Cobordism categories

Masterclass: Building bridges between algebra and geometry

University of Copenhagen, 17-21 February 2025

This masterclass aims to showcase different perspectives of the cobordism categories. The lecture series will explore the interface of cobordism categories, modular ∞-operads, (equivariant) topological modular forms, topological quantum field theories, and fully dualizable objects.

 

 

  • David Reutter (University of Hamburg)
  • Jan Steinebrunner (Cambridge University), 2-dimensional TFTs via modular ∞-operad.
  • Mayuko Yamashita (University of Kyoto), Topological modular forms, its equivariant refinements and supersymmetric quantum field theories

Lecture series:

Title: 2-dimensional TFTs via modular ∞-operad by Jan Steinebrunner

Abstract: 2-dimensional topological field theories (2D TFTs) valued in vector spaces are commutative Frobenius algebras. The goal of this lecture series is to generalise from the 1-category of vector spaces to any symmetric monoidal ∞-category C, i.e. to study symmetric monoidal functors Bord2 --> C . Choosing C to be the (2, 1)-category of linear categories, this recovers a definition of modular functors, and choosing it to be the derived category of a ring yields a notion closely related to cohomological field theories.

I will introduce a notion of modular ∞-operads and algebras over them, construct the modular ∞-operad of surfaces M, and show that algebras over M in C are exactly 2D TFTs valued in C. Along the way we will encounter variants modular ∞-operads (such as cyclic ∞-operads and ∞-properads) as well as a proof of the 1D cobordism hypothesis with singularities. This uses some (mild) ∞-category theory, but no familiarity with (∞-)operads will be assumed.

The main goal will be to filter M by genus to obtain an obstruction-theoretic description of 2D TFTs with general target. Applying this to invertible TFTs one can construct a new spectral sequence exhibiting relations between the cohomology groups of moduli spaces of curves.

 

 

 

 

 

 

 

To appear

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 Steel House Copenhagen (close to city centre), and 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.

 

 

 

 

 

 

 

 

 

 

 

 

Please fill out the registration form here.

We have a limited amount of funding for junior participants. The deadline for funding applications is Dec 20, 2024. 
If you are not applying for funding, please register by Jan 20, 2025.

 

 

 

 

 

 

 

 

 

 

Organisers: Jonathan Clivio, Branko Juran, Fadi Mezher, Azélie Picot, Nathalie Wahl, Adela Zhang

For inquiries, please contact Branko Juran <bj@math.ku.dk> and Adela Zhang <yz@math.ku.dk>.