Palestrante: Roberto Rubio (IMPA)
Título: Higher analogues of symplectic, Poisson and Dirac structures
Resumo: We will first review how symplectic and Poisson structures on a manifold M fit into Dirac structures (integrable Lagrangian subbundles of TM+T*M), and then introduce higher versions of symplectic and Poisson structures. Having a higher version of Dirac structures is more troublesome than expected. Indeed, whereas the graph of a Poisson structure is a Dirac structure, and hence Lagrangian, the graph of a higher-Poisson structure, lying on TM plus the k-th exterior power of T*M, is not necessarily Lagrangian. To fix this, we introduce the notion of a weak-Lagrangian subspace and define higher-Dirac structures as integrable weak-Lagrangian subbundles, so that higher-Poisson structures do correspond to higher-Dirac structures transversal to TM. Higher-Dirac structures are, in particular, Lie algebroids, so their projection to TM is an involutive distribution and hence induces a (singular) foliation on M. A precise linear algebraic description of weak-Lagrangian subspaces allows us to describe the geometry appearing on the induced foliation, which is a non-trivial higher version of the presymplectic foliation of a Dirac structure. This is joint work with H. Bursztyn and N. Martínez-Alba.
Palestrante: Lázaro Rodríguez (Unicamp)
Título: G_2 holonomy manifolds are superconformal.
Resumo: Given a manifold with G_2 holonomy we prove the space of global sections of the chiral de Rham complex contains two commuting copies of the Shatashvili-Vafa superconformal algebra. This algebra appears as the chiral algebra associated to the sigma model with target a G_2 holonomy manifold. I will discuss how the structure and representation theory of this vertex algebra can be used to unravel the geometry and topology of the underlying manifold.
Palestrante: Olivier Glorieux (IMPA)
Título: Critical exponent for globally hyperbolic, anti-de Sitter
Resumo: Critical exponent is a dynamical invariant measuring the exponential growth rate of number of closed geodesics. It has been extensively studied for hyperbolic manifolds. Anti-de Sitter manifolds are Lorentzian manifolds of constant curvature $-1$, it is the Lorentzian counterpart of the hyperbolic space. A subclass of Lorentzian manifolds, called globally hyperbolic, have nice properties making them look like quasi-Fuchsian hyperbolic manifolds. For globally hyperbolic, anti-de Sitter manifolds, we will explain how to define a notion of critical exponent and how it is related, as in the hyperbolic case, to the Hausdorff dimension of the limit set. We will not suppose any backgrounds
on Lorentzian geometry and recall all the basic definitions. Our aim is to explain similarities and differences between Anti-de Sitter geometry and hyperbolic geometry, from a dynamical point of view.