Skein modules: Skein theory assigns to a 3-manifold M the so-called skein module S(M) depending on a quantum parameter q and reductive group G, which in physics corresponds to a 3D Dirichlet boundary in the Kapustin-Witten twist of 4D N=4 SYM. Based on this, Witten conjectured that for generic q and M closed the skein module S(M) is finite dimensional - a statement later proved by [Gunningham-Jordan-Safronov]. This striking result about skein modules motivated us to generalize this to manifolds with non-empty boundary in [Jordan-Romaidis]. The strong version of theorem states that the internal skein module IS(M) of M is holonomic over the internal skein algebra. Rougly, internal allows skeins at a specific point in the boundary while holonomic means finitely generated and supported on a Lagrangian. This generalized finiteness property of skein modules can be applied to study several properties in skein theory, which will appear in future projects.
The role of TQFTs in 3d quantum gravity: One can use TQFTs and defects as a mathematical tool when studying non-topological theories like 2d CFT. In particular, chiral CFT can be described by the modular functor part of a 3d TQFT, i.e. its mapping class group representations. Defects and boundary conditions provide a construction of correlators (at least for rational CFTs). Another surprising appearance of such mapping class group actions is in the context of 3d quantum gravity, where mapping class group averages provide candidate gravity partition functions (see my Thesis and arXiv:2309.14000 ).
Mapping class group representations: Motivated by expectations in quantum gravity, I am interested (for Reshetikhin-Turaev TQFTs) in questions of:
Irreducibility: See arXiv:2106.01454 for a result (with I. Runkel) that uses irreducibility of mapping class group representations to imply that there are no non-trivial surface defects. This result can be related to the no global symmetry conjecture in quantum gravity.
Finiteness: This requires that the representation image of the mapping class group is finite. Note that questions on the size of the representation image are also relevant in Topological Quantum Computation (TQC).
Constructing TQFTs from fusion categories via J-algebras with V. Mulevicius, in preparation.
Finiteness and holonomicity of skein modules with D. Jordan, arXiv:2509.22313.
CFT correlators and mapping class group averages with I. Runkel, Commun. Math. Phys. 405, 247 (2024), arXiv:2309.14000.
Mapping class group representations and Morita classes of algebras with I. Runkel, Quantum Topol. 14.3 (2023), arXiv:2106.01454.
Mapping class group actions and their applications to 3D gravity, Doctoral dissertation, Staats und Universitätsbibliothek Hamburg 2022. [pdf]
Master Thesis: Permutation Orbifolds in Reshetikhin-Turaev TQFT (2019) [pdf]
Notes from STOAT on Higher tensor categories and their extensions: arXiv:2509.10636.