Home page of João Faria Martins

Recent preprints

• Faria Martins J, Porter T : A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from strict omega-groupoids. arXiv:2301.02491 [math.CT]. (Recently updated.)


• Faria Martins J, Meusburger C : A geometrical description of untwisted Dijkgraaf-Witten TQFT with defects. arXiv:2410.18049 [math.QA].

News

• Postdoctoral positions at Leeds! Research Fellow in Pure Mathematics
  Closing date: Friday 07 March 2025. Fixed Term (26 months, to end by 31 July 2027).
  This position is part of our EPSRC Programme Grant, Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology, 2022-2027.


• Workshop recently finished at Leeds:
  • ARTIN in Leeds: Biracks and Biquandles: Theory, applications, and new perspectives, 16-19 December 2024

Miscellaneous • List of publications; • conferences organised; • Yorkshire and Midlands Category Theory Seminar (YaMCATS) My pages at: •Mathematical Reviews; • Scopus; • Web of Science; • scholar.google; • inspire (hep-th); • ORCID; • arXiv; the Leverhulme Trust Research Project Grant: PI: Emerging Physics From Lattice Models of Higher Gauge Theory. (Completed.) Programme Grant (EPSRC): Co-I: Combinatorial Representation Theory: Discovering the Interfaces of Algebra with Geometry and Topology, 2022-2027. Prospective PhD students Do contact me if you are interested in completing a PhD degree at Leeds, in any of my areas of interest. Several projects are possible. Two particular projects are below: Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory Algebraic, geometric and physical underpinnings of topological quantum computation. (Written 3rd December 2025.) There are deadlines coming very soon if you want to be eligible to apply for a PhD scholarship from Leeds. Please contact me (by email) if you want to apply to any of these positions.

Compiling xymatrix diagrams into HTML using Lwarp and XyJax-v3 Quickly compile latex notes containing commutative diagrams coded with xymatrix into (reasonably?) accessible HTML. Comments and suggestions are very welcome. Lwarp with Xy-Jaxv3 (split page version) Lwarp with Xy-Jaxv3 (stand alone page version). Research interests: algebraic and geometric topology in dimensions 3 and 4; extended Topological Quantum Field Theories; algebraic models for homotopy types; higher dimensional category theory; homotopical algebra; Hopf algebras and quantum (q-deformed) Lie groups; differential geometry (higher Lie and gauge theory); mathematical aspects of Chern-Simons theory, BF-theory and generalisations; mathematical models for topological phases of matter; topological quantum computing.

Mini biography before joining Leeds (dates are approximate): 06/2000: Degree in Applied Mathematics and Computation, from Instituto Superior Técnico, of Lisbon. 12/2004: PhD in Mathematics University of Nottingham, supervised by John W. Barrett. 01/2005 - 06/2008: Postoctoral fellow in the Mathematics Department of Instituto Superior Técnico, of Lisbon. I was funded by FCT (Portugal) 07/2008 - 09/2009: Assistant Researcher, Center of Mathematics of University of Oporto, Faculty of Sciences of University of Oporto. 10/2009 - 09/2016: Assistant Professor, Department of Mathematics, Nova School of Sciences and Technology. Lisbon (Portugal)

Selected publications (a full list of publications is here):

• Koppen V, Faria Martins J, Martin P: Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras. To appear in Advances in Theoretical and Mathematical Physics
  
  
• Faria Martins J, Frank W Nijhoff, Daniel Riccombeni Darboux-Kadomtsev-Petviashvili system as an integrable Chern-Simons multiform theory in infinite dimensional space . Physical Review D 109, L021701. Preliminary version.
  
  
• Torzewska F, Faria Martins J, Martin P: Motion groupoids and mapping class groupoids. . Communications in Mathematical Physics 402, 1621–1705 (2023). Preliminary version.


• Damiani C, Martin P, and Faria Martins J: On a canonical lift of Artin's representation to loop braid groups, Journal of Pure and Applied Algebra Volume 225, Issue 12, December 2021, 106760. Postprint version.


• Bullivant A, Martin P, and Faria Martins J: Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. Advances in Theoretical and Mathematical Physics, Volume 23 (2019) Number 7, Pages: 1685--1769. Postprint version. Arxiv version arXiv:1807.09551 [math-ph].


• Bullivant A, Calçada M, Kádár Z, Martin P, and Faria Martins J: Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry. Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020) | Research Paper. Postprint version. Earlier version (arXiv:1702.00868 [math-ph]).


• Cirio L.S., Faria Martins J. Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra. Advances in Theoretical and Mathematical Physics. Volume 21 (2017) Number 1, Pages: 147 - 229 Preliminary version.


• Faria Martins J.: Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy. Journal of Geometry and Physics Volume 99, January 2016, Pages 68-110. Preliminary version.


• Cirio L.S, Faria Martins J.: Infinitesimal 2-braidings and differential crossed modules, Advances in Mathematics, Volume 277, 4 June 2015, Pages 426-491. Preliminary version.


