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05/06/2006, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

Marko Stosic, *Instituto Superior Técnico*

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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

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02/06/2006, 15:30 — 16:30 — Sala P12, Pavilhão de Matemática

Louis Kauffman, *Univ. Illinois, Chicago*

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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.

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02/06/2006, 14:00 — 15:00 — Sala P12, Pavilhão de Matemática

João Faria Martins, *Instituto Superior Técnico*

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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

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31/05/2006, 10:30 — 11:30 — Sala P6, Pavilhão de Matemática

Roger Picken, *IST, Lisbon*

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Introductory session for "Virtual Knot Theory" by Louis Kauffman

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

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14/07/2005, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

Roger Picken, *Instituto Superior Técnico*

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Gerbes and their Parallel Transport

Gerbes are higher-order generalizations of Abelian bundles. They appear in nature, for instance as obstructions to lifting $\mathrm{SO}\left(n\right)$-bundles to $\mathrm{Spin}\left(n\right)$ or ${\mathrm{Spin}}_{c}\left(n\right)$ bundles. It is possible to endow gerbes with connection 1- and 2-forms and curvature 3-forms, and study aspects of the ensuing differential geometry. In particular, gerbes with connection have holonomies and parallel transports along surfaces, as opposed to along loops and paths. Apart from discussing these features, I hope to describe an interesting recent categorification approach to non-Abelian gerbes due to Baez and Schreiber. #### References

- J. Baez and U. Schreiber, Higher gauge theory: 2-connections on 2-bundles, hep-th/0412325.
- M. Mackaay and R. Picken, Holonomy and parallel transport for Abelian gerbes, Adv. Math. 170, 287-339 (2002), math.DG/0007053.
- R. Picken, TQFT's and gerbes, Algebr. Geom. Topol. 4 (2004) 243-272, math.DG/0302065.

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14/07/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

Michael Paluch, *Instituto Superior Técnico*

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Twisted K-Theory

In 1945 Samuel Eilenberg and Norman E. Steenrod set forth the essential properties of a homology theory in terms of seven axioms; the last stipulating that the reduced homology of point is trivial. A number of years later (1957) Alexander Grothendieck introduced K-theory and expressed the Riemann-Roch theorem for nonsingular projective varieties by saying that the mapping $E\to ch\left(E\right)*\mathrm{Td}\left(X\right)$ from ${K}^{0}\left(X\right)$ to ${H}^{*}\left(X\right)$ is a natural transformation of covariant functors. Here ${K}^{0}\left(X\right)$ denotes the Grothendieck group of algebraic vector bundles on $X$, ${H}^{*}\left(X\right)$ denotes a suitable cohomology theory, $ch$ is the Chern character, and $\mathrm{Td}\left(X\right)$ is the Todd class of the tangent bundle of $X$. Michael Atiyah and Friedrich Hirzebruch developed K-theory in the context of topological spaces and showed that topological K-theory satisfies the first six axioms of Eilenberg and Steenrod. Using Bott periodicity one readily shows that the K-theory of a point is infinite cyclic in even degrees and vanishes in odd degrees.

Recently Edward Witten has argued that K-theory is relevant to the classification of Ramond-Ramond (RR) charges as well as noncommutative Yang-Mills theory or open string field theory. In order to consider D-branes with a topologically non-trivial Neveu-Schwarts 3-form field $H$, one needs to work with a twisted version of topological K-theory. If $H$ represents a torsion class, one may use the twisted K-theory developed by Peter Donavan and Max Karoubi. In this talk I shall describe two constructions of twisted K-theory one set forth by Michael Atiyah and Graeme Segal and the other by Daniel Freed, Michael J. Hopkins and Constantin Teleman. Due to personal limitations I shall give a braneless presentation.

#### References

- M Atiyah and F Hirzebruch, Vector bundles and homogenuous spaces, Proc. of Symposia in Pure Maths vol 3, Differential Geometry, Amer. Math. Soc. 1961, 7-38.
- M Atiyah and G Segal, Twisted K-theory, math.kt/0407054.
- A Borel and J-P Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958) 97-136.
- P Donavan and M Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math. IHES 38 (1970) 5-25.
- S Eilenberg and N E Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. USA 31 (1945), 117-120.
- D Freed, M J Hopkins and C Teleman, Twisted K-theory and loop groups representations I, math.AT/0312155.
- E Witten, Overview of K-theory Applied to Strings, hep-th/0007175.

