Abstracts Hauptvorträge

In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, basic examples being the pinch-off of liquid droplets or the shrinkage and disappearance of phases in mean curvature flow. As a consequence, classical solution concepts for such PDEs are naturally limited to short-time existence results or particular initial configurations like perturbations of a steady state. At the same time, the transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions. In particular, for interface evolution equations not subject to a comparison principle - like multiphase mean curvature flow or fluid-fluid interface evolution problems -, the relation between weak and strong solutions has remained unclear.
By introducing a novel concept of "gradient flow calibrations", we establish a weak-strong uniqueness principle for multiphase mean curvature flow: Weak (BV) solutions to multiphase mean curvature flow are unique as long as a classical solution exists. In particular, in planar multiphase mean curvature flow, weak (BV) solutions are unique prior to the first topological change. As basic counterexamples show, the uniqueness of evolutions may fail past certain topology changes, demonstrating the optimality of our result. We establish an analogous weak-strong uniqueness principle for the Navier-Stokes equation for two fluids separated by a sharp free interface.
In the last part of the talk, we discuss further applications of our new concept, including the quantitative convergence of diffuse-interface (Allen-Cahn) approximations for mean curvature flow.

Exclusion processes are interacting particle systems which generalize the basic model of simple random walk. They can model traffic flows or molecules in a low-density gas.
Exclusion processes have been investigated intensively in analysis, statistical mechancis as well as in combinatorics.
We first explain some of the classical questions about such processes: invariant measures, the speed of a tagged particle, the current and its fluctuations. We then turn to more recent results about the convergence of a finite system to its invariant distribution, introducing mixing times and the cutoff phenomenon.
The question about cutoff is of independent interest and may be asked for many (sequences of) Markov chains. We present some recent results for a finite system with open boundaries.
In the end, we mention open questions about the current and about second class particles.
The talk is based on joint work(s) with Nicos Georgiou, Evita Nestoridi and Dominik Schmid.
Introduction Video

Der Vortragstitel erscheint ein wenig provokativ. Eine allgemein akzeptierte Definition von Mathematik gibt es nicht. Was soll dann moderne Mathematik sein?
Das Ziel des Vortrags ist eine Beschreibung von Entwicklungen in der Mathematik. Der Vortrag richtet seinen Fokus auf die letzten fünfzig Jahre und basiert auf Erfahrungen des Vortragenden als Forscher, Hochschullehrer und Wissenschaftsadministrator. Viele Beispiele dienen als Zeugnis signifikanter Veränderungen.
Die klassische Mathematik, die durch fachinterne Fragestellungen getrieben das eigene Strukturgebäude immer wieder erneuert und weiter ausbaut, ist höchst aktuell und sehr erfolgreich. Die Verfügbarkeit von leistungsfähigen Rechnern und vielfältige Einflüsse aus der Informatik haben traditionelle mathematische Fächer z. T. neu ausgerichtet. Die unsinnige Aufspaltung der Mathematik in reine und angewandte Fachgebiete ist im Verschwinden begriffen. Mathematisch relevante Fragestellungen aus Industrie, Gesellschaft und anderen Wissenschaftsdisziplinen haben neue mathematische Forschungsthemen generiert. Das vertrauensvolle Zusammenwirken der Mathematik mit anderen Fachdisziplinen hat sich für alle Seiten als äußerst fruchtbar erwiesen. Neue, von politischer Seite aufgelegte, hochdotierte Förderformate haben die interdisziplinäre Zusammenarbeit in den Mittelpunkt gestellt. Bei der Einwerbung derartiger Drittmittel war die Mathematik sehr erfolgreich.
Fazit: Die heutige Mathematik ist eine lebendige Wissenschaft, die ein unverzichtbarer und zentraler Knoten im Netz der Wissenschaften und vieler Anwendungfelder ist. Sie ist in dem Sinne modern, dass sie ernsthaft und nachhaltig „vielfältige Herausforderungen der Welt“ aufnimmt, zu deren Verständnis durch mathematische Modellierung und konkrete Lösungsvorschläge beiträgt. Hinzu kommt, dass sich das öffentliche Bild der Mathematik – durch erfolgreiche Medienarbeit – sehr positiv gewandelt hat.

Die Königin der Wissenschaften und das Königliche Spiel: Sie sind Jahrtausende alt und haben weltumspannenden Einzug in alle Kulturen gehalten. Der Vortrag befasst sich mit den Ähnlichkeiten und wechselseitigen Beziehungen zwischen diesen beiden Kulturgütern. Viele Mathematiker:innen haben sich durch das Schachspiel zu tiefen Problemen inspirieren lassen. Und nicht wenige Schachspieler:innen begeistern sich auch für mathematische Probleme.
Es gibt schachbezogene Fragestellungen, die mithilfe mathematischer Methoden lösbar sind. Umgekehrt gibt es auch mathematische Fragestellungen, die mit schachlichen Accessoires – also mit Schachbrett und Figuren – gelöst werden können. Besonders das Letztere scheint überraschend. Elegante Beispiele für beide Arten von Problemen sollen besprochen werden. Allgemeinverständlichkeit wird angestrebt.

