Synthetic modular forms

Topological modular forms (tmf), as introduced by Mike Hopkins and his coauthors, have long been the subject of much study and use in homotopy theory. However, a complete proof of the computation of the homotopy groups of tmf has never appeared in the literature. This thesis gives a proof, discovered in collaboration with Christian Carrick and Jack Davies, of the so-called Gap Theorem for topological modular forms, as well as a computation of the homotopy groups of tmf. The crucial technique in our proof is the use of the recently-developed theory of synthetic spectra. More specifically, we introduce an object we call synthetic modular forms, and use this as the centrepiece for our proof and computation. In addition, this thesis contains an introduction to synthetic spectra and spectral sequences, both with the aim of deepening the theory to make the aforementioned proof possible, as well as being a general reference to learn about synthetic spectra.