MPIM Topology Seminar: Miniseries on topological modular forms and synthetic spectra

Topological modular forms (tmf) have long played a big role in modern homotopy theory, with applications ranging from studying bordism, to being a key part of the computation of stable homotopy groups of spheres. However, a complete computation of its homotopy groups has never appeared in the literature. 

 

In this mini-series, we will describe our approach to this computation using modern techniques in synthetic spectra and introducing a synthetic spectrum we call Synthetic modular forms (Smf). In the first talk for a general audience, we will introduce tmf and Smf as well as give an outline of how synthetic spectra naturally appear in our work. In the second talk, a series of generic tools from synthetic spectra will be discussed, giving us general tricks and techniques to compute with these objects. In the final talk, a detailed computation of the spectral sequence associated to Smf, also called the descent spectral sequence, giving a complete calculation of the homotopy groups of tmf.