The titles and abstracts of the speakers at the conference are displayed below. Recordings and/or slides of the talks are available as well in many cases.
Featured talks
Dan BerwickEvans: How do field theories detect the torsion in topological modular forms? 

Since the mid 1980s, there have been hints of a deep connection between 2dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) in which cocycles are 2dimensional supersymmetric field theories. Basic properties of these field theories lead to expected integrality and modularity properties, but the abundant torsion in TMF has always been mysterious. In this talk, I will describe deformation invariants of 2dimensional field theories that realize some of the torsion in TMF. 



Ilka Brunner: Truncated affine RozanskyWitten models as extended defect TQFTs 

I will discuss a construction of RozanskyWitten models with target T^*C^n as extended TQFTs. The starting point is a 3category, whose objects are affine RW models, and whose morphisms are defects of any codimension. By truncation, we obtain a (nonsemisimple) 2category CC of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2dimensional TQFT valued in CC for every affine Rozansky–Witten model. 



Slides  Nils Carqueville: Orbifold completion of 3categories 

We develop a general theory of 3dimensional "orbifold completion", to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3category T with adjoints for all 1 and 2morphisms, we construct the 3category T_orb as a Morita category of certain E_1algebras in T which encode triangulation invariance. We prove that in T_orb again all 1 and 2morphisms have adjoints and that it contains T as a full subcategory. Applications include defect TQFTs for state sum models and ReshethikinTuraev theory. This is joint work with Lukas Müller.  


André Henriques: Lessons from Liouville theory 

Liouville CFT has a number of interesting features, such as the nonexistence of a vacuum vector, and the nonsurjectiveness of the statefield correspondence. I will explain which properties of Liouville CFT one might expect to find in general unitary noncompact functorial field theories. 



Theo JohnsonFreyd: Super duper vector spaces II  The highercategorical Galois group of the complex numbers 

A theorem of Deligne suggests that the complex numbers are not algebraically closed in a 1categorical sense but that their 1categorical algebraic closure is the category sVec of complex super vector spaces. In fact, this property uniquely (up to nonunique isomorphism) characterizes sVec amongst complexlinear symmetric monoidal categories. In these talks, we will outline work in progress on constructing complexlinear symmetric ncategories which are higher categorical analogues of sVec in that they are uniquely characterized by being the ncategorical separable closure of the complex numbers. We will explore the resulting highercategorical absolute Galois group of the complex numbers, and outline a construction of that group very much akin to the surgerytheoretic description of the stable piecewise linear group PL. 



Recording  Catherine Meusburger: TuraevViroBarrettWestbury state sums with defects 

We define a TuraevViroBarrettWestbury state sum model of triangulated 3manifolds with surface defects (oriented 2d surfaces), line defects and point defects (graphs on the defect surfaces). Surface defects are labeled with bimodule categories over spherical fusion categories with bimodule traces, line and point defects with bimodule functors and bimodule natural transformations.




Slides  Recording  Lukas Müller: Reflection positivity for extended topological field theories> 

In quantum field theories, locality and unitarity are essential properties. For functorial field theories, locality is manifested through compatibility with cutting and gluing of manifolds, which can be fully encoded in the definition of fully extended functorial field theories. However, unitarity or reflection positivity (its Euclidean version) has so far only been defined for nonextended or invertible functorial field theories. In this talk, I will address the challenge of defining reflection positivity for extended topological field theories, proposing a definition based on a version of higher dagger categories.  


Dmitri Pavlov: The geometric cobordism hypothesis 

I will explain my recent joint work with Daniel Grady on locality of functorial field theories (arXiv:2011.01208) and the geometric cobordism hypothesis (arXiv:2111.01095). The latter generalizes the Baez–Dolan cobordism hypothesis to nontopological field theories, in which bordisms can be equipped with geometric structures, such as smooth maps to a fixed target manifold, Riemannian metrics, conformal structures, principal bundles with connection, or geometric string structures. Applications include a generalization of the Galatius–Madsen–Tillmann–Weiss theorem, a solution to a conjecture of Stolz–Teichner on representability of concordance classes of functorial field theories, and a construction of power operations on the level of field theories (extending the recent work of Barthel–BerwickEvans–Stapleton). I will illustrate the general theory by constructing the prequantum Chern–Simons theory as a fully extended nontopological functorial field theory. If time permits, I will discuss the ongoing work on the geometric cobordism hypothesis with defects, nonperturbative quantization of functorial field theories, and explicit constructions of field theories.  


