## Institut für Mathematik und Informatik

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Abstract
Cellular stress has been associated with inflammation, yet precise underlying mechanisms remain elusive. In this study, various unrelated stress inducers were employed to screen for sensors linking altered cellular homeostasis and inflammation. We identified the intracellular pattern recognition receptors NOD1/2, which sense bacterial peptidoglycans, as general stress sensors detecting perturbations of cellular homeostasis. NOD1/2 activation upon such perturbations required generation of the endogenous metabolite sphingosine‐1‐phosphate (S1P). Unlike peptidoglycan sensing via the leucine‐rich repeats domain, cytosolic S1P directly bound to the nucleotide binding domains of NOD1/2, triggering NF‐κB activation and inflammatory responses. In sum, we unveiled a hitherto unknown role of NOD1/2 in surveillance of cellular homeostasis through sensing of the cytosolic metabolite S1P. We propose S1P, an endogenous metabolite, as a novel NOD1/2 activator and NOD1/2 as molecular hubs integrating bacterial and metabolic cues.

Phytopathogenic Verticillia cause Verticillium wilt on numerous economically important crops. Plant infection begins at the roots, where the fungus is confronted with rhizosphere inhabiting bacteria. The effects of different fluorescent pseudomonads, including some known biocontrol agents of other plant pathogens, on fungal growth of the haploid Verticillium dahliae and/or the amphidiploid Verticillium longisporum were compared on pectin-rich medium, in microfluidic interaction channels, allowing visualization of single hyphae, or on Arabidopsis thaliana roots. We found that the potential for formation of bacterial lipopeptide syringomycin resulted in stronger growth reduction effects on saprophytic Aspergillus nidulans compared to Verticillium spp. A more detailed analyses on bacterial-fungal co-cultivation in narrow interaction channels of microfluidic devices revealed that the strongest inhibitory potential was found for Pseudomonas protegens CHA0, with its inhibitory potential depending on the presence of the GacS/GacA system controlling several bacterial metabolites. Hyphal tip polarity was altered when V. longisporum was confronted with pseudomonads in narrow interaction channels, resulting in a curly morphology instead of straight hyphal tip growth. These results support the hypothesis that the fungus attempts to evade the bacterial confrontation. Alterations due to co-cultivation with bacteria could not only be observed in fungal morphology but also in fungal transcriptome. P. protegens CHA0 alters transcriptional profiles of V. longisporum during 2 h liquid media co-cultivation in pectin-rich medium. Genes required for degradation of and growth on the carbon source pectin were down-regulated, whereas transcripts involved in redox processes were up-regulated. Thus, the secondary metabolite mediated effect of Pseudomonas isolates on Verticillium species results in a complex transcriptional response, leading to decreased growth with precautions for self-protection combined with the initiation of a change in fungal growth direction. This interplay of bacterial effects on the pathogen can be beneficial to protect plants from infection, as shown with A. thaliana root experiments. Treatment of the roots with bacteria prior to infection with V. dahliae resulted in a significant reduction of fungal root colonization. Taken together we demonstrate how pseudomonads interfere with the growth of Verticillium spp. and show that these bacteria could serve in plant protection.

Diese Dissertation untersucht Zusammenhänge der spieltheoretischen Begriffe des Nash- und Stackelberg-Gleichgewichts in Differenialspielen im N-Spieler-Fall. Weiterhin werden drei verschiedene Lösungskonzepte für das Finden von Gleichgewichten in 2-Spieler-Differentialspielen vorgestellt. Direkte Methoden aus der nichtlinearen Optimierung, der globalen Optimierung und der optimalen Steuerung werden verwendet, um Nash- und Stackelberg-Gleichgewichte in 2-Spieler-Differentialspielen zu finden. Anhand von Anwendungsbeispielen werden die Methoden getestet, analysiert und ausgewertet. Eine Erweiterung des Verfolgungsspiels von Isaacs auf Beschleunigungskomponenten wird betrachtet. Ein bisher unbekanntes Stackelberg-Gleichgewicht wird im Kapitalismusspiel nach Lancaster numerisch berechnet. Zuletzt wird ein Problem aus der Fischerei modelliert und anhand der eingeführten Methoden gelöst.

