Vorträge in der Woche 04.11.2019 bis 10.11.2019

Dienstag, 05.11.2019: Expanding graphs, Hausdorff-Banach-Tarski-paradox, Ruziewicz problem, invariant measures II

Julien Sessler

Dienstag, 05.11.2019: Stochastic Analysis of Nematic Liquid Crystals

Akash Ashirbad Panda (Montanuniversität Leoben)

In this talk, I will be discussing the results obtained for the stochastic evolution equation, which describes the system governing the nematic liquid crystals perturbed by pure jump noise in the Marcus canonical form. A briefing on the existence of a martingale solution in two and three dimensions will be presented. In addition, the pathwise uniqueness of the martingale solution in two dimensions will be presented, from which the existence of a strong solution will be deduced. The final part of the talk concerns the large deviation theory for the above-said model. I start with the stochastic two-dimensional nematic liquid crystal model influenced by multiplicative Gaussian noise. The Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions will be analyzed using the weak convergence method. Then using a similar technique, I will establish a large deviation principle for stochastic nematic liquid crystals driven by pure jump noise in the Marcus canonical form in two dimensions.

Freitag, 08.11.2019: Quotientendarstellungen von Mori Dream Spaces

Lukas Braun

Wir untersuchen Quotientendarstellungen von sogenannten Mori Dream Spaces: algebraischen Varietäten mit endlich erzeugtem Coxring. Im ersten Teil betrachten wir die vom Coxring induzierte natürliche Quotientendarstellung mittels einer reduktiven abelschen Gruppe. Wir betrachten eine Iteration der Coxring-Konstruktion und zeigen, dass diese Iteration für Kawamata log terminale Quasikegel und Varietäten vom Fano-Typ terminiert. Das impliziert eine Darstellung dieser Räume als GIT-Quotient eines faktoriellen Quasikegels mit kanonischer Singularität nach einer reduktiven auflösbaren Gruppe. Im zweiten Teil untersuchen wir, wie solche faktoriellen Quasikegel als Quotienten nach der speziellen linearen Gruppe SLn(C) dargestellt werden können. Ausgehend von klassischer Invariantentheorie entwickeln wir eine grafische Methode zur Berechnung solcher Quotienten. Wir vollenden darüberhinaus die Klassifikation der Darstellungen von SL n (C), deren Invariantenringe vollständige Durchschnitte sind.

Freitag, 08.11.2019: Surgery for extended Ricci flow systems

Florian Johne

We study extended Ricci flow systems in three dimensions. The extended system obtained by coupling Ricci flow to a function satisfying the heat equation is known as List flow in the literature. The motivation to study this system steams from its connection to static solutions to the vacuum Einstein equations and from its relation to four-dimensional Ricci flow with symmetry. Our first result concerns the classification of singularity models of the extended system: The singularity models have non-negative sectional curvature. The main ingredient in the proof is an improved bound on the Hessian along the flow. The second contribution is a convergence result: In constrast to the Ricci flow, a round sphere may become less round under the flow initially. We identify conditions such that the metric converges to a metric of constant positive sectional curvature and the function to a constant. Our conditions reduce to R. Hamilton‘s 1982 result in the limit of vanishing coupling. The third contribution is an adaptation of the surgery procedure due to R. Hamilton and G. Perelman to the extended Ricci flow system. An important point in the construction is to show that the a-priori bound on the energy density is preserved through the surgery procedure.