Talks

Bei der numerischen Simulation physikalischer Prozesse treten häufig große parameterabhängige nichtlineare Gleichungssysteme auf. Zur Verringerung des Rechenaufwands werden oft Reduzierte-Basis-Methoden verwendet, die sich in lokale und globale Methoden unterscheiden lassen, wobei letztere Umkehrpunkte bezüglich des Parameters gewöhnlich nicht zulassen. In dieser Arbeit wird ein globaler, interpolationsbasierter Ansatz für Probleme mit Umkehrpunkten entwickelt und es werden die Vorteile und Grenzen dieser Methode aufgezeigt.

Viele Variationsmethoden in der mathematischen Bildverarbeitung nutzen die 1-Norm des Gradienten, die sogenannte Totalvariation, als Regularisierungsterm. Diese Totalvariation hat die Eigenschaft, Kanten im Bild zuzulassen und zu erhalten. Sie kann aber auch zur Entstehung von unerwünschten Kanten beitragen, dem sogenannten Staircasing-Effekt. Diese Arbeit soll die Huber-Funktion, eine Kombination zweier Normen, als mögliche Alternative vorstellen.

Bachelor-Vortrag

Bachelorvortrag

A linear operator $T$ on a Banach space is called Tadmor-Ritt if its spectrum is contained in the closed unit disc and the resolvent satisfies $C(T)=\sup_{|z|>1} \|(z-1)R(z,T)\|<\infty$. Such operators can be seen as discrete analog for sectorial operators.
We prove corresponding $H^{\infty}$-functional calculus estimates, which generalize and improve results by Vitse. Moreover, they are in conformity with the best so-far known power-bound for Tadmor-Ritt operators in terms of the constant $C(T)$.
We furthermore show the effect of having discrete square function estimates on the derived estimates.

We say a graph $G$ is universal for a set of graphs $\mathcal{H}$ if for each $H\in\mathcal{H}$ we have $H\subset G$. There are several results stating that the random graph $G(n,p)$ is universal for various classes of graphs $\mathcal{H}$, for appropriate functions $p=p(n)$. In order for $p$ not to be very close to one, we need the graphs in $\mathcal{H}$ to be quite sparse. There are then (at least) three natural graph classes one could consider: trees, graphs with bounded degree, and graphs with bounded degeneracy. I will outline the current state of knowledge (mainly due to other people) and sketch one or two proofs

Bachelor-Vortrag

After the introduction of random matrices to nuclear physics by Eugene Wigner in 1955, random quantum systems have grown in popularity. Wigner's idea was to consider families of Hamiltonians that underlie a certain probability distribution to describe overly complicated systems. Of particular interest are, of course, the spectra of these Hamiltonians. In this talk we consider random, in general non-self-adjoint, tridiagonal operators on the Hilbert space of square-summable sequences. To model randomness, we use an approach by Davies that eliminates all probabilistic arguments.

Despite the rising interest, not much is known about the spectra of non-self-adjoint random operators. The Feinberg-Zee random hopping matrix reveals this in a beautiful manner. The boundary of its spectrum appears to be fractal, but a proof has not been found yet. While we can not give a proof either, we present a reason why this is very plausible. Certain tridiagonal operators share remarkable symmetries that allow us to enlarge known subsets of the spectrum by sizeable amounts. In some cases like the Feinberg-Zee random hopping matrix, this implies that the spectrum contains an infinite sequence of Julia sets.