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 Anyonica and, since recently, TensorCategories.jl. I aim to make as much data available online as well on the AnyonWiki.

My main goal is to have a central repository of computational (yet readable) data on fusion categories that is free, open-source, accessible, user friendly for both mathematicians and physicists.

At the moment, the AnyonWiki contains

The $F$-symbols, $R$-symbols, and pivotal coefficients of all multiplicity-free fusion categories up to rank 7. All data is downloadable in numerical format and also available symbolically as part of the Anyonica and, more recently, TensorCategories.jl packages.
A list of all multiplicity-free fusion rings up to rank 9 together with downloadable data of their most important properties, such as Frobenius-Perron dimensions, $S$-matrices, characters, decompositions, etc
Pages on classes of fusion rings corresponding to certain quantum groups.

Take a look at my current projects page to learn more about the future plans for the wiki.

My CV can be downloaded here

Papers

2025

Triangular Prism Equations and Categorification

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

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

PQDT - Global, Jul 2024

Copyright - Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works; Last updated - 2025-05-14

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) 10-2025 – Complete invariants and minimal fields of multiplicity-free fusion categories,
AMS Fall Sectional meeting
(Invited) 10-2025 – The AnyonWiki, a digital repository for fusion categories and anyon models,
AMS Fall Sectional meeting
(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 Fusions Categories,
Quantum groups and monoidal categories seminar at Universite Libre Bruxelles
06-2024 – Anyonica and the AnyonWiki,
Great Lakes Mathematical Physics Meeting
(Invited) 12-2022 – Knots and quantum computing,
Mathematics seminar at Osnabruck
(Invited) 04-2022 – Knots and quantum computing,
Doctoral mini-course on Combinatorial p-Kazhdan-Lustzig 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 2024 — 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
Associate 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