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Roger Picken, IST, Lisbon

Introductory session for "Virtual Knot Theory" by Louis Kauffman

Topological Quantum Field Theory Seminar

Roger Picken, IST, Lisbon

Introductory session for "Virtual Knot Theory" by Louis Kauffman

Introductory session for the minicourse on Virtual Knot Theory by Louis Kauffman.

João Faria Martins, Instituto Superior Técnico

Crossed Modules and Crossed Complexes in Geometric Topology I

This short course aims at describing some applications of crossed modules and crossed complexes to Geometric Topology, and it is based on results by the author. The background is R. Brown and P.J. Higgins beautiful work on crossed modules and crossed complexes. We will give a lot of attention to applications to knotted embedded surfaces in S^4, and we will make explicit use of movie representations of them. Some of the ideas of this work started from Yetter's Invariant of manifolds and subsequent developments. Full summary and references: http://www.math.ist.utl.pt/~rpicken/tqft/kauffman062006/CMGT.pdf

Louis Kauffman, Univ. Illinois, Chicago

Virtual Knot Theory I

Introduction to combinatorial knot theory; Reidemeister moves, moves on virtuals; interpretation of virtual knot theory in terms of knots and links in thickened surfaces; bracket polynomial for virtuals, involutory quandle for virtuals.

Marko Stosic, Instituto Superior Técnico

Khovanov homology of torus knots

In this talk we show that the torus knots ${T}_{p,q}$ for $3\le p\le q$ are homologically thick. Furthermore, we show that we can reduce the number of twists $q$ without changing a certain part of the homology, and consequently we show that there exists a stable homology for torus knots conjectured in [1]. Also, we calculate the Khovanov homology groups of low homological degree for torus knots, and we conjecture that the homological width of the torus knot ${T}_{p,q}$ is at least $p$. References: [1] N. Dunfield, S. Gukov and J. Rasmussen: *The Superpolynomial for link homologies*, arXiv:math.GT/0505056. [2] M. Stosic: *Homological thickness of torus knots*, arXiv:math.GT/0511532

Louis Kauffman, Univ. Illinois, Chicago

Virtual Knot Theory II

Continuing discussion of invariants of virtual knots and links. Biquandles and 0-level Alexander polynomial. Quaternionic biquandle. Weyl algebra and the linear non-commutative biquandles.

João Faria Martins, Instituto Superior Técnico

Crossed Modules and Crossed Complexes in Geometric Topology II

This short course aims at describing some applications of crossed modules and crossed complexes to Geometric Topology, and it is based on results by the author. The background is R. Brown and P.J. Higgins beautiful work on crossed modules and crossed complexes. We will give a lot of attention to applications to knotted embedded surfaces in S^4, and we will make explicit use of movie representations of them. Some of the ideas of this work started from Yetter's Invariant of manifolds and subsequent developments. Full summary and references: http://www.math.ist.utl.pt/~rpicken/tqft/kauffman062006/CMGT.pdf

Louis Kauffman, Univ. Illinois, Chicago

Virtual Knot Theory III

Flat virtuals and long flat virtuals. Khovanov homology and virtual knot theory.

Mark Gotay, Univ. of Hawai at Manoa

Stress-Energy-Momentum Tensors

J. Marsden and I present a new method of constructing a stress-energy-momentum tensor for a classical field theory based on covariance considerations and Noether theory. Our stress-energy-momentum tensor ${T}^{\mu}{}_{\nu}$ is defined using the (multi)momentum map associated to the spacetime diffeomorphism group. The tensor ${T}^{\mu}{}_{\nu}$ is uniquely determined as well as gauge-covariant, and depends only upon the divergence equivalence class of the Lagrangian. It satisfies a generalized version of the classical Belinfante-Rosenfeld formula, and hence *naturally* incorporates both the canonical stress-energy-momentum tensor and the "correction terms" that are necessary to make the latter well behaved. Furthermore, in the presence of a metric on spacetime, our ${T}^{\mu \nu}$ coincides with the Hilbert tensor and hence is automatically symmetric.

**References: **

- Gotay, M. J. and J. E. Marsden [1992], Stress-energy-momentum tensors and the Belinfante-Rosenfeld formula,
*Contemp. Math.***132**, 367-391. - Forger, M. and H. Römer [2004], Currents and the energy-momentum tensor in classical field theory: A fresh look at an old problem,
*Ann. Phys.***309**, 306-389.

Mark Gotay, Univ. of Hawai at Manoa

Obstructions to Quantization 1

Quantization is not a straightforward proposition, as demonstrated by Groenewold's and Van Hove's discovery, sixty years ago, of an "obstruction" to quantization. Their "no-go theorems" assert that it is in principle impossible to consistently quantize every classical polynomial observable on the phase space ${R}^{2n}$ in a physically meaningful way. Similar obstructions have been recently found for ${S}^{2}$ and ${T}^{*}{S}^{1}$, buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so-it has just been proven that there are no obstructions to quantizing either ${T}^{2}$ or ${T}^{*}{R}_{+}$. In this talk we conjecture-and in some cases prove-generalized Groenewold-Van Hove theorems, and determine the maximal Lie subalgebras of observables which can be consistently quantized. This requires a study of the structure of Poisson algebras of symplectic manifolds and their representations. To these ends we review known results as well as recent theoretical work. Our discussion is independent of any particular method of quantization; we concentrate on the structural aspects of quantization theory which are common to all Hilbert space-based quantization techniques. (This is joint work with J. Grabowski, H. Grundling and A. Hurst.)

**References: **

- Gotay, M. J. [2000], Obstructions to Quantization, in:
*Mechanics: From Theory to Computation.*(*Essays in Honor of Juan-Carlos Simo*),*J. Nonlinear Science*Editors, 271-316 (Springer, New York). - Gotay, M. J. [2002], On Quantizing Non-nilpotent Coadjoint Orbits of Semisimple Lie Groups.
*Lett. Math. Phys*.**62**, 47-50.

Mark Gotay, Univ. of Hawai at Manoa

Obstructions to Quantization 2

Let $(L,\nabla )$ be a prequantum line bundle over a symplectic manifold $X$, and $S$ its symplectization. Kostant showed that the classical Poisson bracket on $S$ is simply prequantization on $X$. C. Duval and I have taken this a step farther to obtain a quantization of $X$ using a generalized star-product on $S$.

**References: **

- Kostant, B. [2003], Minimal coadjoint orbits and symplectic induction,
*arXiv*: SG/0312252.