Research interests

My research focuses on quantum topology and the associated (higher) algebraic structures. My thesis consists of the construction of quantum homotopy invariants of 4-manifolds using a state sum method.

Thesis: 3-group extensions of fusion 2-categories and quantum homotopy invariants of 4-manifolds

Quantum topology is a branch of mathematics that emerged in the late twentieth century following groundbreaking work by Jones, Drinfeld, Witten, and Kontsevich, which drastically renewed topology, particularly in low dimensions. Drawing on ideas from quantum physics, their framework has led to the construction of new manifold invariants, called quantum invariants, constructed from relevant algebraic data. One of the early successes in this direction is the Turaev–Viro invariant of 3-dimensional manifolds, defined as a state sum using the algebraic data of a fusion category (constructed for instance from the representations of a quantum group). More recently, Douglas and Reutter have constructed state-sum invariants of 4-dimensional manifolds using fusion 2-categories.

This thesis investigates quantum homotopy invariants, that is, quantum invariants of manifolds equipped with additional structure encoded by a map to a target topological space. More specifically, we generalize the Douglas–Reutter construction to derive invariants of flat 3-bundles over 4-manifolds or, equivalently, of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type. The relevant algebraic input for these quantum homotopy invariants is 3-group extensions of fusion 2-categories, where the 3-group models the target homotopy 3-type. Within this framework, maps to a homotopy 3-type are represented by colorings of triangulations. By proving a colored generalization of Pachner’s theorem, we obtain that these state-sum invariants are independent of the chosen triangulation.

We show through examples that the resulting invariants are nontrivial and can even distinguish phantom maps, that is, maps that are trivial at the level of homotopy groups. Our invariants encompass several previously known 4-dimensional invariants, and are expected to fit into the framework of a homotopy quantum field theory.

Talks

• (future) May 2026 - Algebra seminar, University of Leeds

• Feb 2026 - Algebraic Topology seminar, UCLouvain

• Feb 2026 - Topology seminar, Université de Lille

• Nov 2025 - Quantum Universe Attract. Workshop, Universität Hamburg

• Sep 2025 - Higher Structures: Recent developments and applications, Universität Hamburg (see slides)

• May 2025 - Algebraic Approaches to Mapping Class Groups of Surfaces, The University of Tokyo

• May 2025 - FJ-LMI Seminar, The University of Tokyo (see slides)

• Apr 2025 - Journées de Jeunes Chercheuses et Chercheurs en Topologie, Université Picardie Jules Verne

• Nov 2024 - Séminaire des doctorants et post-doctorants, Université de Lille (see slides in French)

• Jan 2024 - Topology seminar, University of Louisiana at Lafayette (see slides)

• Sep 2023 - Topology seminar, Université de Lille

Conferences attended

• May 2026 - BTW6: The Volume Conjecture: from some TQFTs to recent developments, Villa Clythia Fréjus

• Mar 2026 - Journées de topologie algébrique, géométrique et quantique en Picardie III, Université de Picardie Jules Verne

• Mar 2026 - Winter braids XV, Aix-Marseille Université

• Oct 2025 - Journée de topologie quantique, Université Paris Cité

• Sep 2025 - Higher Structures: Recent developments and applications, Universität Hamburg

• May 2025 - Algebraic approaches to mapping class groups of surfaces,  The University of Tokyo

• Apr 2025 - Journées de Jeunes Chercheuses et Chercheurs en Topologie, Université de Picardie Jules Verne 

• Mar 2025 - Journées de topologie algébrique, géométrique et quantique en Picardie II, Université de Picardie Jules Verne

• Feb 2025 - Winter braids XIV, Université de Bordeaux

• Dec 2024 - Skein days, Journées thématiques de Cergy-Pontoise, CY Cergy Paris Université

• Nov 2024 - Journée de topologie quantique, Université Paris Cité 

• Oct 2024  - Séminaire itinérant de catégories, Université Paris Cité

• Oct 2024 - Higher structures in Noncommutative Geometry and Quantum Algebra, Université de Lille

• Sep 2024 - Scottish Talbot on Algebra and Topology, Cairngorm Lodge, Glenmore, Scotland

• Apr 2024 - Séminaire itinérant de catégories, Université de Lille

• Mar 2024 - Journées de topologie algébrique, géométrique et quantique en Picardie, Université de Picardie Jules Verne

• Mar 2024 - Journée de topologie quantique, Université Paris Cité

• Feb 2024 - Winter braids XIII, Université de Montpellier 

• Oct 2023 - Conférence du GDR Théorie de l'Homotopie et Applications, Université de Lille

• Oct 2023 - Séminaire itinérant de catégories, Université du Littoral à Calais

• Mar 2023 - Building-up differential homotopy theory at Aizu, The University of Aizu

• Feb 2023 - 18th East Asian Conference on Geometric Topology, online

• Nov 2022 - International conference on "Topology and its Applications to Engineering and Life Science", online

Mathematical projects

Notes

• "Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology", 2025

• "Constructions of 2-holonomy functors", ARPE report, 2023, under the supervision of Toshitake Kohno

• "Sur la topologie des surfaces algébriques réelles non singulières de degré au plus 5 dans RP^3", master's thesis, 2022 (in French), under the supervision of Ilia Itenberg

• "The invariance of knot lattice homology", first year master's project, 2021

Interdisciplinary projects

• "Contrefaçon d'une œuvre musicale : comment déterminer le plagiat ?", research project in law and mathematics, 2023, with Thomas Hum (see poster)