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30/06/2005, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

Marko Stosic, *Instituto Superior Técnico*

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Categorification of the chromatic and dichromatic polynomial for
graphs

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30/06/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

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Rasmussen's s-invariant for links

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29/06/2005, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

João Paulo Santos, *Instituto Superior Técnico*

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Instantons, Holomorphic Bundles and the Bar Construction

One version of the Kobayashi-Hitchin correspondence relates moduli spaces of instantons on blow ups of ${C}^{2}$ trivialized at $\infty $, with holomorphic bundles on blow ups of ${\mathrm{CP}}^{2}$ framed on a line. In the rank stable limit these moduli spaces can be described using a bar construction.

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29/06/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

Nuno Romão, *University of Adelaide*

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Gauged Vortices in a Background

I will discuss how a coupling to an external potential can be used to probe the interactions among gauged vortices on a sphere via symplectic localisation. I shall also illustrate how these results can be applied to statistical mechanics on the moduli space of vortices.

References:

- N S Manton & P M Sutcliffe, "Topological Solitons", Cambridge Univ. Press, 2004.
- N M Romão, "Gauged vortices in a background",

hep-th/0503014

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09/06/2005, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática

José Natário, *Instituto Superior Técnico*

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Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes

We explain what asymptotic quasinormal modes are, why there has been considerable recent interest in computing their frequencies, and how to obtain a complete classification of asymptotic quasinormal frequencies for static, spherically symmetric black hole spacetimes in d dimensions. ##### References

- José Natário, Ricardo Schiappa, On the Classification of Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes and Quantum Gravity,

hep-th/0411267

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09/06/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

Ricardo Schiappa, *Instituto Superior Técnico*

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BRST Cohomology and Characters of Pure Spinors

In this talk we shall review the pure spinor approach for the super-Poincaré BRST covariant quantization of the superstring. We will focus on the BRST operator, its cohomology and the computation of central charges in the pure spinor conformal field theory, where the ghosts are constrained to be pure spinors. This will mainly review work due to Nathan Berkovits and will be at a broad/informal level. ##### References

- Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring,

hep-th/0209059

- Nathan Berkovits, Covariant Multiloop Superstring Amplitudes,

hep-th/0410079

- Nathan Berkovits, Nikita Nikrasov, The Character of Pure Spinors,

hep-th/0503075

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11/03/2005, 16:30 — 17:30 — Anfiteatro Pa2, Pavilhão de Matemática

João Faria Martins, *Instituto Superior Técnico*

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Categorical Groups, Knots and Knotted Surfaces

We define an invariant of knots and an invariant of knotted surfaces from any finite categorical group (crossed module of groups). We illustrate its non-triviality by calculating an explicit example, namely the Spun Trefoil. The talk will be based on:

[1] João Faria Martins, Categorical Groups, Knots and Knotted Surfaces.

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11/03/2005, 15:00 — 16:00 — Anfiteatro Pa2, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

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Colored stable Bar-Natan link homology

Khovanov defined several link homologies categorifying the colored Jones polynomial and conjectured relations between them. Unfortunately none of them can be computed with the existing computer programs for link homology. Fortunately Khovanov's constructions are universal in the sense that any Frobenius algebra satisfying Bar-Natan's universal axioms can be plugged into them yielding framed link homologies. Paul Turner and I did this for the stable Bar-Natan Frobenius algebra and computed the colored link homology for this choice completely for any link. In my talk I will review Khovanov's constructions briefly and then explain the results Paul and I obtained for the stable Bar-Natan theory.

[1] Marco Mackaay and Paul Turner, Colored stable Bar-Natan link homology

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09/02/2005, 14:00 — 15:00 — Sala P12, Pavilhão de Matemática

Stephen Sawin, *Fairfield University*

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Witten-Style Nonabelian Localization For a Noncompact Manifold

Witten in [1] offered a clever scheme to express certain integrals over a Hamiltonian (i.e., symplectic, with group action and a moment map) manifold as a sum of local contributions from the critical points of the square of the moment map. In particular this allows one to read off the ring structure of the cohomology of the symplectic reduction (when it is nice enough) from integrating equivariant cohomology classes in the original space. His elegant argument ignores most analytic subtleties and thus is purely heuristic, but Jeffrey and Kirwan in [2] were able to reproduce his key results in the compact case, by relating the question to one accessible by older abelian localization techniques. I will argue that the noncompact case is particularly important by relating to some outstanding cases, and that the abelian localization argument is unlikely to extend here. I will prove Witten's results rigorously using his version of nonabelian localization, and suggest ways to extend these results further. - E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.

