Oporto Meeting on Geometry, Topology and Physics   Oporto Meeting on Geometry, Topology and PhysicsOporto Meeting on Geometry, Topology and PhysicsOporto Meeting on Geometry, Topology and Physics
FCUP - Isntituto Superior Técnico Faculdade de Ciências da Universidade do Porto Instituto Superior Técnico  
XVIIIth Oporto Meeting on
Geometry, Topology and Physics
    9th to 12th July 2009
 

Talks 1:
  • Representations up to homotopy and cohomology of classifying spaces (Camilo  Arias Abad, University of Zurich).
Abstract: 1.We will show how representations up to homotopy appear in the problem of computing cohomology of classifying spaces of Lie groupoids. Specifically, we will show how to generalize a formula of Bott [6] that relates the cohomology of BG to the representations on polynomials in the Lie algebra. I will aslo explain how this formula relates to the models of Getzler and Cartan for equivariant cohomology.
This is based on joint works with Marius Crainic and Benoit Dherin.
  • Geometric quantization of families of degenerating complex structures (Thomas Baier, Instituto Superior Tecnico, Lisboa).
Abstract: Using a weak formulation (on distributions) of the equations of covariant constancy as a unified setting for Geometric Quantization in both positive (i.e. complex) and non-negative (real or mixed) polarizations, we study some aspects of the behaviour of the associated quantum bundle as the polarization degenerates for two families of examples, toric manifolds (based on the pre-print arXiv:0806.0606) and abelian varieties.
  • Yang-Mills Morse theory over a nonorientable surface (Tom Baird, University of Oxford).
Abstract: Atiyah and Bott studied the Morse theoretic properties of the Yang-Mills functional, L: A -> R, defined of the space A of connections over a Riemann surface S. They found that the functional is (gauge equivariantly) perfect, meaning that the Morse inequalities are in fact equalities. This enabled them to compute Betti numbers for the minimizing set for L, which can be identified with the moduli stack of semistable holomorphic bundles over the surface S.
More recently, various authors have considered the situation when S is nonorientable. Surprisingly, it was found that at least for bundles of rank 2 and 3, the Yang-Mills functional is "antiperfect" in the sense that the Morse inequalities are as unequal as possible. This opposite extreme also allows a computation of the Betti numbers of the minimizing set for L. We review these results and discuss some implications.
  • Symplectic Origami (Ana Rita Pires, MIT, USA).
Abstract: An origami manifold is a manifold equipped with a closed 2-form which is symplectic everywhere except on a hypersurface, where it is a folded form whose kernel defines a circle fibration. In this talk, I will explain how an origami manifold can be unfolded into a collection of symplectic pieces and conversely, how a collection of symplectic pieces can be folded (modulo compatibility conditions), into an origami manifold. Using equivariant versions of these operations, we will see how the classic symplectic results of convexity and classification of toric manifolds translate to the origami world. There will be pictures resembling paper origami, but no instructions on how to fold a paper crane. This is joint work with Victor Guillemin and Ana Cannas da Silva.