Termine

Abstract: The study of descent properties of K-theory plays an important role in understanding its values and behaviour in many geometric contexts. In this talk, I will explain a construction of certain diagrams arising from formal gluing situations for which continuous K-theory, and more generally all localising invariants, satisfy descent, from the perspective of dualisable categories. I will also discuss how this encompasses both Clausen?Scholze?s gluing result for analytic adic spaces and an adelic descent result for dualisable categories.

Abstract: In 2006, Arezzo and Pacard proved the existence of cscK metrics on the blow-up at finitely many points of a cscK manifold admitting no non-trivial holomorphic vector fields. In this talk, I will describe the gluing strategy we used to construct balanced constant Chern scalar curvature metrics on the blow-up of a compact balanced Chern-Ricci flat manifold, generalizing Arezzo-Pacard's result to this case. Finally, I will explain how the same gluing strategy can be adapted to construct Chern-Ricci flat balanced metrics on crepant resolutions of Chern-Ricci flat balanced orbifolds. This is a joint work with Federico Giusti.

We consider the dynamics of several vortex rings in an ideal incompressible fluid governed by the 3D Euler equations. While the self-induced motion of a single vortex ring amounts to a translation along its symmetry axis, Helmholtz conjectured in 1858 that coaxial vortex rings may exhibit the so-called leapfrogging dynamics. Specifically, the vortex rings display a periodic motion in which they repeatedly pass through each other by shrinking and widening respectively due to mutual interactions. The phenomenon is well observed numerically and experimentally. We rigorously justify these dynamics on a logarithmic time scale for a general class of initial data corresponding to N sharply concentrated coaxial vortex rings. In order for the leapfrogging to occur, the vortex rings need to be placed at a very small mutual distance leading to singular interactions. The key mathematical difficulty then consists in proving that the sharp concentration of vorticity persists for positive times. We provide new and improved estimates on the localization of the vorticity in strong and weak sense. In particular, this allows us to derive the respective asymptotic motion law. Joint work with M. Donati (Grenoble), C. Lacave (Chambery) and E. Miot (Grenoble).

The diagonal dimension of a C*-subalgebra was introduced by Li, Liao, and Winter, and is a version of nuclear dimension for diagonal C*-subalgebras. In the case of crossed product C*-algebras, it extracts information on the underlying dynamical systems. Since diagonal dimension restricts the study to nuclear algebras, we propose a generalisation of the dimension that also treats non-nuclear algebras. Our motivation comes from Roe algebras, which are C*-algebras that bridge the study of coarse geometry and operator algebras. The generalised diagonal dimension can be used to recover the asymptotic dimension of a metric space from its (non-uniform) Roe algebra.

The talk will start with an introduction to curvature and a discussion of some related problems. We will then turn to the Dirac equation which was introduced in the context of relativistic quantum physics. We will see how curvature and other geometric quantities influence the eigenvalues of the Dirac operator. This insight will be used to draw some geometric conclusions which will allow us to answer the questions from the beginning of the lecture.