Vorträge

Axons are micrometer-thin cables that transmit signals across the brain. Their size affects how fast signals travel, making axon diameter a key determinant of brain function -- and, when altered, a potential marker of disease. In theory, MRI is sensitive to axon size through the physics of water diffusion, but this sensitivity has remained unproven in real-world settings for decades. In this talk, I'll present recent advances in validating MRI-based axon radius estimates using experimental MRI and high-resolution microscopy of more than 46 million axons across the human brain.

Zoomlink:
https://tuhh.zoom.us/j/87285771127?pwd=bjlWT3AyQncwajZQN0l3dVd1WXJmZz09

In this talk, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are L^\infty for non-negative measure spaces. This shows once again that $L^\infty$-spaces have to be treated differently.

Atmospheric motion covers a broad range of temporal and spatial scales. Low- and high-pressure systems can influence us for days or even weeks, and they extend up to hundreds of kilometers. In contrast, sound waves pass by in seconds with wavelengths of centimetres to meters. Implicit-explicit (IMEX) time stepping methods can help avoid drastic limitations on the time step induced by this variety of scales without requiring computationally expensive fully nonlinear implicit solves.
We discuss (IMEX-) Spectral Deferred Correction (SDC) methods in the context of Runge-Kutta methods (RKM) and apply SDC to test cases, which are relevant to numerical weather prediction. First, we use RKM theory to:
1. Construct new SDC methods that increase convergence order by 2 per iteration, in contrast to the increase of 1 in general SDC methods;
2. Construct SDC methods that conserve quadratic invariants;
3. Show that SDC can be symmetric but not symplectic for finite iterations.
Second, we apply implicit-explicit (IMEX) SDC to fluid dynamical problems that are relevant to numerical weather prediction. In particular, we compare IMEX-SDC, multistep and RKM time integrators for the Galewsky test case using the Python spectral method framework Dedalus. We demonstrate that SDC methods have superior stability properties and can provide a shorter time to solution for comparable errors. In addition, we outline strategies that could further reduce simulation times by using the SDC residual to minimise the computational effort. Finally, we apply SDC to the compressible Euler equations using compatible finite element methods, demonstrating their applicability to more complex atmospheric models.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

In this talk I will show how transfer matrix techniques can be used to compute the asymptotic spectra of non-Hermitian tridiagonal finite-block Toeplitz matrices. In this way one can recover Widoms results on the asymptotic spectra of such matrices. Special attention will be given to topological eigenvalues arising from matrices with a chiral symmetry and the associated bulk-boundary correspondence. Going beyond Widoms theory, I will also show how the transfer matrix approach can be used to compute the asymptotic spectra of Toeplitz matrices with a perturbation on a finite number of sites. One can then see how the different parts of the spectra depend on the perturbations. Varying the ranks of the perturbations one can now also interpolate between open and closed boundary conditions. The results will be illustrated by numerics.

This is joint work with Hermann Schulz-Baldes.

We explore examples of Dirac operators on bounded domains exhibiting an interval of essential spectrum. In particular, we consider three-dimensional Dirac operators on Lipschitz domains with critical electrostatic and Lorentz scalar shell interactions supported on a compact smooth surface. Unlike typical bounded-domain settings where the spectrum is purely discrete, we show that the criticality of these interactions can generate a nontrivial essential spectrum interval, whose position and length are explicitly controlled by the coupling constants and surface curvatures.

Based on joint work with J. Behrndt (TU Graz), M. Holzmann (TU Graz), and K. Pankrashkin (Univ. Oldenburg).