About me

My research focuses on finding and probing properties of (multiplicity-free) fusion categories. To do so, I develop software using computational algebra to solve consistency equations arising from commutative diagrams. The data obtained this way is collected as part of packages such as

FusionRings,
Lyctor.jl (still in heavy development),
Anyonica,
TensorCategories.jl, and
CategoryData.jl, which is used by TensorKit.jl.

My main goal is to have a central repository of computational (yet readable) data on fusion categories that is free, open-source, accessible, and user-friendly for both mathematicians and physicists. Therefore, I co-founded (with J.K. Slingerland) and help build the AnyonWiki: an open-source database of fusion rings and fusion categories for mathematical physics research.

At the moment, the AnyonWiki

contains a page for (what we believe are) all multiplicity-free pivotal (braided) fusion categories up to rank 7. All F-symbols, R-symbols, and P-symbols are available both numerically and symbolically, and if the category is modular, so are its S-matrix and topological spins,
contains more than 25000 fusion rings, including a complete catalog of all multiplicity-free fusion rings up to rank 10. Each fusion ring has a dedicated page displaying its most important properties such as Frobenius-Perron dimensions, $S$-matrices, characters, decompositions, etc
automatically fetches its data from an open-source database that
- Has all data stored in JSON files that are readable by any programming language
- Contains all data on rings and categories in both numeric and symbolic format
- Has explanations on the interpretation of the data as part of the data
has download buttons for each piece of data
stores references for each individual piece of data as part of the data

Many people have put a lot of effort into the AnyonWiki so if you use it, please consider citing it.

Take a look at my current projects page to learn more about the future plans for the wiki and the packages that support it.

CV

My CV can be downloaded here

Papers

2026

AnyonWiki.org

Colleen Delaney, Ralph Razzouk, Joost Slingerland, and 1 more author

2026
On Low-Rank Multiplicity-Free Complex Fusion Categories

Gert Vercleyen and Joost Slingerland

Communications in Mathematical Physics, Sep 2026

Abs DOI

This paper explains the methods and algorithms we used to obtain explicit F-symbols, R-symbols, and pivotal coefficients of what we believe are all multiplicity-free pivotal complex fusion categories up to rank 7. Particular attention is given to how the individual steps for solving the consistency equations are done with Anyonica, a software package we developed for working with fusion categories. Gauge and automorphism equivalence are reviewed and a method to remove equivalent categories via a gauge-split basis is given. A census of all multiplicity-free fusion categories up to rank 7, along with all compatible braided and pivotal structures, is presented, and comments on their general properties are given.
F-symbols and R-symbols for the Drinfeld center of the Haagerup subfactor

Fabian Mäurer, Ulrich Thiel, and Gert Vercleyen

Sep 2026

2025

Triangular Prism Equations and Categorification

Zhengwei Liu, Sebastien Palcoux, Yunxiang Ren, and 1 more author

Sep 2025
Extending the planar theory of anyons to quantum wire networks

Tomasz Maciazek, Mia Conlon, Gert Vercleyen, and 1 more author

SciPost Phys., Feb 2025

DOI
Tables of practical invariants for distinguishing multiplicity-free fusion categories up to rank 7

Gert Vercleyen

Jul 2025

DOI

2024

On Low-Rank Multiplicity-Free Fusion Categories

Gert Vercleyen

Jul 2024

Abs

This thesis explains the methods and algorithms we used to obtain explicit 𝐹 symbols, 𝑅 symbols, and pivotal coefficients of all multiplicity-free pivotal fusion categories up to rank 7. The thesis starts by introducing the concept of a unitary modular fusion system via two applications: modeling anyons for topological quantum computation and calculating braid group representations. Next, the notions of a pivotal, spherical, braided, ribbon, and modular fusion system are introduced. These are arithmetic counterparts to fusion categories with the respective structure. Unitarity and its implications on the pivotal structure are discussed as well. The next part of the thesis is devoted to algorithms for finding fusion systems and compatible structures. First, an algorithm to find low-rank fusion rings is explained, and its results are given. Special attention is given to the structure of non-commutative fusion rings and the construction of songs, which are generalizations of the Tambara-Yamagami and Haagerup Izumi fusion rings, is given. Then, the algorithms used to find fusion systems are discussed. Particular attention is given to how the individual steps for solving the consistency equations are done with Anyonica, a software package we developed for working with fusion systems. Gauge and automorphism equivalence are reviewed, and algorithms that put solutions in a unitary gauge and remove redundant solutions are given. Some results on the categorification of all multiplicity-free pivotal fusion rings up to rank 7 are presented. The final part of the thesis is devoted to building models of anyons on graphs and how their behavior differs from those in the plane. The appendices contain a minimal mathematical exposition on fusion categories with their relation to fusion systems, a list of all multiplicity-free fusion rings up to rank 9 with information on categorifiability, a list of all multiplicity-free fusion categories up to rank 7, and data on some graph-braid models.

