Topological modular forms (aftern Hopkins, Miller, and Lurie)

TL;DR

This paper explains the connection between homotopy theory and algebraic geometry via the Hopkins-Miller-Lurie theorem on topological modular forms, highlighting the realization of the moduli stack of elliptic curves in derived algebraic geometry.

Contribution

It elucidates the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie theorem, emphasizing the derived algebraic geometry perspective.

Findings

Realization of the moduli stack of elliptic curves in derived algebraic geometry

Explanation of the Hopkins-Miller-Lurie theorem on topological modular forms

Connection between homotopy theory and algebraic geometry

Abstract

This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie theorem on topological modular forms, from which we learn that the Deligne-Mumford moduli stack for elliptic curves is canonically realized as an object in derived algebraic geometry.