Math 549. Elliptic cohomology. Fall, 2003.The purpose of this course is to introduce elliptic cohomology, a subject of interest to topologists, number theorists and mathematical physicists. I will begin with an expository introduction to elliptic curves (aka Riemann surfaces of genus 1), formal group laws and modular forms. This will include an explanation of why elliptic curves are called elliptic; it has very little to do with actual ellipses. Elliptic curves have been central objects in mathematics for the past century or so. They figure prominently in Wiles' proof of Fermat's Last Theorem and in cryptography, neither of which will be covered in this course
After that I will spend the rest of the semester explaining how they are related to algebraic topology. This will entail an introduction to stable homotopy theory, generalized homology theories, and complex cobordism. Much of this material is covered in Frank Adams' book Stable Homotopy and Generalised Homology.