Kolloquium Wilhelm Killing

Get an insight into the research of seven new postdoctoral researchers of Mathematics Münster. In short scientific presentations they will introduce their topics.
After the talks, there will be the opportunity to exchange ideas while enjoying tea, coffee and cake in the Common Room.

Due to its catastrophic character, brittle fracture has been an important object of investigation in fracture mechanics. The idea of a fracture energy proportional to the measure of the fracture surface is due to Griffith. The revisitation of the subject by Francfort and Marigo offered a genuinely variational formulation of the problem.
For the mathematical community, the interest in fracture problems originated from its reformulation as a free discontinuity problem (FDP), following the approach by De Giorgi and Ambrosio to analyze the Mumford and Shah image segmentation functional.
After introducing the mathematical setting for FDPs, I will survey on several basic problems to allow the use of variational methods to establish existence of weak/strong minimizers and their phase-field approximation, both for Griffith type energies and for cohesive fracture models. The latter have been the subject of much attention recently and are based on FDPs with surface densities that are concave and bounded functions of the jump amplitude.

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The Anti-de-Sitter (AdS) space is the analogue in Lorentzian geometry to the hyperbolic space of negative constant curvature familiar to Riemannian geometers. The study of the Anti-de-Sitter space and more generally, of solutions to the Einstein equations which ressemble the Anti-de-Sitter space has many applications in high energy physics and provides mathematicians with problems lying at the intersection of Lorenzian geometry and hyperbolic pdes. I will start with a general introduction to the study of the Einstein equations, with an emphasis on the evolution problem. I will then briefly discuss the well-posedness for wave equations, including the Einstein equations, in the case of AdS type boundaries. In the second part of the talk, I will present several results concerning the linear or non-linear perturbations of the AdS space for the vacuum Einstein equations and some related toy models.

Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where individuals keep full memory of their forebearers, and procreation involves the reactivation of a gene picked at random on the ancestral lineage.
By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.

We introduce a variation of the classic ballistic deposition model in which vertically falling blocks can only stick to the top or the upper right corner of growing columns. We establish that the fluctuations of the height function at points near the time axis are given by the GUE Tracy-Widom limiting distribution, confirming that the strong KPZ universality conjecture is satisfied in this model. The proof is based on a graphical construction of the process in terms of a directed Last Passage Percolation model. This is a joint work with Pablo Groisman, Santiago Saglietti, and Sebastián Zaninovich.

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