Winter 2002, MWF 2:30-3:20
Although formal groups are of
interest to various sorts of algebraists, topologists first became interested
in the subject after the work
of Quillen in the 1960's, who
discovered a remarkable connection between complex bordism and formal group
laws. This connection has been
exploited so that it now plays a key role in our understanding of global
phenomena in stable homotopy theory. In
this course, I intend to discuss formal groups from the viewpoint of an algebraic
topologist, although I will try to present other viewpoints as well. I will mention the connections to homotopy
theory; however, the treatment of formal groups will be algebraic. I intend to focus on one dimensional
commutative formal groups and hope to cover the following topics: Lazard's theorem on the structure of the universal
formal group law, the classification of formal groups over algebraically closed
fields and the automorphisms of such formal groups, and the Lubin-Tate theory of
lifts of formal groups. As time
permits, we will study the Dieudonn\'e module associated to a formal group and
outline Cartier's solution to the lifting problem.
Prerequisites: certainly 504-6, and probably 545 should be
taken at least concurrently. The more
algebraic topology you know, the
more meaningful this course
may be; however, the theory of formal groups themselves will be presented
without algebraic topology.