Elliptic cohomology and dualizability, I

The ring of topological modular forms arises as functions on the moduli stack of elliptic curves, suitably interpreted in a homotopical sense. To see more of the geometry of elliptic cohomology, we would like to interpret the cohomology of classifying spaces similarly, in terms of schemes related to symmetric powers of elliptic curves.

In this minicourse, we will begin by recalling the link between elliptic curves and chromatic homotopy theory via formal groups, as well as the related but simpler case of K-theory and the multiplicative group. Next, we will describe how an orientation of the formal group allows one to extend elliptic cohomology from topological spaces to topological stacks, and describe the computation of the elliptic cohomology of circle-bundles. Finally, we will consider more general groups, and indicate why elliptic cohomology see the of the stack of principal G-bundles as dualizable, in contrast to the case of K-theory.

The prerequisites for this course are a basic understanding of algebraic topology, algebraic geometry, and category theory. We will not assume familiarly with spectral algebraic geometry, though we will introduce this language in an informal way.