The goal of this course is to understand the construction of "genuinely equivariant" cohomology theories from an algebraic point of view. We will survey the original topological contexts of these theories and discuss the emerging perspective of cohomology as functions on loop spaces.
The course will be roughly divided into two parts. In the first part we focus on the topological context in which generalized cohomology theories were originally conceived, essentially following the work of Quillen, Atiyah, Segal, and Grojnowski :
- Formal group laws
- Complex cobordism
- Construction of "Borel equivariant" elliptic cohomology theories.
In the second part, we turn to modern constructions of "genuinely equivariant" cohomology theories using tools from (mildly derived) algebraic geometry. Some topics that will be discussed include :
- Algebraic analogues of loop spaces
- The Tate construction
- Comparison of the algebraic and topological approaches
- Loop master: Eric Yen-Yo Chen
- Teacher: Kamil Rychlewicz