2023

On low rank fusion rings

G. Vercleyen and J. K. Slingerland

Journal of Mathematical Physics, Sep 2023

Abs DOI

We present a method to generate all fusion rings of a specific rank and multiplicity. This method generated exhaustive lists of fusion rings up to order 9 for several multiplicities. We introduce a class of non-commutative fusion rings based on a group with transitive action on a set. This construction generalises the Tambara–Yamagami (TY) and Haagerup-Izumi (HI) fusion rings. We give an example of two such rings that are categorifiable and not of TY or HI type. The structure of non-commutative fusion rings with a subgroup is reviewed, and the one- and two-particle extensions of groups are classified. A website containing data on fusion rings is introduced, and an overview of a Wolfram Language package for working with these rings is given. We also applied several categorifiability criteria to the fusion rings. We provide a table of all multiplicity-free fusion rings up to rank 9 with info on the categorifiability of each ring.

Talks

(Invited) 09-2026 – 989 Fusion Categories Walk Into a Search Bar, Quantum Theory Neighborhood Seminar, Purdue University
05-2026 – Minimal fields and the center of the Haagerup fusion category, QuaSy-Con IV, Illinois State University
(Invited) 02-2026 – On Low-Rank Multiplicity-Free Fusion Categories, Quantum Symmetries Seminar, Ohio State University
(Invited) 10-2025 – Complete invariants and minimal fields of multiplicity-free fusion categories,
AMS Fall Sectional meeting on “Advances in Quantum Topology, Tensor Categories, and Diagrammatic Algebras”
(Invited) 10-2025 – The AnyonWiki, a digital repository for fusion categories and anyon models,
AMS Fall Sectional meeting on “New Horizons for Quantum Error Correction: Categorical Approaches to Topological Codes”
(Invited) 07-2025 – Techniques for categorifying multiplicity-free fusion rings,
Mathematics seminar Kaiserslautern
04-2025 – Anyonica and the AnyonWiki,
Hopf25 Conference on Hopf algebras, quantum groups, monoidal categories and related structures
12-2024 – Anyonica and the AnyonWiki,
ARTIN in Leeds, Conference on Biracks and Biquandles: Theory, applications, and new perspectives
(Invited) 11-2024 – Knots and quantum computing,
Topology Seminar Purdue
(Invited) 09-2024 – Intro to Anyonica,
Quantum Research Group Seminar at Purdue
06-2024 – Anyonica and the AnyonWiki,
Thematic Program in Field Theory and Topology at Notre Dame University
(Invited) 05-2024 – On Low-Rank Multiplicity-Free Fusion Categories,
Quantum groups and monoidal categories seminar at Université Libre Bruxelles
06-2024 – Anyonica and the AnyonWiki,
Great Lakes Mathematical Physics Meeting
(Invited) 12-2022 – Knots and quantum computing,
Mathematics seminar at Osnabrück
(Invited) 04-2022 – Knots and quantum computing,
Doctoral mini-course on Combinatorial p-Kazhdan-Lusztig Theory 
(Invited) 03-2022– Geometric Algebra,
Maynooth theoretical physics seminar

Posters

I like being creative with my research posters. In particular, I love to combine the topics I’m researching with other media I’m engaging with. The poster on fusion rings is inspired by The Lord of The Rings, my favorite childhood books and movies. The poster on fusion categories is based on the Doom video games I was playing during my PhD.

Both posters were created with the free and open source application Inkscape.

The images shown below are of limited quality but you can download the pdf files here

Note that (especially for the fusion categories poster) it might take some time for your pdf viewer to load the whole image.

Teaching

Fall 2026 — Lectures on Ordinary Differential Equations (MA266) at Purdue University.
Spring 2026 — Lectures on Differential Equations and Linear Algebra (MA262) at Purdue University.
Fall 2024 to fall 2025 — Lectures on Ordinary Differential Equations (MA266) at Purdue University.
2021 — Tutorials on Special Relativity for 3rd-year physics students in Maynooth
2021 — Tutorials on Probability and Statistics for 3rd-year engineering students at Maynooth University
2021 — Tutorials on Quantum Mechanics for 2nd-year physics students in Maynooth
2020 — Tutorials on Vector Algebra and Matrices for first-year engineering students in Maynooth
2019 — Tutorials on Quantum Computing for graduate students at Ghent University. Also wrote and marked the exercises of the exam papers

Referees

Colleen Delaney
Assistant Professor
Department of Mathematics & Department of Physics and Astronomy
Purdue University
email: colleend (at) purdue.edu
Eric Samperton
Assistant Professor
Department of Mathematics & Department of Computer Science
Purdue University
email: eric (at) purdue.edu
Eric Rowell
Professor, Presidential Impact Fellow
Department of Mathematics
Texas A & M
email: rowell (at) math.tamu.edu
Steven Simon
Professor of Physics and Professorial Fellow of Somerville College
Rudolf Peierls Centre for Theoretical Physics
Oxford University
email: steven.simon (at) physics.ox.ac.uk
Eddy Ardonne
Professor
Department of Physics
Stockholm University
email: ardonne (at) fysik.su.se

Purdue Website

The following is my Purdue Webpage.