Die Theorie „minimaler Modelle“ hat sich in den letzten Jahrzehnten zu einem Kernthema der algebraischen Geometrie entwickelt. Leider steht das Gebiet nicht ganz zu unrecht im Ruf, recht technisch zu sein und es dem Außenstehenden nicht leicht zu machen.
Der Vortrag erläutert in nicht-technischer Weise die Ziele und die Entwicklung des Programms minimaler Modelle und gibt einen kleinen Einblick in aktuelle Ergebnisse und Forschungsprobleme.

The interplay of the geometric and the spectral properties of Riemannian manifolds is highly influencial in essentially all areas of mathematics, but far from being fully understood. Some of the recent advancements in understanding this relation could be achieved by means of transfer operators. I shall overview some recent developments in this area with a focus on hyperbolic surfaces, automorphic functions, resonances and the dynamics of the geodesic flow and with an emphasis on insights and heuristics.

The Schatten p-classes are among the most fundamental operator ideals studied in functional analysis. These matrix spaces are non-commutative versions of classical $\ell_p$ sequence spaces with which they share several structural charecteristics. However, while $\ell_p$ spaces are quite well understood from an analytic, geometric and probabilistic point of view, this often cannot be said about the Schatten p-classes. In this talk, we shall present and discuss some recent results concerning their asymptotic structure.

I will motivate why it is useful to look at random submatrices where each atom is not drawn with the same probability but some atoms are more likely. I will then provide conditions under which such submatrices are well-conditioned with high probability, sketch the proof and show the crux of the proof in more detail. Finally I will give an example application of the results and show how they can be used to decide what a good sensing matrix for compressed sensing is and how we can precondition an pre-determined one.
Joint work with Simon Ruetz.

Practitioners are often reluctant applying mathematical results. Many reasons for this exist. One is that traditional methods are "stable" and practitioners know how to adapt them to changing requirements. And they are right: Most real-world problems contain parameters which are not known at the time a decision is to be made. Data may not be measurable in the precision needed or may depend on future developments. An optimal solution which does not take such an uncertainty into account often becomes bad or even infeasible for the scenario which is finally realized.
In robust optimization one specifies the uncertainty as a scenario set and tries to find a solution which is good enough, no matter which scenario occurs. Classical robust optimization aims at finding a solution which minimizes the costs in the worst case. It is a well-studied concept, but it is known to be very conservative: A robust solution comes with a high price in its nominal objective function value.
This motivated researchers to introduce less conservative robustness concepts. In the first part of this talk, several definitions of robustness will be shown. Two of the less conservative robustness approaches will be discussed in more detail: Light robustness and a scenario-based approach to recovery robustness.
The second part of the talk goes one step further: How to handle uncertain optimization problems in which more than one objective function is to be considered? This yields a robust multi-objective optimization problem, a class of problems only recently introduced. Concepts on how to define robust Pareto solutions will be developed. Mathematical properties will be derived as well as first approaches on how to compute robust efficient solutions.
All concepts will be illustrated on real-world problems which are currently tackled at Fraunhofer ITWM.

Dynamical imaging – the treatment of videos and multimodal images – leads to mathematical modeling by optical flow, image metamorphosis, and optimal transport. We present recent adaptations of the three concepts for manifold-valued image processing.
The optimal flow approach is driven by applications in material sciences, in particular electron backscatter diffraction.
While metamorphosis can be seen from an optimal control point of view, there is also a geometric concept which endows the space of images with a nonlinear Riemannian structure, which can be used for diffeomorphism estimation by minimizing the path energies of a corresponding geodesics.
In optimal transport, we are interested in the multimarginal, unbalanced setting.

New infectious diseases emerge regularly, parasites mutate, and often become infectious again for the previously immunized host population. History provides many unfortunate examples. Abel and Riemann died from tuberculosis, C.G.J. Jacobi from smallpox, Kowalewskaja, Kummer and Weierstrass from influenza, and Boole and Descartes died from pneumonia. Fermat fortunately survived the last great outbreak of plague in Toulouse.
Mathematics itself has successfully played a longstanding and important role in understanding the dynamics of epidemics. Due to the structural similarities of infection processes an abstract approach is natural. This has been realized and exploited a long time ago.
In this talk some of the mathematical tools available for epidemiology are summarized and phenomena like epidemic waves and inter-epidemic periods, heterogeneous populations with variable infectivity, as well as vaccination schemes with subcritical bifurcations are discussed. Finally we have a look at some actual data.