Organizer: Gil Cavalcanti
|14.00-15.00||Chris Wendl, Humboldt-Universität, Berlin
On symplectic manifolds with boundary, or "when is a Stein manifold merely symplectic?"
|15.00-16.00||Oscar Garcia Prada, CSIC and ICMAT, Madrid
Higgs bundles and higher Teichmüller components
Chris Wendl: On symplectic manifolds with boundary, or "when is a Stein manifold merely symplectic?"
Abstract: Symplectic manifolds are widely known as the natural geometric setting for Hamiltonian mechanics, but they also arise in many other contexts, e.g. in complex geometry, one sometimes considers Stein manifolds, which have natural symplectic structures containing contact hypersurfaces. The goal of this talk will be to explain why, for a large class of Stein manifolds in complex dimension two, complex geometry is largely irrelevant and symplectic topology is all that matters, i.e. two such manifolds have deformation-equivalent complex structures if and only if their symplectic structures are deformation equivalent. One can think of this as a kind of middle ground between the extremes of "rigidity" and "flexibility" that typically characterize important insights about symplectic manifolds. The proof is based on a topological characterization of Stein structures in terms of Lefschetz fibrations, together with an analytical result presenting the latter as foliations by pseudoholomorphic curves. Along the way, I'll explain how contact structures arise on boundaries of symplectic manifolds, and give some examples of interesting questions that contact topologists think about, and the methods they use to attack them. There will be lots of pictures, and some of them will move.
Oscar Garcia-Prada: Higgs bundles and higher Teichmüller components
Abstract: Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this talk, I will describe examples of such ‘exotic’ components in moduli spaces of G-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into a semisimple Lie group G. These exotic components are then related to higher Teichhmüller theory, and the notion of positive Anosov representations recently developed by Guichard-Wienhard.