Home page of João Faria Martins

Contacts:

Dr João Faria Martins,
(also known as Joao Goncalves Faria Martins)

Associate Professor (Teaching and Research)
Algebra Group, Department of Pure Mathematics.

~~Equality~~ Equity, Diversity and Inclusion Lead.

School of Mathematics,
University of Leeds,
Leeds, LS2 9JT, UK.

E-mail: j.fariamartins@leeds.ac.uk
Telephone: (+44) (0113) 343 5130.

My official page at the Faculty of Engineering and Physical Sciences at Leeds. • Conferences; • list of publications; • compiling xymatrix diagrams into HTML using Lwarp Lwarp with Xy-Jaxv3 (split page version) and Lwarp with Xy-Jaxv3 (stand alone page version).

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.

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.

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.

Prospective PhD students

Combinatorial and higher-categorical techniques in low dimensional topology and topological quantum field theory

Algebraic, geometric and physical underpinnings of topological quantum computation.

Postdoctoral positions at Leeds (closing Date: Thursday 06 July 2023):

Research Fellow in Representation Theory and/or Low Dimensional Topology (one or two positions available).

Main scientific publications (a full list of publications is here):

Torzewska F, Faria Martins J, Martin P: Motion groupoids and mapping class groupoids. arXiv:2103.10377 [math-ph]. (Updated version to appear in Communications in Mathematical Physics.)
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]. (235 pages: preprint only.)

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]).
Bullivant A, Calçada M, Kádár Z and Martin P, Faria Martins J: Topological phases from higher gauge symmetry in 3+1 dimensions. PHYSICAL REVIEW B 95, 155118 (2017). Preliminary version.
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.

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. Preliminary version. Very preliminary 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. Preliminary version. Very preliminary 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

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

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

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.

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)