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.
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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: 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.