João Faria Martins
Faculty of Sciences and Technology
jn.martins -at-
fct dot unl dot pt
NOT UP-TO-DATE: I am officially on-leave. This is my current webpage.
I
am an assistant professor at the Mathematics
Department of the Faculty of
Sciences and Technology of Universidade
Nova de Lisboa, in
Lisbon, Portugal.
I am also an Executive Committee member of the research unit CMA (Centro de Matemática e Aplicações da Universidade Nova de Lisboa), and an official collaborator at the research unit Grupo de Física Matemática da Universidade de Lisboa. Previously I was an assistant researcher at the Centro de Matemática da Universidade do Porto, in Oporto, Portugal. I was offered that position under the programme Ciência 2007 of Fundação para a Ciência e Tecnologia.
Before that I was a postdoctoral fellow at the Mathematics
Department of Instituto Superior
Técnico, in
Lisbon, Portugal,
affiliated with the Center
for Mathematical Analysis, Geometry, and Dynamical Systems, and
financially supported by Fundação
para a Ciência e Tecnologia.
I did my PhD at the
School of Mathematical Sciences of Nottingham
University. My supervisor was John
W. Barrett.
Here are my pages at Mathematical Reviews,
scholar.google;
inspire (hep-th);
ORCID;
ResearcherID.
Official page at the Mathematics
Department of the Faculty of
Sciences and Technology of Universidade
Nova de Lisboa.
Since in 2010, I have been involved in the organisation of the summer “Matnova”. This is a summer school in
Mathematics conceived for high achieving secondary school students, with the aim of encouraging
excellent students to go on to study for a degree in Mathematics. Each edition (of 5 days) revolves
around seminars and courses. You can see the individual pages of each edition in
Matnova2010 ,
Matnova2011 ,
Matnova2012 ,
Matnova2013 ,
Matnova2014 ,
Matnova2015 .
We welcomed 83 participants in the 2015 edition.
Research
Interests:
-Algebraic and Geometric Topology in dimensions 3 and 4.
-Higher dimensional category theory.
-Applications to condensed matter Physics and quantum computing.
-Quantum groups and low dimensional topology.
-Algebraic models for homotopy types.
-Differential geometry (higher Lie and gauge theory).
-Mathematical aspects of Chern-Simons Theory and BF-theory.
Preprints
(comments are very welcome):
-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.
Publications:
Crossed modules of Hopf algebras and of associative algebras and two-dimensional holonomy.
To appear in Journal of Geometry and Physics. Preliminary version.
Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras, with İ.İlker Akça and Kadir Emir. To appear in Homology,
Homotopy and Applications. Preliminary version.
Link invariants from finite categorical groups, with Roger
Picken. To appear in Homology,
Homotopy and Applications. 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.
PhD Thesis:
-Quantum Topology and the Lorentz Group.
Other work (comments
are very welcome):
-On
2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces.
This is an archaic version of "The Fundamental Crossed Module of
the Complement of a Knotted Surface", using movies with bands
and spanners rather than hyperbolic splittings of knotted surfaces in
the construction. May still be of interest.
Conferences / workshops (past and future):
- XVIIIth Oporto Meeting on Geometry, Topology and Physics (July 8-12, 2009) at the Oporto University
Main Theme: Symplectic and Poisson Geometry
-Higher Gauge Theory, TQFT and Quantum Gravity, Lisbon, 10-13 February, 2011 (Workshop), 7-13 February, 2011 (School)
- Recent Advances in Topological Quantum Field Theory (TQFT-2012) A one-week workshop on Topological Quantum Field Theory and Quantum Gravity, to be held at the University of Lisbon on September 10–14, 2012
Talk slides (comments are very welcome):
Categorifying the Knizhnik-Zamolodchikov Connections 3Quantum: Algebra Geometry Information, Tallinn 10-13 July 2012 , session: Geometry and topology in quantum gravity