Tilman Bauer: Formal plethories and Dieudonné modules

Given some cohomology theory K, its value K^*(X) on a space X has a rich structure: it is a module over the coefficient ring K^*(pt), an algebra (under the cup product), and the operations of K act on it, i.e. the natural transformations of functors K^i \to K^j. The best way to formulate all of this structure uses the language of algebraic geometry, and I will describe how; the central object of study here is a commutative affine ring scheme with additional structure, called ``formal plethory''. For nice cohomology theories K, the operations form a graded bicommutative Hopf algebra; such Hopf algebras form an abelian category, which is equivalent to the category of modules over some ring. I will describe these Dieudonné modules and show how they can be used to give a compact description of unstable operations. I will also touch on how one uses this machinery to compute unstable homotopy groups.