This is a learning seminar on manifolds and homotopy theory organized by Azélie Picot, Sangmin Ko and myself.

We aim to learn a breadth of applications of homotopy theory to the study of manifolds. Some topics we will cover include homological stability results for the moduli space of genus g surfaces or the general linear groups over a field, and ways in which K theory can relate to manifolds. We also will vary topics based on the interest of participants and may include applications to representation theory and arithmetic statistics.

Location: 528
Meeting Time: Mondays 5pm-6pm
Syllabus: Found here

1. Introduction

Azélie Picot 01/26/2026

Homological stability is a property of sequences of groups. In this introductory talk, I will define homological stability and state diverse examples that exhibit this phenomena. Then, we will discuss the current plan for the seminar and will decide among the different options we could explore.

2. A spectral sequence argument

Azélie Picot 02/02/2026

In this talk, I will sketch the proof of homological stability for symmetric groups, following an inductive argument originally due to Quillen.

3. Madsen-Weiss Theorem

Lisa Faulkner Valiente 02/09/2026

We motivate and state the Madsen-Weiss Theorem, and explain some of the key ingredients in its proof, including the scanning map.

4. Quillen's framework for homological stability

Yiming Song 02/16/2026

We'll see how the proof of homological stability for symmetric groups generalizes to broader sequences of groups, following Quillen's framework. Notes are available here.

5. Ek Algebras

Carlos Andrés Alvarado Álvarez 03/02/2026

We define the Ek algebras and compute their homology.