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15/07/2010, 14:00 — 15:00 — Room P3.10, Mathematics Building

Rafael Diaz, *Universidad Sergio Arboleda, Colombia*

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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.

Support: FCT, CAMGSD, New Geometry and Topology

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07/12/2009, 11:30 — 12:30 — Room P3.10, Mathematics Building

Branislav Jurco, *Max Planck Institute for Mathematics, Bonn, Germany*

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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.

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30/11/2009, 14:30 — 15:30 — Room P3.10, Mathematics Building

Camille Laurent, *University of Coimbra*

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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.

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04/12/2008, 16:30 — 17:30 — Room P3.10, Mathematics Building

Tim Porter, *University of Wales, Bangor, UK*

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The "Crossed" technology and its applications - IV

(see Lecture 1)

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04/12/2008, 14:30 — 15:30 — Room P3.10, Mathematics Building

Tim Porter, *University of Wales, Bangor, UK*

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The "Crossed" technology and its applications - III

(see Lecture 1)

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03/12/2008, 16:30 — 17:30 — Room P3.10, Mathematics Building

Tim Porter, *University of Wales, Bangor, UK*

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The “Crossed” technology and its applications - II

(see Lecture 1)

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03/12/2008, 11:30 — 12:30 — Room P3.10, Mathematics Building

Tim Porter, *University of Wales, Bangor, UK*

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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

- The pdf file at the start of the following page, which includes discussions: n-Category Café
- Slightly updated version of the crossed menagerie file: Crossed Menagerie

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

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14/07/2008, 11:30 — 12:30 — Room P3.10, Mathematics Building

John Baez, *Department of Mathematics, University of California at Riverside*

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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 $[M,B\mid G\mid ]$, where $B\mid G\mid $ is the classifying space of the geometric realization of the nerve of $G$. Applying this result to the ''string 2-group" $\mathrm{String}(G)$ of a simply-connected compact simple Lie group $G$, we obtain a theory of characteristic classes for principal $\mathrm{String}(G)$-2-bundles.

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14/07/2008, 10:00 — 11:00 — Room P3.10, Mathematics Building

Louis Kauffman, *Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago*

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Virtual Knot Theory

New developments in Virtual Knot Theory.

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12/03/2008, 16:00 — 17:00 — Room P3.10, Mathematics Building

George Zoupanos, *Physics Department, National Technical University, Athens*

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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.

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05/03/2008, 16:00 — 17:00 — Room P3.10, Mathematics Building

Atle Hahn, *Group of Mathematical Physics of the University of Lisbon*

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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.

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06/12/2007, 15:30 — 16:30 — Room P3.10, Mathematics Building

Ugo Bruzzo, *International School for Advanced Studies (SISSA), Trieste*

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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 $X$.

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30/11/2007, 14:00 — 15:00 — Room P4.35, Mathematics Building

Yassir Dinar, *International School for Advanced Studies (SISSA), Trieste*

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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.

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08/06/2007, 16:00 — 17:00 — Amphitheatre Pa3, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques*

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New methods in renormalization theories - III

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05/06/2007, 16:30 — 17:30 — Amphitheatre Pa3, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques*

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New methods in renormalization theories - II

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05/06/2007, 11:00 — 12:00 — Amphitheatre Pa3, Mathematics Building

Pierre Cartier, *Institut des Hautes Études Scientifiques*

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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.

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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.
```

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23/02/2007, 14:45 — 15:45 — Room P3.10, Mathematics Building

Rui Carpentier, *Instituto Superior Técnico*

```
Sexta 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.
```

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23/02/2007, 14:00 — 15:00 — Room P3.10, Mathematics Building

João Martins, *Instituto Superior Técnico*

```
Quinta 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.
```

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22/02/2007, 16:45 — 17:45 — Room P3.10, Mathematics Building

Marco Mackaay, *Universidade do Algarve*

```
Quarta 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.
```