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Schedule

Venue
School of Mathematics, University of Leeds.

Provisional schedule

Thursday, 26th March

  • 9:00 -- 10:00: Christoph Schweigert. Dijkgraaf-Witten theories in 3d and boundaries (MALL 1)
  • 10:00 -- 10:30: Coffee break.
  • 10:30 -- 11:00 Alea Hofstetter. The 2-categorical S-matrix of a braided fusion 1-category is a character table (MALL 1)
  • 11:10 -- 11:40: George Altmann. Crossed modules and counting invariants for welded knots (MALL 1)
  • 11:40 -- 14:00: Lunch.
  • 14:00 -- 16:00: Discussion session. (MALL 2)
  • 16:00 -- 16:30: Coffee break.
  • 16:30 -- 17:30: K​ürşat Sözer. Kuperberg-type invariants of maps from 3-manifolds to homotopy 2-types (MALL 1)
  • 17:30 -- onwards. Open discussions / pub / dinner.

Friday, 27th March

  • 9:30 -- 10:30: Max-Niklas Steffen. A boundary characterization of Turaev-Viro TQFTs (MALL 1)
  • 10:30 -- 11:00: Coffee break.
  • 11:00 -- 12:00: João Faria Martins. Homotopical methods for extended TQFTs and defect TQFTs (MALL 1)
  • 12:00 -- 13:30: Lunch.
  • 13:30 -- 14:20: Lucy Byles. Non-Clifford Braiding of $D(S_3)$ Anyons on a Quantum Computer (MALL 1)
  • 14:30 -- 16:00: Discussion session. (MALL 1)
  • 16:00 -- 16:30: Coffee break.
  • 16:30 -- 17:30: Jack Romö. (Online) From Homotopy Finite Spaces to Modular Tensor Categories
  • 17:30 -- Onwards: Open discussions / pub / dinner.

Abstracts

  • Christoph Schweigert (Hamburg).
    • Title: Dijkgraaf-Witten theories in 3d and boundaries.
    • Abstract: We explain a geometric construction of $3d$ Dijkgraaf-Witten theories on manifolds on manifolds with boundaries. Our discussion includes data from group cohomology and their interpretation as topological Lagrangians.
      (Based on work with J. Fuchs and A. Valentino.)
  • Alea Hofstetter (Hamburg).
    • Title: The 2-categorical S-matrix of a braided fusion 1-category is a character table.
    • Abstract: 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 explain how this 2-categorical S-matrix applies to $Z(Mod(C))$ and that it reduces to the character table of the Müger center of the braided fusion 1-category $C$. This is based on joint work with Christoph Schweigert.
  • George Altmann (Leeds).
    • Title: Crossed modules and counting invariants for welded knots
    • Abstract: Classical knot invariants arise from counting group homomorphisms from the fundamental group of the complement to a finite group. This talk generalises these ideas to welded knots, a diagrammatic theory corresponding to ribbon knotted tori in 4-space. To each welded link diagram, we can combinatorially associate a crossed module valued invariant, yielding higher counting invariants. We introduce a topological interpretation of this invariant, arising from $\pi_1$ and $\pi_2$ of the complement, and generalise the peripheral system and Eisermann invariants to this crossed module via the free loop space.
  • K​ürşat Sözer (Lille).
    • Title: Kuperberg-type invariants of maps from 3-manifolds to homotopy 2-types
    • Abstract: Topological quantum field theories (TQFTs) provide a powerful framework for constructing invariants of manifolds. A natural extension of this framework is given by homotopy quantum field theories (HQFTs), where manifolds are equipped with maps to a fixed target space. In this talk, I will start with a brief overview of HQFTs. I will then review some known results in dimension three, focusing on the case of targets given by homotopy 1-types (i.e. spaces of the form $K(G,1)$) and discussing more recent developments for homotopy 2-type targets. Next, I will describe the algebraic structures that arise in this context, including crossed modules and their associated tensor-categorical and Hopf-algebraic counterparts. Finally, I will present joint work with Alexis Virelizier, where we extend Kuperberg-type invariants to 3-manifolds equipped with maps to homotopy 2-types. The construction is based on a combinatorial description using Heegaard diagrams together with suitable algebraic data, and recovers known invariants in special case.
  • Max-Niklas Steffen (Hamburg).
    • Title: A boundary characterization of Turaev-Viro TQFTs
    • Abstract: Boundaries and boundary conditions can provide useful insights into the structure of three-dimensional topological quantum field theories (TQFTs). In this talk, based on the preprint arXiv:2508.13759 [math.QA], I will discuss how introducing defect graphs in the free boundary of three-dimensional cobordisms gives rise to a characterization of Turaev-Viro TQFTs in terms of a set of boundary locality conditions. In particular, non-degeneracy of the theory naturally follows from these conditions. Starting with a brief review of Dijkgraaf-Witten theory with defects as discussed by Faria Martins and Meusburger, I will then explain how this framework provides an effective route for establishing the equivalence of Dijkgraaf-Witten TQFTs and Turaev-Viro theories for group-graded vector spaces in the presence of boundary defects.
  • João Faria Martins (Leeds).
      • Title: Homotopical methods for extended TQFTs and defect TQFTs.

