Cobordism categories

• 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:

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.

Lecture notes can be found here.

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

Abstract: This lecture is about the recent developments on the Segal-Stolz-Teichner proposal. This proposal relates 2-dimensional N=(0, 1) supersymmetric quantum field theories and elliptic cohomology theories. Moreover there are important refinements of the original statements taking symmetries of field theories into account. The classifying objects for such field theories are expected to be equivariant elliptic cohomology and more generally to be "categorical equivariant elliptic cohomology" which is yet to be constructed mathematically. 

**: Note that the location for the Monday exercise session, pizza reception, and the Gong show will be at the Department of Mathematical Sciences, Universitetsparken 5, 2100 København Ø. We will depart together from the museum to the department, with bus, biking, and walking options.

The conference/masterclass will take place at Statens Naturhistoriske Museum, Øster Voldgade 5 – 7, 1350 København K.

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.

The registration has been closed.

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>.