Abstract
The expected signature is an analogue of the Laplace transform for probability measures on rough paths. A key question in the area has been to identify a general condition to ensure that the expected signature uniquely determines the measures. A sufficient condition has recently been given by Chevyrev and Lyons and requires a strong upper bound on the expected signature. While the upper bound was verified for many well‐known processes up to a deterministic time, it was not known whether the required bound holds for random time. In fact, even the simplest case of Brownian motion up to the exit time of a planar disc was open. For this particular case we answer this question using a suitable hyperbolic projection of the expected signature. The projection satisfies a three‐dimensional system of linear PDEs, which (surprisingly) can be solved explicitly, and which allows us to show that the upper bound on the expected signature is not satisfied.

Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic
(2020)

Abstract
With the advent of molecular genetic methods, an increasing number of morphologically cryptic taxa has been discovered. The majority of them, however, remains formally undescribed and without a proper name although their importance in ecology and evolution is increasingly being acknowledged. Despite suggestions to complement traditional descriptions with genetic characters, the taxonomic community appears to be reluctant to adopt this proposition. As an incentive, we introduce QUIDDICH, a tool for the QUick IDentification of DIgnostic CHaracters, which automatically scans a DNA or amino acid alignment for those columns that allow to distinguish taxa and classifies them into four different types of diagnostic characters. QUIDDICH is a system‐independent, fast and user‐friendly tool that requires few manual steps and provides a comprehensive output, which can be included in formal taxonomic descriptions. Thus, cryptic taxa do not have to remain in taxonomic crypsis and, bearing a proper name, can readily be included in biodiversity assessments and ecological and evolutionary analyses. QUIDDICH can be obtained from the comprehensive R archive network (CRAN, https://cran.r-project.org/package=quiddich).

Twisted topological K-theory is a twisted version of topological K-theory in the sense of twisted generalized cohomology theories. It was pioneered by Donavan and Karoubi in 1970 where they used bundles of central simple graded algebras to model twists of K-theory. By the end of the last century physicists realised that D-brane charges in the field of string theory may be studied in terms of twisted K-theory. This rekindled interest in the topic lead to a wave of new models for the twists and new ways to realize the respective twisted K-theory groups. The state-of-the-art models today use bundles of projective unitary operators on separable Hilbert spaces as twists and K-groups are modeled by homotopy classes of sections of certain bundles of Fredholm operators. From a physics perspective these treatments are not optimal yet: they are intrinsically infinite-dimensional and these models do not immediately allow the inclusion of differential data like forms and connections.
In this thesis we introduce the 2-stack of k-algebra gerbes. Objects, 1-morphisms and 2-morphisms consist of finite-dimensional geometric data simultaneously generalizing bundle gerbes and bundles of central simple graded k-algebras for k either the field of real numbers or the field of complex numbers. We construct an explicit isomorphism from equivalence classes of k-algebra gerbes over a space X to the full set of twists of real K-theory and complex K-theory respectively. Further, we model relative twisted K-groups for compact spaces X and closed subspaces Y twisted by algebra gerbes. These groups are modeled directly in terms of 1-morphisms and 2-morphisms of algebra gerbes over X. We exhibit a relation to the K-groups introduced by Donavan and Karoubi and we translate their fundamental isomorphism -- an isomorphism relating K-groups over Thom spaces with K-groups twisted by Clifford algebra bundles -- to the new setting. With the help of this fundamental isomorphism we construct an explicit Thom isomorphism and explicit pushforward homomorphisms for smooth maps between compact manifolds, without requiring these maps to be K-oriented. Further -- in order to treat K-groups for non-torsion twists -- we implement a geometric cocycle model, inspired by a related geometric cycle model developed by Baum and Douglas for K-homology in 1982, and construct an assembly map for this model.