Preprints (comments are very welcome):
- Joao Faria Martins, Frank W Nijhoff, Daniel Riccombeni: The Darboux-KP system as an integrable Chern-Simons multiform theory in infinite dimensional space. arXiv:2305.03182 [math-ph]
- 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, preliminary version.)
- Koppen V, Faria Martins J,
Martin
P: Exactly solvable models for 2+1D topological phases derived from crossed modules of semisimple Hopf algebras. arXiv:2104.02766 [math-ph]
Publications (comments are very welcome as well):
- Motion groupoids and mapping class groupoids. arXiv:2103.10377 [math-ph], with Fiona Torzewska
and Paul Martin. To appear in Commmunications in Mathematical Physics.
-On a canonical lift of Artin's representation to loop braid groups.
With Celeste Damiani and Paul Martin.
Journal of Pure and Applied Algebra Volume 225, Issue 12, December 2021, 106760.
Preliminary version.
-Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory. With Alex Bullivant and Paul Martin. Advances in Theoretical and Mathematical Physics.(To appear.)
Preliminary version.
-Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry. With Alex Bullivant, Marcos Calçada,
Zoltán Kádár and Paul Martin. Reviews in Mathematical Physics, Vol. 32, No. 04, 2050011 (2020)| Research Paper. Postprint version. Earlier version (arXiv:1702.00868 [math-ph]).
-Topological phases from higher
gauge symmetry in 3+1D. With Alex Bullivant, Marcos Calçada,
Zoltán Kádár and Paul Martin. PHYSICAL REVIEW B 95, 155118 (2017). Preliminary version.
-Categorifying the $sl(2,C)$ Knizhnik-Zamolodchikov Connection via an
Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra,
with
Lucio Simone Cirio. Advances in Theoretical and Mathematical Physics. Volume 21 (2017)
Number 1, Pages: 147 - 229.
Preliminary version.
-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.
-Two-Fold Homotopy of 2-Crossed
Module Maps of Commutative Algebras. With İ.İlker Akça and Kadir Emir. Communications in Algebra 47 (2019), no. 1, 289–311. Preliminary version.
-Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras, with İ.İlker Akça and Kadir Emir. Homology,
Homotopy and Applications18 (2016), no. 1, 99–128. Preliminary version.
-Link invariants from finite categorical groups, with Roger
Picken. Homology, Homotopy & Applications. 2015, Vol. 17 Issue 2, p205-233. 29p. Preliminary version. This is a very reduced version of: Link invariants from finite categorical groups and braided crossed modules.
-Infinitesimal 2-braidings and differential crossed modules, with Lucio Cirio.
Advances in Mathematics,
Volume 277, 4 June 2015, Pages 426–491. Preliminary version.
-Pointed homotopy and pointed lax homotopy of 2-crossed module maps,
with Björn Gohla, Advances in Mathematics
Volume 248, 25 November 2013, Pages 986–1049. Preliminary version. Very preliminary version.
-Spin Foam State Sums and Chern-Simons Theory, with Aleksandar
Miković . Chern-Simons gauge theory: 20 years after, 277–284, AMS/IP Stud. Adv. Math., 50, Amer. Math. Soc., Providence, RI, 2011.
Preliminary version.
-Categorifying the Knizhnik–Zamolodchikov connection,
with
Lucio Simone Cirio, Differential Geometry and its Applications, Volume 30, Issue 3, June 2012, Pages 238–261. Preliminary version.
-Four-Dimensional Spin Foam Perturbation Theory, with Aleksandar
Miković .
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA).
7 (2011), 094, 22 pages.
-The fundamental 2-crossed complex of a reduced CW-complex.
Homology,
Homotopy and Applications, Vol. 13 (2011), No. 2, pp.129-157.
-Lie crossed modules and
gauge-invariant actions for 2-BF theories, with Aleksandar
Miković. Advances
in Theoretical and Mathematical Physics, Volume 15, Number 4 (August 2011) p.1059-1084.
Preliminary version
-Surface
Holonomy for Non-Abelian 2-Bundles via Double Groupoids, with
Roger Picken.
Advances
in Mathematics
Volume
226, Issue 4, 1 March 2011, Pages 3309-3366. Preliminary version.
Very preliminary version.
-The
fundamental Gray 3-groupoid of a smooth manifold and local
3-dimensional holonomy based on a 2-crossed module, with Roger
Picken. Differential
Geometry and its Applications Volume
29, Issue 2, March 2011, Pages 179-206. Preliminary version. Very preliminary version.
-On
Two-Dimensional Holonomy, with Roger
Picken. Transactions of the
American Mathematical Society, 362 (2010), 5657-5695.
-Spin
Foam Perturbation Theory for Three-Dimensional Quantum Gravity,
with Aleksandar
Miković . Communications
in Mathematical Physics. Volume 288, Number 2 / June,
2009.
-The
Fundamental Crossed Module of the Complement of a Knotted Surface.
Transactions of the American
Mathematical Society. 361 (2009), 4593-4630.
-Invariants
of Welded Virtual Knots Via Crossed Module Invariants of Knotted
Surfaces, with Louis
H. Kauffman. Compositio
Mathematica. Volume 144, Issue 04, July 2008, pp
1046-1080.
-Invariants
of Spin Networks Embedded in Three-Manifolds,
with Aleksandar
Miković. Communications
in Mathematical Physics, Volume 279, Number 2/ Abril
2008, pages 381-399.
-Observables
in the Turaev-Viro and Crane-Yetter models, with John
W. Barrett and J. Manuel Garcia Islas. Journal
of Mathematical Physics, 48, 093508 (2007) (18
pages).
-On
Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant
to Categorical Groups, with Timothy
Porter. Theory and
Application of Categories, Vol. 18, 2007, No. 4, pp 118-150.
-On the
Homotopy Type and the Fundamental Crossed Complex of the Skeletal
Filtration of a CW-Complex. Homology
Homotopy and Applications, Vol. 9
(2007), No. 1, pp.295-329.
-Categorical
Groups, Knots and Knotted Surfaces, J.
Knot Theory Ramifications 16 (2007), no 9, 1181-1217.
Other work:
-On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces.
Note:
This is an archaic version of "The Fundamental Crossed Module of the Complement of a Knotted Surface, Trans AMS. 361 (2009), 4593-4630", using movies of knotted surfaces with bands and spanners rather than hyperbolic splittings of knotted surfaces for constructing a representation of the algebraic homotopy 2-type of the complement of a knotted surface. May still be of interest.
PhD Thesis:
-Quantum Topology and the Lorentz Group.
Thesis related publications
-Knot
Theory with the Lorentz Group, FUNDAMENTA MATHEMATICAE
188 (2005),
59-93. (Special volume based on papers submitted by the participants of
the Second International Conference “Knots in Poland 2003”).
-On
the Analytic Properties of the $z$-coloured Jones Polynomial, J.
Knot Theory Ramifications 14 (2005), no. 4,
435--466.
