Modular forms are certain functions in the upper-half of the complex
plane (or q-series) that transform nicely under the action of SL(2,Z).
They can be thought as an analog (on the moduli space of elliptic
curves) of polynomials (on the Riemann sphere). Many relations between
them follow from the fact they form a finite-dimensional vector space,
with beautiful consequences in many different areas of mathematics, not
limited to arithmetic. They also appear in physics, e.g. in
low-dimensional topology, conformal field theory and string theory.
The
first session will consist of a presentation by the lecturer and
discussion for the planning of the next talks by the participants. The
first half of the semester will cover the basics. Later on, the
participants will select a few advanced topics for presentation, for
instance among: Hecke theory and L functions ; Viazovska's theorem on
optimal sphere packing in dimension 8 ; mock and quantum modular forms ;
modular forms in conformal field theory and the monstruous moonshine ;
Bloch-Okounkov theorem and enumeration of branched covering of the
torus.
Prerequisite: complex analysis. The necessary knowledge on elliptic curves will be introduced in the talks.
Validation by regular attendance and delivering one talk during the term.
- Course owner: Gaetan Borot