170625 Sven Nigtevecht

Modular forms have long played an important role in number theory. More recently, an analogue of modular forms has been constructed in algebraic topology, forming an object known as topological modular forms (tmf). It has played an important role in many important results in algebraic topology, in particular in computations of homotopy groups of spheres. However, although tmf was constructed in the 90s, a complete and linear proof of certain foundational computations of it have not appeared in the literature. Using the recent technology in homotopy theory of synthetic spectra, in joint work with Christian Carrick and Jack Davies, we define an object called synthetic modular forms, and use this to complete the computation of topological modular forms. In this talk, I will motivate tmf and synthetic spectra, giving their broader context, as well as a glimpse of how these are used computationally.