A symplectic generalization of 3-dimensional taut foliations
Abstract: A taut foliation on a closed 3-manifold is a foliation by surfaces which has a transverse cycle meeting every leaf. Equivalently, it has a closed 2-form which induces an area form on each leaf.
In arbitrary dimensions, and from a symplectic view point, it makes sense to look at codimension one foliations with a closed 2-form making each leaf symplectic. We shall recall how Donaldson approximately holomorphic theory for symplectic manifolds can be adapted to this class of foliations, to produce "Donalson divisors'' which generalize the transverse cycles in the 3-dimensional case. Finally, we shall discuss a foliated version of the Lefschetz hyperplane theorem which implies that Donaldson divisors capture entirely the transverse geometry of the ambient foliation.
The Kaehler geometry of the Weinstein construction
Abstract: We discuss the Weinstein construction of symplectic bundles in the framework of Kaehler manifolds. In particular, we give examples of csc Kaehler metrics which are not Einstein-Kaehler. We finally use the Weinstein construction to give a local characterization of Kaehler manifolds admitting holomorphic, totally geodesic and homothetic foliations. This a joint work with Paul-Andi Nagy.
On the classification of Bochner-Kahler metrics
Abstract: A Bochner-Kahler metric is a Kahler metric whose Bochner component of the curvature tensor vanishes. In this talk I will explain how Bryant's classification of such metrics can be simplified and improved using Lie algebroids and Lie groupoids, leading to a better understanding of the moduli space of Bochner-Kahler metrics. This is based on joint work with Ivan Struchiner (USP).
The contribution of Jean François Le Gall to Brownian Geometry
Abstract: J.F. Le Gall has been awarded the Wolff prize in 2019 "for his profound and elegant works on stochastic processes". In this talk we wish to introduce to a large audience to Le Gall's contribution to the subject of Random Geometry (his main object of focus in the last 15 years). Our starting point is the following question:
"Is there a good notion of random sphere ?"
or more precisely:
"Is there a natural way to choose at random a manifold among all those that are homeomorphic to the sphere?"
In order to give a more precise meaning to the question and to explain the elegant answer brought to the above question by Le Gall and Miermont, we will make a detour to the world of discrete random geometry and the notion Quadragulations of the Sphere, and explore the path that lead to the construction of the Brownian Sphere via Brownian Motion, Brownian Tree, and Brownian Snake.