Masterclass: Exit paths and stratified homotopy types

University of Copenhagen
24-28 June, 2024

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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.

We apologize that the masterclass will not be recorded or zoomed.

Peter offered us the following beautiful notes of his lectures (it will be further updated): Exodromy beyond conicality

Lecturers/Invited speakers

Peter Haine (University of California, Berkeley)

Mikala Ørsnes Jansen (Københavns Universitet)

Hiro Lee Tanaka (Texas State University)

Titles and abstracts

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.

Peter: Exodromy beyond conicality

Generalizing the monodromy equivalence, the exodromy equivalence says that the ∞-category of constructible sheaves on a nice enough stratified space (X,P) is equivalent to functors out of the exit-path ∞-category of (X,P). Up until recently, the meaning of “nice enough” was quite restrictive; specifically, the exodromy theorem required the stratification of X to be conical. Unfortunately, many stratifications naturally arising in geometry are not conical. In this lecture series, we’ll explain joint work with Mauro Porta and Jean-Baptiste Teyssier, building on work of Dustin Clausen and Mikala Ørsnes Jansen, that allows us to extend the exodromy theorem to a much larger class of stratified spaces and even stacks. Examples include: stratifications that can be locally refined by conical stratifications, subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. The tools we use are from ∞-topos theory, and we’ll explain the necessary background to understand both the statements of the general result and its proof. We’ll also explain some applications. The first is that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exit-path ∞-categories; this refines classical theorems of Lefschetz–Whitehead, Łojasiewicz, and Hironaka on the finiteness of the underlying homotopy types of these spaces. The second is to use these finiteness results to prove representability results for moduli of constructible and perverse sheaves. These tools are key inputs to Porta and Teyssier’s recent work on the moduli of Stokes structures.

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.

Programme

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

Lunches are in the Bio Centre Canteen.

Wednesday dinner at Food Club, Sortedam Dossering 7c 2200 København Ø. Starts at 17:30 and leave together at 17:00 from the campus.

	Monday	Tuesday	Wednesday	Thursday	Friday
9:30-10:30	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

Venue and transportation

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen. See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

Accommodation

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.