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Description: 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. |
