**Algebraic groups, ellipic curves, and the
stable homotopy groups of spheres**

Abstract: The problem of computing the stable homotopy groups of spheres has been central to algebraic topology at least since the 1930s, but it is

only in the last 30 years that we have progressed from intricate and confusing case-by-case calculations to a wider understanding of more

global phenomena. At the core of this understanding is the idea that we should be computing with families of complex orientable cohomology

theories -- those with a good theory of Chern classes. These families can be built using the algebraic geometry of formal groups, which in turn can

be studied using families of groups from algebraic geometry and algebraic number theory. For example, much of our current knowledge can be captured

using cohomology theories built from elliptic curves by Hopkins and his coauthors. In this talk, I will explain these interconnections, examine some

classical calculations from this point of view, and give an overview of the current directions.