We held a (not quite) week-long meeting at the Institut de Mathématiques de Bourgogne of the Université Bourgogne Europe in Dijon from August 31 to September 4, 2026, with the last day being a departure day without talks. The goal was to bring together some experienced, but also young researchers in quantum topology in a setting allowing for plenty of time for discussions.
List of speakers:
Jennifer Brown (Edinburgh/Hamburg)
Francesco Costantino (Toulouse)
Quentin Faes (Louvain)
Benjamin Haïoun (Edinburgh/Hamburg)
Sebastian Halbig (Marburg)
Alea Hofstetter (Hamburg)
Catherine Meusburger (Erlangen-Nürnberg)
Lukas Müller (LMU München)
Florian Naef (Dublin)
Kursat Sözer (Lille)
Jan Steinebrunner (Cambridge)
Markus Zetto (Hamburg)
The venue is the Salle Réné Baire on the 4th floor of the Wing A of the building Sciences Mirande of the Université Bourgogne Europe.
Titles, abstracts and the schedule are below.
A conference dinner will take place on September 2.
Monday
Tuesday
Wednesday
Thursday
9h30-10h30
Hofstetter
Steinebrunner
Naef Meusburger
10h45-11h45
Halbig
Faes
Haïoun** Brown
14h-15h00
Zetto
Müller*
Haïoun
15h30-16h30
Sözer
Müller
Costantino
* introductory lecture on cyclic and modular operads;
** introductory lecture on skein theory.
TITLES AND ABSTRACTS
Jennifer Brown: Topological defects and quantum cluster coordinates
Cluster coordinates give a combinatorial quantisation of certain decorated character varieties of surfaces. Skein theory gives an independent approach to quantisation. The two approaches are known to sometimes agree, but so far proofs have relied on careful understanding of the specifics of the reductive group. Wanting to generalise this correspondence, there has been a recent push to implement features of quantum cluster algebras directly in skein theory, in a group-agnostic way. The language of topological defects has been an essential tool in this effort.
Francesco Costantino: Stated skein TQFTs and differential calculi
In this talk I will start by recalling the notion of stated skein algebras of surfaces and some of their properties.
Then I will rapidly state a result (joint with M. Faitg) providing a braided monoidal functor from a category of surfaces and their cobordisms to a suitable category of algebras and their bimodules.
Finally I will recall the notion of differential calculus on an algebra and provide examples based on O_q(sl_2) and on U(1).
I will interpret these examples in terms of stated skeins and exhibit a new stated skein quantum differential calculus for these examples.
(Joint with M. Young).
Quentin Faes: Torsion in the Lie algebra of homology cylinders
The Torelli subgroup of the mapping class group of a surface S embeds naturally into the monoid of homology cylinders, providing a useful bridge between 2-dimensional topology and 3-dimensional topology. The "Y-filtration" of this monoid is an analogue of the lower central series of the Torelli group, and is intimately related to Goussarov and Habiro's clasper calculus, and hence to the theory of finite-type invariants of 3-dimensional manifolds. The "Lie algebra of homology cylinders" is the graded Sp(H_1(S))-module associated with this filtration.
In this talk, I will discuss the odd-degree part of the 2-torsion of this Lie algebra and its structure as an Sp(H_1(S))-module. Our approach relies on clasper calculus and on the use of the LMO functor as a universal finite-type invariant, that provide elegant diagrammatic methods. I will also briefly discuss some consequences of these results for the study of the lower central series of the Torelli group. Joint work with G. Massuyeau and M. Sato.
Benjamin Haïoun: Skein theory in Non-semisimple Settings
Skein theory in the semisimple realm is a well-developed tool for constructing TQFTs. I will review different approaches to replicate this story in the non-semisimple cases, and will try to relate them. If time allows i will discuss internal skein algebras and partial dualizabiliy of braided tensor categories. Based on past and ongoing work with Jennifer Brown.
Sebastian Halbig: Towards a homological Kitaev model
In 1997, Alexei Kitaev proposed with the toric code one of the most important models for fault-tolerant quantum computation. This model (and its generalisations) mix combinatorial topology---via CW-decompositions of connected sums of tori---with complex semisimple Hopf algebras. While the Hilbert space of physical qudits and its Hamiltonian depend on the 0-, 1-, and 2-cells, the logical qudits (described by the Hamiltonian's ground state) are a topological invariant, leading to their protection against a wide range of errors.
In this talk, based on joint work with U. Krähmer, we present a generalisation of this model that allows arbitrary Hopf algebras with bijective antipode as inputs. Two challenges prevent a straightforward approach. First, the construction of the space of physical qubits relies on an involutive antipode---a condition equivalent to the underlying Hopf algebra being semisimple. Second, topological invariance is proven using projectors assembled from (co)integrals. Since we do not have these tools at our disposal, we follow a different idea.
We introduce involutive Hopf bimodules as new coefficients to the theory and, instead of considering trivial submodules (the algebraic equivalent to the ground state of the system's Hamiltonian), our code spaces (logical qudits) arise as bitensor products---combinations of cotensor and tensor products.
Furthermore, towards formulating a fully homological Kitaev model, higher bitensor products are needed and we discuss how these arise from canonical maps between cohomology and homology theories associated to paracyclic objects.
