Contents/conteúdo

Topological Quantum Field Theory Seminar   RSS

Past sessions

Newer session pages: Next 6 5 4 3 2 1 Newest 

16/07/2010, 11:30 — 12:30 — Room P3.10, Mathematics Building
Rafael Diaz, Universidad Sergio Arboleda, Colombia

Categorification, Feynman Integrals, and Quantization - II

We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

  1. R. Diaz, E. Pariguan, Super, Quantum, and Noncommutative Species, African Diaspora J. Math.  8  (2009)  90-130.
  2. 2. H. Blandin, R. Diaz, Rational Combinatorics, Adv. Appl. Math 40 (2008) 107-126.
  3. E. Castillo, R. Diaz, Rota-Baxter Categories, Int. Elect. J. Alg. 5 (2009) 27-57.
Support: FCT, CAMGSD, New Geometry and Topology

16/07/2010, 10:00 — 11:00 — Room P3.10, Mathematics Building
Yasuyoshi Yonezawa, Univ. of the Algarve and IST

Quantum ( sln , Vn ) link invariant and matrix factorizations.

M. Khovanov and L. Rozansky constructed a homology for a link diagram whose Euler characteristic is the quantum link invariant associated to the quantum group Uq(sl n) and its vector representation V n by using matrix factorizations. In my thesis, I study a generalization of the Khovanov-Rozansky homology for the quantum link invariant associated to Uq(sl n) and its fundamental representations V n. In this talk, I will define a new link invariant derived from the generalization of Khovanov-Rozansky homology.
Support: FCT, CAMGSD, New Geometry and Topology

15/07/2010, 16:15 — 17:15 — Room P3.10, Mathematics Building
Jeffrey Morton, University of Western Ontario

2-Linearization, Discrete and Smooth

This talk will introduce the construction "2-linearization", an extension of the "groupoidification" program of Baez and Dolan which seeks to interpret constructions in linear algebra in terms of groupoids and spans of groupoids. The 2-linearization construction finds this as a special case of a 2-functor Λ:Span(Gpd)2 Vect, where 2 Vect is the 2-category of Kapranov-Voevodsky 2-vector spaces. This 2-functor is constructed in terms of pairs of ambi-adjoint functors associated to each groupoid homomorphism, the "push" and "pull" operations, closely related to restriction and induction maps in representation theory, Grothendieck's 6-operation framework for sheaves, among other examples. We will begin with the discrete case, and consider generalizations to smooth groupoids. Finally we will consider some applications.
Support: FCT, CAMGSD, New Geometry and Topology

15/07/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building
Rafael Diaz, Universidad Sergio Arboleda, Colombia

Categorification, Feynman Integrals, and Quantization - I

We review the general notion of categorification in order to study quantization -- in the sense of deformation quantization -- and Feynman integrals from the viewpoint of category theory. The fairly abstract setting that we propose leads to a rather down to earth approach to these often elusive notions.

  1. R. Diaz, E. Pariguan, Super, Quantum, and Noncommutative Species, African Diaspora J. Math.  8  (2009)  90-130.
  2. 2. H. Blandin, R. Diaz, Rational Combinatorics, Adv. Appl. Math 40 (2008) 107-126.
  3. E. Castillo, R. Diaz, Rota-Baxter Categories, Int. Elect. J. Alg. 5 (2009) 27-57.
Support: FCT, CAMGSD, New Geometry and Topology

07/12/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building
Branislav Jurco, Max Planck Institute for Mathematics, Bonn, Germany

Introduction to nonabelian bundle 2-gerbes

We introduce nonabelian bundle 2-gerbes related to Lie 2-crossed modules and discuss their properties. Such bundle 2-gerbes are defined in terms of bundle 1-gerbes, which play a role similar to that of transition functions in the theory of principal bundles.

30/11/2009, 14:30 — 15:30 — Room P3.10, Mathematics Building
Camille Laurent, University of Coimbra

Introduction to Abelian and non-Abelian gerbes

We shall use the language of Lie groupoids and Morita equivalence to introduce gerbes, and connections on them. The advantage of this presentation is that it makes it possible to use ordinary differential geometry, without going to "higher" objects - the default being the need to check the invariance of all constructions under Morita equivalence. I shall try to relate this construction with previous ones, especially the constructions of Breen and Messing.

04/12/2008, 16:30 — 17:30 — Room P3.10, Mathematics Building
, University of Wales, Bangor, UK

The "Crossed" technology and its applications - IV

(see Lecture 1)

04/12/2008, 14:30 — 15:30 — Room P3.10, Mathematics Building
, University of Wales, Bangor, UK

The "Crossed" technology and its applications - III

(see Lecture 1)

03/12/2008, 16:30 — 17:30 — Room P3.10, Mathematics Building
, University of Wales, Bangor, UK

The “Crossed” technology and its applications - II

(see Lecture 1)

03/12/2008, 11:30 — 12:30 — Room P3.10, Mathematics Building
, University of Wales, Bangor, UK

The “Crossed” technology and its applications

In this short series of lectures I will introduce crossed modules and crossed complexes, both algebraically and topologically, will look at their relevance for combinatorial group theory, the theory of syzygies and group cohomology, and then will head for higher order objects namely 2-crossed modules, and related complexes. In the final parts I will introduce some of the constructions of non-Abelian cohomology, sheaves, torsors and Bitorsors and consider the interaction between the crossed gadgetry of the earlier lectures and this area.
References

Schedule:
Lecture 1: Wednesday, 3rd December, 2008, 11h30 - 13
Lecture 2: Wednesday, 3rd December, 2008, 16h30 - 18
Lecture 3: Thursday, 4th December, 2008, 14h30 - 16
Lecture 4: Thursday, 4th December, 2008, 16h30 - 18

