Masterclass: Exit paths and stratified homotopy types

University of Copenhagen
24-28 June, 2024

The stratified homotopy theory of space and exit path categories have gone through major development in recent years. They are becoming powerful tools in other parts of mathematics like higher algebra, K-theory, geometric representation theory and symplectic topology. The purpose of this masterclass is to discuss recent developments in different aspects of the theory and illustrate interesting examples of application.

A limited number of funding is available for participants.



Peter Haine (University of California, Berkeley)
Mikala Ørsnes Jansen (Københavns Universitet)
Hiro Lee Tanaka (Texas State University)



Mikala: Calculating stratified homotopy types

In this lecture series, I'll review some concrete tools for explicitly calculating the stratified homotopy type of stratified topological spaces.

We'll begin by exhibiting two different approaches to looking at a stratified space: we can zoom in by studying the local topology or zoom out by studying the category of constructible sheaves. The latter approach uses some standard proper descent results for sheaf categories. In order to exploit these proper descent results, we'll study a functorial analogue that allows us to calculate certain colimits of infinity-categories. With the basic tools in place, we'll go through an explicit calculation: the stratified homotopy type of the reductive Borel-Serre compactification of a locally symmetric space. This will combine proper descent of sheaf categories and the functorial analogue. If time permits, we'll see how the stratified homotopy type of the reductive Borel-Serre compactification generalises to provide a model for unstable algebraic K-theory.

Hiro: Broken techniques for higher algebra in geometry
In these lectures, I will explain how moduli stacks of broken objects (e.g., holomorphic maps from disks to a point, gradient trajectories to a point) encode fundamental objects in algebra (planar operads, associative algebras, respectively). This observation is meant to give a moral explanation of why certain geometric settings (e.g., Morse theory, Floer theory for symplectic manifolds) give rise to rich algebra (Morse complexes, Fukaya categories, respectively) along with a practical, deformation-theoretic approach to computing stable homotopy theory lifts of such algebraic outputs. Most of this is joint work with Jacob Lurie.








All lectures are in Auditorium 6 of the HCØ building Universitetsparken 5, 2100 København Ø.

Lunches are in the Bio Centre Canteen.

Monday Tuesday Wednesday Thursday Friday

Registration/ Coffee and Fruit

Coffee and fruit Coffee and fruit Coffee and fruit Coffee and fruit
10:30-11:30 Mikala Mikala Mikala Mikala Mikala
11:30-13:30 Lunch Lunch Lunch Lunch Lunch
13:30-14:30 Peter Peter Peter Peter Peter
14:30-15:00 Coffee and Cake Coffee and Cake Coffee and Cake Coffee and Cake Coffee and Cake
15:00-16:00 Hiro Hiro Hiro Hiro Hiro
17:30 Dinner
18:00 Reception
















We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and  Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.












Registration has closed.











Academic programme:

Qingyuan Bai 

(if you need assistance wth travel document or invitation letter you can write to

Oscar Harr

Branko Juran

Florian Riedel


Jan Tapdrup