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06/06/2006, 15:00 — 16:00 — Sala P9, Pavilhão de Matemática
, 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.

05/06/2006, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
Marko Stosic, Instituto Superior Técnico

Khovanov homology of torus knots

In this talk we show that the torus knots Tp,q for 3pq 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 Tp,q is at least p.
[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

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

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

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

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

Gerbes and their Parallel Transport

Gerbes are higher-order generalizations of Abelian bundles. They appear in nature, for instance as obstructions to lifting SO (n) -bundles to Spin (n) or Spin c (n) 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.


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

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

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 ch(E)*Td(X) from K 0 (X) to H * (X) is a natural transformation of covariant functors. Here K 0 (X) denotes the Grothendieck group of algebraic vector bundles on X , H * (X) denotes a suitable cohomology theory, ch is the Chern character, and Td (X) 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.


  1. 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.
  2. M Atiyah and G Segal, Twisted K-theory, math.kt/0407054.
  3. A Borel and J-P Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958) 97-136.
  4. P Donavan and M Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math. IHES 38 (1970) 5-25.
  5. S Eilenberg and N E Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. USA 31 (1945), 117-120.
  6. D Freed, M J Hopkins and C Teleman, Twisted K-theory and loop groups representations I, math.AT/0312155.
  7. E Witten, Overview of K-theory Applied to Strings, hep-th/0007175.

30/06/2005, 15:30 — 16:30 — Sala P3.10, Pavilhão de Matemática
Marko Stosic, Instituto Superior Técnico

Categorification of the chromatic and dichromatic polynomial for graphs

Session of the reading and discussion seminar on Khovanov homology.

30/06/2005, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
Marco Mackaay, Universidade do Algarve

Rasmussen's s-invariant for links

Session of the reading and discussion seminar on Khovanov homology.

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

Instantons, Holomorphic Bundles and the Bar Construction

One version of the Kobayashi-Hitchin correspondence relates moduli spaces of instantons on blow ups of C2 trivialized at , with holomorphic bundles on blow ups of CP2 framed on a line. In the rank stable limit these moduli spaces can be described using a bar construction.

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

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.


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

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

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

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

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.
  1. Nathan Berkovits, ICTP Lectures on Covariant Quantization of the Superstring,
  2. Nathan Berkovits, Covariant Multiloop Superstring Amplitudes,
  3. Nathan Berkovits, Nikita Nikrasov, The Character of Pure Spinors,

11/03/2005, 16:30 — 17:30 — Anfiteatro Pa2, Pavilhão de Matemática
João Faria Martins, Instituto Superior Técnico

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.

11/03/2005, 15:00 — 16:00 — Anfiteatro Pa2, Pavilhão de Matemática
Marco Mackaay, Universidade do Algarve

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

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

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.
  1. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), no. 4, 303-368.
  2. L. C. Jeffrey, F. Kirwan, Localization for nonabelian group action, Topology 34 (1995) no. 2, 291-327.

21/05/2004, 11:00 — 12:00 — Sala P3.10, Pavilhão de Matemática
Marco Mackaay, Universidade do Algarve

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:
  1. Dror Bar-Natan, "On Khovanov's categorification of the Jones polynomial", Algebraic and Geometric Topology 2 (2002) 337-370; math.QA/0201043.
  2. Mikhail Khovanov, "A functor-valued invariant of tangles", Algebr. Geom. Topol. 2 (2002) 665-741; math.QA/0103190.
  3. Mikhail Khovanov, "A categorification of the Jones polynomial", math.QA/9908171.

21/05/2004, 10:00 — 11:00 — Sala P3.10, Pavilhão de Matemática
Pedro Vaz, Universidade do Algarve

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.

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

17/12/2003, 14:00 — 15:00 — Sala P3.10, Pavilhão de Matemática
J. Scott Carter, Univ. South Alabama

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.

06/11/2003, 17:00 — 18:00 — Sala P3.31, Pavilhão de Matemática
Pedro Lopes, Instituto Superior Técnico

Nós e os Quandles II

Apresentação dos resultados da tese de doutoramento.


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

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Organizadores correntes: Roger Picken, Marko Stošić.

Projecto FCT PTDC/MAT-GEO/3319/2014, Quantization and Kähler Geometry.