Humboldt-Universität zu Berlin Math.- Nat. Fakultät Institut für Physik in english

4020205188 Modular forms and applications       

Präsenzkurs

Unterrichtssprache

DE

Voraussetzungen

complex analysis. The necessary knowledge on elliptic
curves will be introduced in the talks.

Gliederung / Themen / Inhalte

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.

Zugeordnete Module

P27 P28

Umfang, Studienpunkte; Modulabschlussprüfung / Leistungsnachweis

2 SWS, 4 SP/ECTS (Arbeitsanteil im Modul für diese Lehrveranstaltung, nicht verbindlich)
Validation by regular attendance and delivering a talk.

Sonstiges

Tuesdays 2pm-3.45pm. Starting date: 03 November 2020.