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): Hopf algebras in bordism categories and the homotopy quantum groups of a topological field theory
  • 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:

David Reutter: Hopf algebras in bordism categories and the homotopy quantum groups of a topological field theory
 
Abstract: The goal of this lecture series will be to explain various Hopf algebraic structures in bordism categories, and how they arise naturally from the interplay of higher categorical structure and adjunctions. 
 
This will be used to associate a list of Hopf algebras to any topological field theory with a boundary theory. I will argue that these Hopf algebras should be thought of as the `homotopy quantum groups' of the field theory: Indeed, if the field theory is built from a sufficiently finite topological space, they precisely recover the ordinary homotopy groups of the space. 
 I will also explain how to build a Puppe long exact sequence from the homotopy quantum groups of a relative TFT, generalizing the classical Puppe sequence associated to a map of spaces. This will then be used to relate various characterizations of invertibility of a topological field theory.
 
Along the way, I will introduce the Gray tensor and smash product of higher categories, explain how bialgebras and Hopf algebras fall out of the interplay of the Gray smash product and adjunctions, and how this relates to Tannakian reconstruction.
 
Based on joint work with Theo Johnson-Freyd.

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

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.

Mayuko Yamashita: Topological modular forms, its equivariant refinements and supersymmetric quantum field theories

Abstract: tba

 

 

 

 

 

 

 

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>.