hep-th/9204083 - L. C. Jeffrey, F. Kirwan, Localization for nonabelian group action, Topology 34 (1995) no. 2, 291-327.

alg-geom/9307001

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21/05/2004, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática

Marco Mackaay, *Universidade do Algarve*

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Khovanov's categorification of the Jones polynomial

Following Bar-Natan's down-to-earth approach, I will explain
Khovanov's construction which associates to a knot a certain
complex of graded vector spaces. If two knots are ambient isotopic
their complexes are homotopy equivalent (grading is preserved).
Therefore the cohomology groups of the knot complex are
knot-invariants. It turns out that the Jones polynomial of a knot
equals the graded Euler characteristic of the knot cohomology.
Khovanov derived a more general polynomial from his complexes which
is a more powerful knot invariant, as has been shown by explicit
computations. References:
- Dror Bar-Natan, "On Khovanov's categorification of the Jones
polynomial", Algebraic and Geometric Topology 2 (2002) 337-370;
math.QA/0201043.
- Mikhail Khovanov, "A functor-valued invariant of tangles",
Algebr. Geom. Topol. 2 (2002) 665-741;
math.QA/0103190.
- Mikhail Khovanov, "A categorification of the Jones polynomial",
math.QA/9908171.

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21/05/2004, 10:00 — 11:00 — Sala P3.10, Pavilhão de Matemática

Pedro Vaz, *Universidade do Algarve*

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Induced representations and geometric quantization of coadjoint
orbits

There is a well known correspondence between the orbit method in
geometric quantization and the theory of unitary irreducible
representations of a Lie group. We show that the pre-quantization
of a coadjoint orbit of a connected Lie group G arises as the
infinitesimal version of an induced representation of G. With the
aid of a polarization, this procedure allow us to construct unitary
irreducible representations that are also the infinitesimal version
of an induced representation. As an example, we construct the
corresponding (infinite dimensional) unitary representations of the
Lie group SL(2,C), the universal cover of the Lorentz group.

References:
- I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin. Generalized
Functions volume 5, "Integral Geometry and Representation Theory".
Academic Press,New York, 1966.
- A.A. Kirillov. Elements of the Theory of Representations
Springer-Verlag, 1976.
- N. Woodhouse. Geometric Quantization. Oxford, 1991.

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17/12/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática

J. Scott Carter, *Univ. South Alabama*

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Quandle homology theories and cocycle invariants of knots

Cohomology theories have been developed for certain
self-distributive groupoids called quandles. Variations of
invariants of knots and knotted surfaces have been defined using
quandle cocycles and the state-sum form. We review these
developments, and also discuss quandle modules and their relation
to generalizations of Alexander modules, and topological
applications of these invariants.

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06/11/2003, 17:00 — 18:00 — Sala P3.31, Pavilhão de Matemática

Pedro Lopes, *Instituto Superior Técnico*

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Nós e os Quandles II

Apresentação dos resultados da tese de
doutoramento.

References:

- P. Lopes, Quandles at finite temperatures I, J. Knot Theory
Ramifications, 12(2):159-186 (2003),
math.QA/0105099
- F. M. Dionisio and P. Lopes, Quandles at finite temperatures
II, J. Knot Theory Ramifications to appear,
math.GT/0205053
- J. Bojarczuk and P. Lopes, Quandles at finite temperatures III,
submitted to J. Knot Theory Ramifications

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30/10/2003, 16:30 — 17:30 — Sala P3.10, Pavilhão de Matemática

Pedro Lopes, *Instituto Superior Técnico*

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Nós e os Quandles I

Apresentação dos resultados da tese de
doutoramento.

References:

- P. Lopes, Quandles at finite temperatures I, J. Knot Theory
Ramifications, 12(2):159-186 (2003),
math.QA/0105099
- F. M. Dionisio and P. Lopes, Quandles at finite temperatures
II, J. Knot Theory Ramifications to appear,
math.GT/0205053
- J. Bojarczuk and P. Lopes, Quandles at finite temperatures III,
submitted to J. Knot Theory Ramifications