• Cirio L.S, Faria Martins J.: Categorifying the Knizhnik–Zamolodchikov connection: Lucio Simone Cirio, Differential Geometry and its Applications, Volume 30, Issue 3, June 2012, Pages 238–261. Preliminary version.


• Gohla B.; Faria Martins J.: Pointed homotopy and pointed lax homotopy of 2-crossed module maps, Advances in Mathematics Volume 248, 25 November 2013, Pages 986-1049. Preliminary version. Very preliminary version (results proved in a different way).


• Faria Martins J, Mikovic A. :Lie crossed modules and gauge-invariant actions for 2-BF theories, Advances in Theoretical and Mathematical Physics, Volume 15, Number 4 (August 2011) p.1059-1084. Preliminary version


• Faria Martins J, Picken R..: Surface Holonomy for Non-Abelian 2-Bundles via Double Groupoids, Advances in Mathematics Volume 226, Issue 4, 1 March 2011, Pages 3309-3366. Very preliminary version (the title is different from the published version).


• Faria Martins J, Picken R..: The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module, Differential Geometry and its Applications Volume 29, Issue 2, March 2011, Pages 179-206. Very preliminary version (it contains more materials than the published version).


• Faria Martins J., Picken R..On Two-Dimensional Holonomy, Transactions of the American Mathematical Society, 362 (2010), 5657-5695.


• Faria Martins J.: The Fundamental Crossed Module of the Complement of a Knotted Surface.  Transactions of the American Mathematical Society.   361 (2009), 4593-4630.


• Faria Martins J., Kauffman L.H.: Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces, Compositio Mathematica. Volume 144, Issue 04, July 2008, pp 1046-1080.


• Faria Martins J, Mikovic A.: Invariants of Spin Networks Embedded  in Three-Manifolds,  Communications in Mathematical Physics, Volume  279, Number 2/ Abril 2008, pages 381-399.
  


• Faria Martins J, Porter T : On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups,  Theory and Application of Categories, Vol. 18, 2007, No. 4, pp 118-150.

Some selected talk videos

• A categorification of Quinn's finite total homotopy TQFT with application to TQFTs and once-extended TQFTs derived from discrete higher gauge theory. Invited talk at: Twinned workshop on "Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras", MPIM Bonn/Germany and CMSA, Harvard/USA. (Tue, 08 Jul 2025).


• A Lagrange multiform for the Darboux-KP system derived from infinite dimensional Chern-Simons theory Invited talk at: Lagrangian Multiform Theory and Pluri-Lagrangian Systems (23w5043) Hangzhou, China, October 22 to October 27, 2023.


• A once-extended TQFT categorifying Quinn's finite total homotopy. Invited talk at Emergent Geometries from Strings and Quantum Fields/ Galileo Galilei Institute for Theoretical Physics. (Florence) Jun 12, 2023 - Jul 28, 2023.


• Quinn Finite Total Homotopy TQFT as a once-extended TQFT Geometry, Topology, and Physics (GTP) Seminar, Center for Quantum and Topological Systems, NYU Abu Dhabi. Slides. 29/03/2023.


• Quinn Finite Total Homotopy TQFT as a once-extended TQFT. Topological Quantum Field Theory Club (Lisbon). Slides. 24/09/2021, Friday, 17:00.


• Crossed modules, homotopy 2-types, knotted surfaces and welded knots. Topological Quantum Field Theory Club (Lisbon). Slides. 30/10/2020, Friday, 17:00.

Some selected talk slides

• Invariants of knots, loop braids and knotted surfaces derived from finite 2-groups, contributed talk at Knots and Braids in Norway 2019, NTNU (Trondheim). (16th May 2019)


• Discrete Hamitonian models for 3+1 D topological phases derived from higher gauge theory , seminar at Department of Physics, University of Oxford. (24th April 2019).


• Topological Phases Derived from Discrete Higher Gauge Theory and Representations of the Loop Braid Group, invited talk at Quantum Computation and Information Workshop, Institute for Applied Mathematics and Computational Science, Texas A&M University. (22nd September 2018)


• Categorifications of the 4-term relations via infinitesimal 2-braidings. Seminar at the Max Plank Institute for Mathematics, Bonn. (7th May 2014)

Participation in some grant projects of FCT (Portugal):

• HIGHER STRUCTURES AND APPLICATIONS PTDC/MAT-PUR/31089/2017 , 07/2018 - 06/2022
• NEW GEOMETRY AND TOPOLOGY (PTDC/MAT/101503/2008), 01/2010 - 06/2013.
• TOPOLOGICAL INVARIANTS VIA DIFFERENTIAL GEOMETRY (PTDC/MAT/098770/2008), 03/2010 - 09/2013.
• MATHEMATICAL PHYSICS (PTDC/MAT/69635/2006), 09/2007 - 02/2011.