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.

6. Power Operations

Sangmin Ko 03/23/2026

In this talk, we introduce power operations arising from multiplicative structure. We explain how they are given by the homology of free algebras over an operad. As a key example, we discuss the Dyer–Lashof operations for E_k-algebras.

7. Cellular E_k algebras and derived indecomposables

Azélie Picot 03/30/2026

In this talk, I will introduce a recent approach to homological stability due to Galatius-Kupers-Randal-Williams, using the concept of cellular E_k algebras. Our guiding example will be mapping class groups of surfaces.

8. Beyond the stable range?

Azélie Picot 04/06/2026

The E_k-algebra point of view led to new kinds of stability results such as secondary stability for mapping class group. Then, following work of Randal-Williams, I will try to explain what “secondary stability” shall mean in general.

9. Homological stability and arithmetic statistics

Kevin Chang 04/20/2026

I will discuss applications of homological stability to arithmetic statistics, following work of Ellenberg-Venkatesh-Westerland and Landesman-Levy. This talk assumes minimal prerequisites for algebraic geometry and number theory.

10. Koszul Duality for E_k-algebras

Sangmin Ko 04/20/2026

We introduce the bar construction for E_k-algebras and the cobar construction for E_k​-coalgebras. We then state the Koszul (bar-cobar) duality theorem. Time permitting, we discuss the relationship between the bar construction and derived indecomposables.

11. Construction of equivariant Seiberg-Witten-Floer stable homotopy type

Julius 04/20/2026

I will explain the construction of Seiberg-Witten-Floer stable homotopy type by Ciprian Manolescu using finite dimensional approximation and the connections to disproving the Triangulation Conjecture, and introducing the ideas of Gauge Theory in general.

12. Lifts over the sphere?

Carlos Andrés Alvarado Álvarez 05/04/2026

Ordinary rings all live over Z... can we find a lift over S? In other words, given a classical ring R can we find a ring spectrum S_R so that S_R \otimes_S Z = R? Come find out! This is based on notes by Maxime Ramzi.