Cohomology Theories, Equivariance, and Loop Spaces

MATH-749 / 2 credits

Teacher:

Language: English

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.

Resources

Moodle Link

In the programs

  • Exam form: Oral presentation (session free)
  • Subject examined: Cohomology Theories, Equivariance, and Loop Spaces
  • Courses: 20 Hour(s)
  • TP: 16 Hour(s)
  • Type: optional

Reference week

DateTimeRoomCourse
Monday 07.09.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 14.09.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 28.09.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 05.10.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 12.10.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 26.10.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 02.11.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 09.11.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 16.11.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 23.11.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 30.11.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 07.12.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
Monday 14.12.202614:15-15:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces
15:15-16:00MAA112MATH-749 Cohomology Theories, Equivariance, and Loop Spaces

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