14/07/2008, 11:30 — 12:30 — Room P3.10, Mathematics Building
, Department of Mathematics, University of California at Riverside

Topological 2-Groups and Their Classifying Spaces

Categorifying the concept of topological group, one obtains the notion of a topological 2-group. This in turn allows a theory of "principal 2-bundles" generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the Cech cohomology H 1 (M,G) or the set of homotopy classes [M,BG], where BG is the classifying space of G. Here we review work by Bartels, Jurco, Baas-Bökstedt-Kro, and others generalizing this result to topological 2-groups. We explain various viewpoints on topological 2-groups and the Cech cohomology H 1 (M,G) with coefficients in a topological 2-group G, also known as "nonabelian cohomology". Then we sketch a proof that under mild conditions on M and G there is a bijection between H 1 (M,G) and [M,BG], where BG is the classifying space of the geometric realization of the nerve of G. Applying this result to the ''string 2-group" String(G) of a simply-connected compact simple Lie group G, we obtain a theory of characteristic classes for principal String(G)-2-bundles.

14/07/2008, 10:00 — 11:00 — Room P3.10, Mathematics Building
, Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

Virtual Knot Theory

New developments in Virtual Knot Theory.

12/03/2008, 16:00 — 17:00 — Room P3.10, Mathematics Building
, Physics Department, National Technical University, Athens

Dimensional Reduction of Gauge Theories over continuous and fuzzy coset spaces

We review the dimensional reduction of N=1 higher dimensional Gauge Theories over Coset Spaces with emphasis on the possibility to obtain four-dimensional GUTs with chiral fermions and softly broken supersymmetry. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We emphasize some striking features emerging such as (i) the appearance of non-abelian gauge theories in four dimensions starting from an abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory.

05/03/2008, 16:00 — 17:00 — Room P3.10, Mathematics Building
Atle Hahn, Group of Mathematical Physics of the University of Lisbon

From the Chern-Simons path integral to the Reshetikhin-Turaev invariant

The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to quantum topology. Firstly, the perturbative approach based on the CS path integral in the Lorentz gauge and, secondly, the "quantum group approach" by Reshetikhin/Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if/how one can derive the relevant R-matrices or quantum 6j-symbols directly from the CS path integral. In my talk, which summarizes the results of a recent preprint, I will sketch a strategy that should lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by Blau/Thompson for the study of the partition function of CS models. I will show that the formulas of Blau/Thompson can be generalized to Wilson lines and that the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which Turaev's shadow invariant is defined. Finally, I will sketch how one can obtain a rigorous realization of the path integral expressions appearing in this treatment.

06/12/2007, 15:30 — 16:30 — Room P3.10, Mathematics Building
, International School for Advanced Studies (SISSA), Trieste

Instantons and framed bundles on rational surfaces

The talk concerns a correspondence between framed instantons on the one-point compactification of an affine complex surface X, and framed holomorphic bundles on a projective completion of X. This correspondence is known for X the affine plane (Donaldson) and X the affine plane blown up at a point (King). After reviewing these cases, I will discuss possible generalizations (basically, when the projective completion is a rational surface). I will also spend some words on instanton countings on these surfaces. Physically this corresponds to studying the Nekrasov partition function for topological super Yang-Mills theories on X.

30/11/2007, 14:00 — 15:00 — Room P4.35, Mathematics Building
Yassir Dinar, International School for Advanced Studies (SISSA), Trieste

Algebraic Frobenius manidolds and primitive conjugacy classes in Weyl group

We develop the theory of generalized bi-Hamiltonian reduction. Applying this theory to the loop algebra proved to be equivalent to a generalized Drinfeld-Sokolov reduction. This gives a way to construct new examples of algebraic Frobenius manifolds.

08/06/2007, 16:00 — 17:00 — Amphitheatre Pa3, Mathematics Building
, Institut des Hautes Études Scientifiques

New methods in renormalization theories - III

05/06/2007, 16:30 — 17:30 — Amphitheatre Pa3, Mathematics Building
, Institut des Hautes Études Scientifiques

New methods in renormalization theories - II

05/06/2007, 11:00 — 12:00 — Amphitheatre Pa3, Mathematics Building
, Institut des Hautes Études Scientifiques

New methods in renormalization theories - I

The first occurence of the ideas of renormalisation in physics is due to Green, around 1850, who used such methods to study the motion of a pendulum in a fluid. The same kind of methods was proposed by J. Oppenheimer around 1930, to take in account the so-called radiative corrections to the spectral lines of atoms. Like the previous attempts in classical electrodynamics, this approach led to unphysical infinite quantities. As it si well-known, the new methods of Bethe, Schwinger, Tomonaga, Feynman and Dyson solved in principle the problem of infinities around 1950. But a conceptual breakthrough occured ten years ago when A. Connes and D. Kreimer introduced Hopf algebraic methods in this game. We propose to explain our own verion of these methods, emphasizing a certain infinite-dimensional group, the so-called dressing group. A striking feature is the deep analogy with groups introduced by Grothendieck under the name of motivic Galois groups. These lectures shall begin with a short historical review, followed by a description of the standard calculations, and then we shall describe in detail the new methods.

23/02/2007, 15:45 — 16:45 — Room P3.10, Mathematics Building
Roger Picken, Instituto Superior Técnico

Sétima palestra dum Mini-Encontro do projecto "Topologia Quântica", onde haverá apresentações curtas e informais de membros do projecto sobre assuntos de interesse actual. Todos os interessados bem-vindos.

Older session pages: Previous 8 9 10 11 12 Oldest


Current organizers: Roger Picken, Marko Stošić.

FCT Projects PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry, PTDC/MAT-PUR/31089/2017, Higher Structures and Applications.

 

CAMGSD FCT