Alea Hofstetter: The 2-categorical S-matrix of a braided fusion 1-category is a character table
The semisimple module categories over a braided fusion 1-category C form a connected fusion 2-category Mod(C). Its 2-categorical Drinfeld center is a braided fusion 2-category 𝑍(Mod(C)). To any braided fusion 2-category, Johnson-Freyd and Reutter have associated a matrix-valued invariant, the 2-categorical S-matrix. In this talk, I will apply this 2-categorical S-matrix to 𝑍(Mod(C)) and explain that it reduces to the character table of the Müger center of the braided fusion 1-category C. Furthermore, I will explicitly compute the 2-categorical S-matrix in the example class of C being the 1-category of G-graded vector spaces with arbitrary braiding and associator for a finite group G. This is based on joint work with Christoph Schweigert.
Catherine Meusburger: Trisection invariants of 4-manifolds
We use Gay and Kirby’s description of 4-manifolds in terms of
trisections and trisection diagrams to define a 4-manifold invariant.
The algebraic data are an indecomposable finite semisimple
bimodule category with a bimodule trace over a pair of spherical fusion
categories and a pivotal functor from another spherical fusion category
into the spherical fusion category of its bimodule endofunctors and
natural transformations between them.
The resulting 4-manifold invariant is formulated in terms of
diagrammatic calculi and includes the earlier invariants of Bärenz and
Barrett and of Chaidez, Cotler and Cui as special cases.
This is joint work with
Vincentas Mulevičius and Fiona Torzewska, arXiv:2511.19384
Lukas Müller: Skein theoretic constructions of modular and ansular functors
Modular functors are consistent systems of mapping class group representations that are compatible under cutting and gluing of surfaces. Ansular functors are the corresponding concept replacing mapping class groups by 3-dimensional handlebody groups. These have close ties to representation theory and quantum algebra, admitting classifications in terms of related higher algebraic structure. In my talk I will explain how to construct modular and ansular functors using versions of admissible skein theory for non-semisimple categories and how these connect to general classification results. The talk is based on joint work with Christoph Schweigert, Lukas Woike, and Yang Yang.
Florian Naef: Homotopy Frobenius Algebra Models for Configuration Spaces
Given a manifold M, a theorem of Campos-Willwacher explains that the collection of configuration space integrals is equivalently encoded in the rational homotopy type of the configuration spaces of points together with its right E_d-module structure. I will explain how configuration space integrals can also be interpreted as structure constants of a homotopy Frobenius algebra structure on the cohomology of M. Furthermore, such an interpretation gives an equivalence between the infinity categories of homotopy Frobenius algebras and E_d right modules of configuration space type. In particular, this implies in one direction that the A-infinity structure on the cohomology of a manifold naturally extends to a homotopy Frobenius algebra of degree d, and in the other direction that one can build a model for the configuration spaces of points in terms of such a homotopy Frobenius algebra structure which is essentially a construction due to Lambrechts-Stanley.
This talk is based on ongoing joint work with Thomas Willwacher.
Kursat Sözer: From group gradings to crossed module gradings in 3-dimensional quantum topology
Many quantum invariants of 3-manifolds admit refinements for manifolds equipped with maps to BG=K(G,1), with the algebraic input correspondingly replaced by G-graded fusion categories or Hopf group-coalgebras. I will discuss a further extension in which BG is replaced by a connected homotopy 2-type B\chi, modeled by a crossed module \chi.
On the algebraic side, this leads to monoidal and fusion categories graded by crossed modules, as well as Hopf \chi-coalgebras. On the topological side, these structures yield 3-dimensional HQFTs and quantum invariants of homotopy classes of maps M \to B\chi, or equivalently, flat principal 2-bundles. I will emphasize how these maps can be encoded combinatorially using crossed-module labelings of presentations of 3-manifolds.
This is joint work with Alexis Virelizier.
Jan Steinebrunner: Open 2D TFT admit initial open-closed extensions.
One can define functorial 2D TFTs as symmetric monodial functors from category of 1-manifolds (with boundary) and 2-dimensional bordisms (with corners) into a target symmetric monoidal category. Taking the target to be the category of vector spaces this recover the Atiyah-Segal definition, but many interesting examples arise if we allow the target to be a higher category. For example, 2D TFTs valued in linear categories are related to (derived) modular functors and 2D TFTs valued in chain complexes to cohomological field theories.
It is often easier to construct only the “open” part of a 2D TFT, i.e. its restriction to a subcategory whose objects are disjoint unions of intervals. In fact, it is possible to classify
such open 2D TFTs in simpler terms, which has been done by Costello and Müller–Woike certain settings.
I will explain some joint work with Shaul Barkan and Adela Zhang: generalizing work of Costello we show that, under mild assumption on the target, every open 2D TFT admits a
universal extension to an open-closed theory. The key ingredient is to show that open-closed bordism category can be obtained from the open bordism category by formally adjoining certain colimits. This can be thought of as formally adding an object that is the Hochschild homology, or “cocenter” of the interval.
Markus Zetto: Cauchy-completions and higher idempotents
An enriched category is said to be Cauchy-complete if it admits all absolute colimits — those weighted colimits that commute with every enriched functor. For instance, an ordinary (Set-enriched) category is Cauchy-complete precisely iff it is idempotent complete, while an Ab-enriched category is so iff it is both idempotent complete and additive.
I will extend this notion to enriched (∞,n)-categories and explain how, assuming the cobordism hypothesis, it can be used to construct framed fully extended topological field theories. In particular, it leads to the notion of higher idempotents, also known as condensations in the sense of Gaiotto and Johnson-Freyd. Joint work in progress with David Reutter.
The conference is financed through the accompagnement CPJ.
Institut de Mathématiques de Bourgogne
UMR 7586 CNRS
Université Bourgogne Europe
Faculté des Sciences Mirande
9 Avenue Alain Savary
F-21000 Dijon