Stable Homotopy Theory, Formal Groups, and Elliptic Curves

One of the most challenging problems in algebraic topology is to compute the stable homotopy groups of spheres. There are many tools we have developed to compute these groups, but at first glance these groups don’t seem to follow any patterns. We now know that this is far from the case and that, in actuality, the stable homotopy groups of spheres are comprised of various "periodic families." In this talk, I will give an overview of this story and, time permitting, I will discuss some current questions on the relationship between some of these patterns and elliptic curves.