Abstract: I will report on new joint work with Franc Forstnerič about introducing ideas and methods from Oka theory into complex contact geometry. This is a new research area in which only a few papers have been published so far (by Alarcón, Forstnerič, López, and myself). More specifically, we study holomorphic Legendrian curves in the projectivised cotangent bundle X=PT*Z of a complex manifold Z of dimension at least 2. Such a manifold X carries a natural complex contact structure. We prove several approximation and general position theorems, as well as some h-principles, for holomorphic Legendrian curves in X. I will start with a very brief introduction to complex contact geometry.
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Abstract: It is a well-known result that the only spheres which can admit almost complex structures occur in dimensions two and six. We show that if one instead considers rational homology spheres (manifolds with the same rational homology as a sphere), then the same conclusion holds: they can only admit almost complex structures in dimensions two and six. However, there are infinitely many six-dimensional rational homology spheres and not all of them admit almost complex structures. We will explain how to construct infinitely many examples which do admit almost complex structures and infinitely many which do not. This is joint work with Aleksandar Milivojevic.
Abstract: I will explain the role of stacks and descent in the theory of bundle gerbes, and talk about various examples and features. I will then describe a new application of non-abelian gerbes to T-duality, in which their stacky nature is of crucial importance.
Abstract: In 2004 Fornaess gave the the first example of a “short C^2”: a domain of C^2 which is an increasing union of (biholomorphic images of) balls and whose Kobayashi metric vanishes, but which is not biholomorphic to C^2 since it admits a nonconstant bounded plurisubharmonic function. Such a domain appears as the basin of attraction at the origin of a nonautonomous sequence of holomorphic automorphisms of C^2. In this talk I will explain how to construct a short C^2 as a Fatou component of a single holomorphic automorphism (a transcendental Hénon map). By the classical Rosay-Rudin result it is clear that such a Fatou component cannot be the basin of attraction of a point, and indeed it follows from the construction that the Fatou component is wandering and oscillating. This is a joint work with Luka Boc Thaler and Han Peters.
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Abstract: In this somewhat coals-to-Newcastle exercise I will describe joint work with Chris Kottke in which we introduce the notion of a bigerbe. Michael Murray's bundle gerbes are smoother versions of the Čech objects due to Brylinski and Hitchin, all giving realizations of integral 3-classes in cohomology. Our bigerbes are special 2-gerbes in the sense of Stevenson, so realizing integral 4-classes, but more constrained and are direct generalizations of bundle gerbes to a bisimplicial setting. Motivating examples include a `Brylinski-McLauglin' bigerbe arising from the transgression of a principal G-bundle.
Abstract: Periodic Floer Homology(PFH) is a Gromov type invariant for fibered 3-manifold, which is defined by counting of J-holomorphic curves. Given a symplectic 4-manifold whose boundaries are fibered 3-manifolds, it is expected that this 4-manifold induces a homomorphism (cobordism map) between the periodic Floer homologies of its boundaries. Due to certain technical difficulties, the cobordism map haven't been defined completely yet. Motivated by defining the cobordism map, I will discuss a concrete example that the 4-manifold is a elementary Lefschetz fibration. I will start with a brief introduction of PFH and ECH index.