The focus of this summer school is the homotopy theoretic approach to the study of moduli spaces. We will focus on the developments that followed Ib Madsen and Michael Weiss' proof in 2002 of a generalized version of a conjecture of Mumford on the stable cohomology of the moduli space of curves.

## Program

An outline of the main series of lectures (mostly to be given by participants) is found below. A detailed description of the content of each of the planned talks, including references, is available here. We will also have several lectures on more specialized topics, as well as exercise sessions to practice the material covered in lectures. These will be announced later.

#### What is a moduli space?

- Introduction. By the organizers.
- Teichmüller theory and moduli spaces. By Luba Stein (Bonn).
- Pontryagin-Thom theory I. By Hiroaki Tanaka (Northwestern).
- Pontryagin-Thom theory II. By Sven Führing (Munich).

#### Gromov's h-principle

- Gromov's theorem: statement. By Ben Cooper (Virginia).
- Gromov's theorem: applications and the easy part of the proof. By Emanuele Dotto (Copenhagen).
- Gromov's theorem: the difficult part of the proof. By Urs Fuchs (Purdue).

#### Homological stability of mapping class groups

- Homological stability. By Julie Bergner (UC Riverside).
- The spectral sequence argument. By Martin Palmer (Oxford).
- Connectivity arguments. By Chad Giusti (Oregon).

#### The homotopy type of the cobordism category

- The cobordism category. By Matsuoku Takuo (Northwestern).
- Sheaves and their realization. By Joseph Cheng (Stanford).
- Sheaves of categories. By John Lind (Chicago).
- The group completion theorem. By Marcy Robertson (UIC).
- The positive boundary category. By Nathan Stiennon (Stanford).

## Recommended reading

Here is some recommended background reading. If you have questions about the program and the talks, the organisers can be contacted via wcatss-org@lists.uoregon.edu.