## Homotopy Theory of Moduli Spaces

During the Fall of 2010 at UBC I co-organized with José Gómez this seminar around the proof of Mumford's conjecture following the paper *"The homotopy type of the cobordism category"* by Galatius et al.

Our seminar followed the program of the 2010 West Coast Algebraic Topology Summer School which happened at the University of Oregon very closely. Notes from these Summer school talks can be found here (scroll down to "Summer 2010 - Various: West Coast Algebraic Topology Summer School").

Below is a schedule of talks and some notes (click on talk title) taken from them. More details about content and references for each talk can be found here.

#### What is a moduli space?

- [13/09] Teichmüller theory and moduli spaces. By Maxim Stykow.
- [20/09] Pontryagin-Thom theory I. By Maxim Stykow.
- [27/09] Pontryagin-Thom theory II. By Maxim Stykow.

#### Gromov's h-principle

- [04/10] Statement. By Cihan Okay.
- [18/10] Applications and the easy part of the proof. By José Gómez.
- [25/10] The difficult part of the proof. By José Gómez.

#### Homological stability of mapping class groups

- [01/11] Homological stability. By Jeff Smith.
- [08/11] The spectral sequence argument. By Jeff Smith.
- [15/11] Connectivity arguments. By Jeff Smith.

#### The homotopy type of the cobordism category

- [22/11] A sheaf model for Ω
^{∞}MTO(d). By Maxim Stykow. - [29/11] Sheaves of categories and the proof of the main theorem. By Maxim Stykow.
- [6/12] Group completion, positive boundary category and the generalized Mumford conjecture. By José Gómez.