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Stratified Homotopy Types
Provider: Faculty of Science

Activity no.: 7001-24-07-31
Enrollment deadline: 31/05/2024
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time24.06.2024, at: 09:00 - 28.06.2024, at: 16:00
Regular seats50
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserNathalie Wahl    E-mail address: wahl@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Semester/BlockSummer
Exam formContinuous assessment
Course workload
Course workload categoryHours
Lectures20.00
Theory exercises15.00
Preparation35.00
Sum70.00
Content
The idea of investigating stratified homotopy types associated to geometric objects was envisioned by MacPherson as a method of extracting intricate homological algebraic information. This idea was implemented as ‘exit path’ in the 2-categorical setting by David Treumann and then in the infinity-categorical setting by Jacob Lurie. As it turns out, the formalism of ‘exit path’ has a very interesting theory in itself. In particular it connects to many different mathematical objects such as: geometry of singular spaces and stratification, manifold topology, moduli space and higher structure, symplectic topology, algebraic geometry, geometric representation theory and so on.

The purpose of the masterclass is to bring together active researchers in this area to lecture on recent developments in the foundational aspects of the theory and provide interesting examples of application. Tentative course contents will be given below:

Hiro Lee Tanaka will lecture about constructible sheaves on moduli stack of broken lines and higher structure of associative algebra. Then he will explain the application of the theory to Morse homotopy type and Floer homotopy type. Time permitting, the lecturer will talk about the very recent progress on categorical invariants in symplectic topology via constructible sheaves.

Peter Haine will talk about the foundation of exit path category, constructible sheaves and stratified homotopy type in the modern setting, i.e., through the infinity-categorical language. He will also talk about microlocal perspective of the theory as a way to translate between constructible sheaves on manifold and microlocal sheaves on cotangent bundles. Time permitting, the lecture will include very recent result how to on extending exit path formalism beyond ‘conical’ setting.

Mikala Jansen (local speaker) will talk about stratified homotopy types of moduli spaces and computations. The focus will be on how the stratified spaces arise naturally in different context, and how to explicitly compute their homotopy types. Connections to K-theory and algebraic topology will also be discussed.

Formel requirements
Basic familiarity with manifold topology, homotopy theory and category theory will suffice.

Learning outcome
Knowledge:
• Understand the stratified homotopy type in modern language
• Computation of exit path category in concrete examples

Skills:
• Be able to implement ideas from stratified homotopy types to research in other areas of mathematics

Competences:
• Have a comprehensive understanding of technical details of exit path formalism
• Be able to understand and participate in the frontier research of the field

Literature
Higher algebra by Jacob Lurie.

Teaching and learning methods
The masterclass will run from Monday to Friday. The schedule for a typical day will be:
Lectures 9:00 - 11:00
Exercise session 11:00 - 12:00
Lunch 12:00 - 13:00
Lectures 13:00 - 15:00
Exercise and Discussion session 15:00 - 17:00
There will also be coffee breaks and discussion in between for exchange of ideas. A social event will be organized to promote communications.

Lecturers
Hiro Lee Tanaka, Texas State University
Peter Haine, University of California, Berkeley
Mikala Ørsnes Jansen, University of Copenhagen

Hiro Lee Tanaka is currently assistant professor in Texas State University. He is a leading expert in this field of research, famous for his research on relating exit path formalism on moduli spaces to higher structures. He also has unique insight into applying the theory to understand manifold geometry and symplectic topology. His research paper in this field is published in famous math journals. Very recently, he received the prestigious Sloan Research Fellowship.
Hiro Lee Tanaka will deliver a lecture course on the research topic that he works on.

Peter Haine is currently an NSF postdoctoral fellow in UC Berkeley. He is very famous for his research on building foundations of stratified homotopy theory and exit path category, in both topological and algebr-geometric context. Very recently, he pushes the exit path category beyond Lurie’s approach, which has potentially very interesting applications. Peter also works on microlocal aspects of the theory and has deep understanding of the subtlety of microlocal sheaf theory expressed in the modern infinity-categorical language.
Peter Haine will deliver a lecture course on the research topic that he works on.

Mikala Jansen is currently a postdoc of department of Mathematical Sciences at University of Copenhagen. She wrote her PhD thesis on determining stratified homotopy types of compactifications of various moduli spaces which has very interesting applications.
Mikala Jansen will deliver a lecture course on the research topic that she works on.

Remarks
Organizing PhD students:
Qingyuan Bai, qb@math.ku.dk
Oscar Harr, obh@math.ku.dk
Branko Juan, bj@math.ku.dk
Florian Riedel, fmr@math.ku.dk

Read more and SIGN UP at the course webpage:
https://www.math.ku.dk/english/calendar/events/stratified-homotopy-types/

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