    Abstract: I will report on recent work with Tim Porter and Catherine Meusburger, the former on once-extended TQFTs derived from homotopy finite spaces, vastly generalising Dijkgraaf-Witten theory, the latter on defect 3d Dijgraaf-Witten theory. Both constructions are underpinned by the usage of homotopy-theoretical techniques (and a lot more), which allows us to bypass the usage of Cerf theory, Pachner moves and the Cobordism Hypotheses.

  • Lucy Byles (Leeds).
    • Title: Non-Clifford Braiding of $D(S_3)$ Anyons on a Quantum Computer
    • Abstract: Quantum computation schemes based on topologically ordered systems provide a promising route towards fault-tolerant information processing. In two-dimensional systems, such order gives rise to anyons—quasiparticles with exotic exchange statistics—whose braiding may enact unitary transformations on a logically encoded fusion space. In this work, we consider Kitaev’s quantum double model $D(G)$, focusing on the smallest non-Abelian instance $D(S_3)$, which can be realised on a lattice of $d=6$ qudits. We develop a minimal protocol in which braiding and fusion are simulated solely through operators that create and measure anyons, demonstrating that the braiding of $D(S_3)$ anyons may enact the non-Clifford operations necessary for universal quantum computation. Finally, a dense encoding scheme is presented, rendering the minimal realisation of this protocol accessible with current quantum technologies and a circuit model for realisation within the framework of linear optics is presented. Notably, error analysis of this encoding scheme reveals an inherent passive protection provided by the topological encoding, extending beyond conventional notions of topological fault tolerance.
  • Jack Romö.
    • Title:From Homotopy Finite Spaces to Modular Tensor Categories
    • Abstract. The relationship between once-extended oriented $(1, 2, 3)$-dimensional topological quantum field theories and modular tensor categories is well-understood; given such a TQFT Z valued in 2-vector spaces, the value of $Z$ on the circle, $Z(S^1)$, identifies a modular tensor category. This connection draws a thread between many well-known constructions, connecting Drin'feld doubles of finite groups to Dijkgraaf-Witten theories as a key example.The explicit construction and study of both TQFTs and modular tensor categories remains an active area of research, as well as understanding what MTCs relate to which TQFTs. It is common to start with a modular tensor category and infer a TQFT from it. In this talk, we proceed in the opposite direction, beginning with the once-extended $(1, 2, 3)$-dimensional version of Quinn's finite total homotopy TQFT as constructed by Faria Martins and Porter. This TQFT takes as its input a homotopy finite space $B$ and yields a TQFT whose value on a closed 3-manifold $X$ is the homotopy cardinality of the space of maps from $X$ to $B$. We take an interest in the modular tensor category this TQFT yields, and will present the results of ongoing work with Joao Faria Martins on explicitly constructing this category. Our description is both highly concrete and homotopical in nature.

      If time permits, we will discuss current efforts to present this MTC as the representation category of a particular ribbon weak Hopf algebra.