Termine

A p-Kähler structure on a complex manifold is a closed, transverse (p,p)-form. For the extremal values of p, we get the well known Kähler and balanced manifolds, however, for each other possible p, such structures have no metric meaning. The aim of the talk, after discussing similarities and differences between p-Kähler structures and Kähler and balanced structures, is to present some results on the existence of p-Kähler structures, with a focus on Lie groups and their compact quotients. Based on a work in progress with A. Fino and G. Grantcharov.

Markus Stroppel will give a lecture series on advanced topics in locally compact groups. The lectures take place on Monday at 14:15 in room SR 1D They will start on Monday, November 17. The planned topics are: * the scale function and tidy subgroups * automorphism groups of trees

Abstract: Chern characters are natural transformations from K-theory like coarse homology theories to coarse homology theories derived from chain-complex constructions. Transgressions are natural transformations from coarse homology theories to functors applied to the Higson corona. I will explain instances of such constructions and how they are possibly related. One goal of this talk is to highlight the richness of the world of coarse homology theories and its connection with interesting developments, classical and modern, in other fields.

Rapoport--Zink spaces associated with unitary groups play a central role in arithmetic geometry. In this talk, I will introduce two types of unitary Rapoport--Zink spaces: one arising from global Shimura varieties and another with a simpler local structure. I will then explain how these two spaces are related. This talk is based on my joint work with A. Mihatsch and Z. Zhang, as well as some forthcoming results of my own.

Topological k-spaces form a convenient category for constructions in algebraic topology. The category of topological k-groups is more elusive, but arises rather naturally from a categorical viewpoint. I will describe some elementary properties of these categories and give an example of a pro-Lie group whose k-fication is discrete. This is joint work with K.H. Hofmann.

We propose a generalization of K-theory to operator systems. Motivated by spectral truncations of noncommutative spaces described by C*-algebras and inspired by the realization of the K-theory of a C*-algebra as the Witt group of hermitian forms, we introduce new operator system invariants indexed by the corresponding matrix size. A direct system is constructed whose direct limit possesses a semigroup structure, and we define the K0-group as the corresponding Grothendieck group. This is an invariant of unital operator systems, and, more generally, an invariant up to Morita equivalence of operator systems. For C*-algebras it reduces to the usual definition. We illustrate our invariant by means of the spectral localizer.

In this talk, we investigate efficient and robust solution strategies for the incompressible Navier-Stokes equations, which are specifically designed to predict transient flow behaviors on modern hardware architectures. The main idea of the framework is to block the individual linear systems of equations at each time step into a single all-at-once saddle-point problem. To address the challenges associated with the zero block, we employ a global Schur complement technique, implicitly eliminating all velocity unknowns and leading to a problem for all pressure unknowns only. This so called pressure Schur complement (PSC) equation is then solved using a space-time multigrid algorithm with tailor-made preconditioners for smoothing purposes. However, as the Reynolds number increases, the accuracy of the PSC preconditioner deteriorates, leading to convergence issues in the overall multigrid solver. To overcome this disadvantage, we enhance the solution strategy by using the Augmented Lagrangian methodology in a global-in-time fashion. This approach guarantees a rapid convergence by drastically increasing the accuracy of the adapted PSC preconditioner by means of a strongly consistent modification of the velocity system matrix. For sufficiently large stabilization parameters, even the global-in-time Schur complement iteration by itself converges fast and makes the coarse grid acceleration of the multigrid algorithm redundant. The Augmented Lagrangian technique comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned for large stabilization parameters so that standard iterative solvers prove to be practically unusable. This calls for more sophisticated iterative solvers, like a specialized multigrid solver utilizing modified intergrid transfer operators and block diagonal preconditioners. In the limit of large stabilization parameters, the Augmented Lagrangian approach implies that all intermediate velocity solutions satisfy the discrete incompressibility condition. A straightforward consequence is therefore the use of discretely divergence-free finite elements, eliminating the pressure variable in an a priori manner. Therefore, no saddle-point structure exists and a wide range of efficient and accurate preconditioners becomes applicable. After illustrating the potential of this approach for two-dimensional global-in-time problems, we give some insight into the rarely investigated extension to three dimensions.