## November 2018

Abstract: The Biot–Savart operator maps the vorticity of an incompressible fluid into its associated velocity field. I will explain the construction of an analog operator for bounded domains. In addition, I will briefly present some results concerning Runge-type approximation theorems and their application to construct solutions of PDEs with some prescribed properties.

Abstract: We review an abstract and simple uncertainty principle from

‘A. Boutet de Monvel, D. Lenz and P. Stollmann: An uncertainty

principle, Wegner estimates and localization near fluctuation

boundaries. Math. Z. 2010′ and its application to graphs and

divergence form operators with rough coefficients based on

‘D. Lenz, P. Stollmann and Gunter Stolz: An uncertainty principle and

lower bounds for the Dirichlet Laplacian on graphs.

arXiv: 1606.07476′ and ‘P. Stollmann and Gunter Stolz: Lower bounds

for Dirichlet Laplacians and uncertainty principles, arXiv:1808.04202′

Prof. Bernd Kirstein – “Hans-Joachim Girlich – 62 Jahre am Mathematischen Institut der Universität Leipzig”

Recent empirical research has revealed striking regularities in growth that apply from algae to elephants and across levels of organization from embryos to ecosystems. Across these very distinct systems, growth scales with mass raised to a power near ¾, suggestive of a universal dynamical process that is not well understood. I will review the ubiquity of these patterns and highlight their surprising implications in ecological and evolutionary theory. I will also outline some directions we are exploring to understand this pattern, including a simple model that could offer insight into the basic processes generating these scaling laws.

The Hurwitz space H_(d,g) parametrizes d-sheeted simply branched covers of the projective line by smooth curves of genus g. In this talk, I will survey our knowledge on the unirationality of Hurwitz spaces and we will discuss some results in this direction.

We study conditional independence of sets of coordinates in a multivariate Gaussian distribution, with an interest in the different conditional independence models that are possible. In one natural case, corresponding to certain memoryless processes that generalise Markov chains, a complete answer can be obtained using the tools of Schubert varieties and determinantal ideals. This is joint work with Jenna Rajchgot and Seth Sullivant.

The condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is important for the stability analysis of algorithms that solve trigonometric mo- ment problems, e.g. Prony’s method. In the univariate case and with well-separated nodes, this condition number is well understood, but if nodes are nearly-colliding, the situation becomes more complicated. After recalling Prony’s method, results for the condition number of Vandermonde matrices with pairs of nearly-colliding nodes are presented. This is joint work with Stefan Kunis.

Geometric singular perturbation theory (GSPT) founded by Fenichel has been successfully used in many areas of mathematical biology, e.g. mathematical neuroscience and calcium signaling. However, slow-fast analysis and GSPT of mathematical models arising in cell biology is much less established. The main reason seems to be that the corresponding models typically do not have an obvious slow-fast structure of the standard form. Nevertheless, many of these models exhibit some form of hidden slow-fast dynamics, which can be utilized in the analysis.

In this talk I will explain some of the main concepts of GSPT in the context of a non-trivial application. I will present a geometric analysis of a novel type of relaxation oscillations involving two different switches in a model for the NF − $kappa$B signaling pathway.

The spectral dimension, a positive real number related to the probability a random walk on a network will eventually return to where it started, is often finite in geometric networks e.g. the k-nearest neighbour graph on uniformly random points on the torus, so the appearance of finite spectral dimension in a growing network model is often considered to be a “geometric” property. So is the appearance of a non-trivial distribution of node “curvatures” (related to the incidence of triangles/simplicial complexes at a point, see e.g. the recent work of J. Jost, as well as M. Gromov and O. Knill), as well as a non-trivial community structure, much higher clustering than that of random networks, and the famous six degrees of separation i.e. “small world” property. A further example introduced recently is the random topology of the network’s clique complex, where we build a topological space by face-including (gluing together at edges/faces) the many-body interactions e.g. triangles, 4-cliques etc in a network data set, then compare its homology to those of random geometric complexes like the Vietoris-Rips or the Čech complex, introduced by e.g. Linial, Meshulam, Farber, Bianconi and Kahle. We thus introduce and discuss recent progress made on determining to what extent these properties emerge in models of complex networks.