1998 seminars

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.

References:
  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.

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.