Algebraic topology and modular forms

TL;DR

This paper discusses the development and recent advances in topological modular forms, a topological refinement of classical modular forms, and explores their applications in algebraic topology and related fields.

Contribution

It presents the completion of a program relating topological modular forms to manifold invariants, homotopy groups, and classical modular forms, advancing the understanding of their interplay.

Findings

Completion of the program relating topological modular forms to manifold invariants.

New directions in the study of topological modular forms and their applications.

Enhanced understanding of the role of modular forms in algebraic topology.

Abstract

Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new directions have emerged. In this talk I will describe this recent work and how it informs our understanding of both algebraic topology and modular forms.