Slides  Recording  David Reutter: Super duper vector spaces I  The highercategorical algebraic closure of the complex numbers 

A theorem of Deligne suggests that the complex numbers are not algebraically closed in a 1categorical sense but that their 1categorical algebraic closure is the category sVec of complex super vector spaces.
In fact, this property uniquely (up to nonunique isomorphism) characterizes sVec amongst complexlinear symmetric monoidal categories.
In these talks, we will outline work in progress on constructing complexlinear symmetric ncategories which are higher categorical analogues of sVec in that they are uniquely characterized by being the ncategorical separable closure of the complex numbers. We will explore the resulting highercategorical absolute Galois group of the complex numbers, and outline a construction of that group very much akin to the surgerytheoretic description of the stable piecewise linear group PL. 



Recording  Gregor Schaumann: Fusion quivers and retracts of tensor categories 

A motivation for derived field theory comes from the deformation of the algebraic input data. To study such one first needs to specify the ""ambient space"" of the deformation. We study the case of tensor categories and investigate retracts of tensor categories A in larger tensor categories A_z parametrized by objects z in the Drinfeld center of A. In case A is semisimple, the categories A_z correspond to (nonsemisimple) categories of quiver modules. Thus the theory provides a construction of quivers with a rigid monoidal structure on their modules. For such fusion quivers, the moduli spaces carry interesting structures. As applications we consider relations on the quiver modules that are compatible with the rigid monoidal structures and study preprojective algebras.  


Recording  Chris SchommerPries: Topological Field Theories vs Stable Diffeomorphism 

This talk will discuss generalized DijkgraafWitten topological field theories and their ability to detect stable diffeomorphism classes of manifolds. Like a gemstone, there are many facets to explore. DijkgraafWitten theories belong to a class of topological field theories obtained by “finite path integration”. The categorical underpinnings of this construction, which could be called “rational ambidexterity”, allow one to take highly structured TQFTs and integrate out the structure. The classification of invertible TQFTs in terms of stable homotopy gives a rich source of inputs for this construction. Then there is Kreck’s classification of manifolds up to stable diffeomorphism. I will try to touch on these topics, explain how they come together, and what they tell us about TQFTs.  


Christoph Schweigert: Traces and higher structures 

Quantum topologists are used to thinking about traces in the framework of pivotal tensor categories and thus in a twodimensional context to which a twodimensional graphical calculus can be associated. We explain that traces are already naturally defined for twisted endomorphisms of linear categories, i.e. in a onedimensional context. The endomorphisms are twisted by the Nakayama functor which, for a module category over a monoidal category, is a twisted module functor and hence an inherently threedimensional object. This naturally leads to a threedimensional graphical calculus. This calculus also has applications to Turaev–Viro topological field theories with defects.  


Stefan Stolz: Partition function determines invertible field theory for dimensions not equal 2 

This is joint work with Matthias Kreck and Peter Teichner, and my talk builds on Peter's talk. The partition function of a functorial field theory Z of dimension d (with tangential structure and values in super vector spaces) is the simplest invariant of the field theory. It assigns a complex number Z(X) to any closed dmanifold X equipped with tangential structure. We show that an invertible field theory (i.e., objects/morphisms of the bordism category map to invertible objects/morphisms) is determined by its partition function, unless Z has dimension 2, in which case we provide a counter example. Furthermore, we characterize those numerical invariants for closed dmanifolds that occur as partition functions of invertible field theories: they are SKK invariants, which were studied 50 years ago.  


Peter Teichner: Invertible TFTs with builtin higher structures 

Certain types of invertible TFTs seem to describe exotic materials in condensed matter physics. After a brief introduction to this topic, we'll focus on relating various kinds of invertible TFTs and show that for a certain target locality, usually described by higher categories, is automatically builtin. Including Stephan's talk, we will then classify such TFTs by their partition function, and characterize those closed manifold invariants that arise.  


