Topological Modular Forms

Samstag, 14. Dezember 2019, 9:30–18 Uhr, Universität Bayreuth

Saturday, December 14, 2019, 9:30 am to 6 pm, Bayreuth University


Leon Hendrian (Universität Münster), Timo Keller (Universität Bayreuth)

Short description

The goal of this Bayerische Arbeitsgemeinschaft is to introduce interested (aspiring) Algebraic Geometers and Algebraic Topologists to the spectrum called TMF.

TMF stands for Topological Modular Forms, which are in some sense the \enquote{higher version} (in the sense of „Higher Algebra“) of the classical ring of modular forms, which is an interesting object of study in Arithmetic Geometry. There, they arise as generalised functions on the moduli space of elliptic curves.

In the first talk, we will revise some fundamental notions of central importance such as formal group laws and their moduli stack $\mathcal{M}_\text{FG}$.

After that, we will spend the second and third talk learning about the algebro-geometric theory of elliptic curves, their moduli problem and its solutions via different kinds of moduli schemes and stacks. In the third talk, we shall encounter modular forms from multiple perspectives, e.g. classically as certain functions on the upper-half plane and as sections of the sheaf $\omega^{\otimes k}$ on certain moduli spaces of elliptic curves.

In the fourth talk, the object $TMF$ shall finally be introduced. Its construction is difficult and relies on $\mathcal{O}^{top}$, a sheaf of $E_\infty$-ring spectra whose existence relies on a theorem of Goerss-Hopkins-Miller.

Finally, in the fifth talk we will try to compute (some of) the homotopy groups of $TMF$ using tools such as the descent and the Adams-Novikov spectral sequence.

Programme (English):
Poster (English)

All talks should last at most one hour.

  • Talk 1: Prelude (Leon Hendrian)
  • Talk 2: Elliptic Curves (Timo Keller)
  • Talk 3: Moduli Spaces of Elliptic Curves and Modular Forms (Timo Keller)
  • Talk 4: Topological Modular Forms (Stefano)
  • Talk 5: Computation of π_*(TMF) via the Descent and Adams-Novikov Spectral Sequences (Aras Ergus)