Cohomology Theories, Equivariance, and Loop Spaces
MATH-749 / 2 crédits
Enseignant:
Langue: Anglais
Remark: Fall 2026
Frequency
Only this year
Summary
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.
Content
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
Keywords
K-theory, elliptic cohomology, loop spaces, equivariant cohomology theories.
Learning Prerequisites
Required courses
Familiarity with ordinary equivariant cohomology, derived categories, and algebraic stacks.
Learning Outcomes
By the end of the course, the student must be able to:
- Describe topological and algebraic constructions of equivariant cohomology theories.
Dans les plans d'études
- Forme de l'examen: Exposé (session libre)
- Matière examinée: Cohomology Theories, Equivariance, and Loop Spaces
- Cours: 20 Heure(s)
- TP: 16 Heure(s)
- Type: optionnel