular forms of higher level (Aug 30)
- Lecture 7: Petersson inner product (Aug 31)
- Lecture 8: Hecke operators of higher level (Sept 1)
- Lecture 9: Dirichlet series and modular forms (Sept 2)
- Lecture 10: Special topics (Sept 3)
In addition to the evaluation based on the
tutorials, an exam will be conducted on the day after lectures are
completed. The text for the course is “Problems in the theory of
modular forms" by M. Ram Murty, M. Dewar and H. Graves, which is Volume
1 of the Lecture Notes series of the Institute of Mathematical Sciences
in Chennai, India.
Picture courtesy : Matthias Beck, http://math.sfsu.edu/beck/