Constantin Teleman: The Quantum GIT conjecture (aka 2D Quantization commutes with Reduction) 

I will review the context, and recent proof, joint with Dan Pomerleano, of the 2dimensional version of Quantization commutes with reduction, describing the quantum cohomology of (smooth) symplectic quotients of Fano manifolds.  


Alexis Virelizier: Homotopy Quantum Field Theories 

Homotopy quantum field theories (HQFTs) generalize topological quantum field theories (TQFTs). The idea is to use TQFTs' techniques to study principal fiber bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space X. In particular, such a HQFT associates a scalar invariant under homotopies to each map from a closed manifold to X. It is well known that groups are algebraic models for 1types, and with Vladimir Turaev we used group graded fusion categories to construct 3dimensional HQFTs with a 1type target. Generalizing groups, crossed modules model 2types. In this talk, I will introduce the notion of a crossed module graded fusion category which generalizes that of a group graded fusion category. Then, using such categories, I will construct 3dimensional state sum HQFTs with a 2type target. This is joint work with Kursat Sozer.  


Slides  Recording  Lukas Woike: Modular functors and factorization homology 

Modular functors are systems of mapping class group representations. The notion plays an important role in the representation theory of quantum groups and conformal field theory and is closely related to threedimensional topological field theory. After summarizing the classical constructions, I will outline an approach to modular functors using cyclic and modular operads, as well as factorization homology, that leads to a classification of modular functors. The idea is to state the main result relatively quickly and then to move on to applications (order of Dehn twists in the representations) and interesting special cases (Drinfeld centers). This is based on joint work with Adrien Brochier and Lukas Müller. 
Contributed talks
Recording  Thibault Decoppet: Morita equivalence classes of fusion 2categories 

I will argue that the Morita 4category of braided fusion 1categories is equivalent to the Morita 4category of fusion 2categories. Thanks to the work of Brochier, Jordan, and Snyder, this implies that every fusion 2category is fully dualizable, and therefore yields a fully extended framed 4D TFT. Further, this equivalence of 4categories can be leveraged to study the equivalence classes of these TFTs. 



Slides  Recording  Patrick Kinnear: Varying the nonsemisimple CraneYetter theory over the character stack 
Associated to a certain subquotient of the category of representations of the small quantum group at a root of unity is an invertible 4d TQFT known as Crane–Yetter. In fact, the nonsemisimplified representation category is invertible in the Morita theory of braided tensor categories: under the cobordism hypothesis this defines a nonsemisimple invertible TQFT. Such an invertible theory assigns to a closed 3manifold a 1dimensional vector space. In this talk, we define a relative TQFT which can be seen as varying the nonsemisimple CraneYetter theory over the character stack: it assigns to a closed 3manifold a line bundle on its Gcharacter stack. We construct this theory by analysing invertibility of a 1morphism in the Morita theory of symmetric tensor categories, coming from representations of Lusztig's quantum group at a root of unity regarded as a bimodule for Rep(G) using the quantum Frobenius map. 



Slides  Recording  Eugenio Landi: String bordism invariants in dimension 3 from U(1)valued TQFTs 


Veronica Pasquarella: Drinfeld Centers from Magnetic Quivers 

In this talk I will show that magnetic quivers encode the necessary information for determining the Drinfeld center in the symmetry TFT associated to a given absolute theory. The crucial argument resides in their common aim of generalising homological mirror symmetry.  


Slides  Recording  Jan Pulmann: Nerves of Hopf algebras and ChernSimons theory 
ChernSimons theory for the double of a Lie bialgebra has two natural boundary conditions, given by the Lie algebra and its dual. Using these boundary conditions for punctured disks and bordisms between them as a motivation, we are led to a definition of a nerve of Hopf algebra. We prove an equivalence between Hopf algebras and such nerves and give an application to quantization, using Drinfeld associators. Based on joint work with Pavol Ševera arXiv:1906.10616. 



Will Stewart: A geometric construction of relative field theory 

In this talk I will present a construction of a relative bordism category. It has the property that symmetric monoidal functors out of this category are classified by an oplax natural transformation whose source is the trivial theory (thus recovering the usual notion of a relative (or twisted) field theory). I will provide some examples and a sketch of this property. 