Further research projects of Research Area C members
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GRK 3027: Rigorous Analysis of Complex Random Systems The Research Training Group is dedicated to educating mathematicians in the field of complex random systems. It provides a strong platform for the development of both industrial and academic careers for its graduate students. The central theme is a mathematically rigorous understanding of how probabilistic systems, modelled on a microscopic level, behave effectively at a macroscopic scale. A quintessential example for this RTG lies in statistical mechanics, where systems comprising an astronomical number of particles, upwards of 10^{23}, can be accurately described by a handful of observables including temperature and entropy. Other examples come from stochastic homogenisation in material sciences, from the behaviour of training algorithms in machine learning, and from geometric discrete structures build from point processes or random graphs. The challenge to understand these phenomena with mathematical rigour has been and continues to be a source of exciting research in probability theory. Within this RTG we strive for macroscopic representations of such complex random systems. It is the main research focus of this RTG to advance (tools for) both qualitative and quantitative analyses of random complex systems using macroscopic/effective variables and to unveil deeper insights into the nature of these intricate mathematical constructs. We will employ a blend of tools from discrete to continuous probability including point processes, large deviations, stochastic analysis and stochastic approximation arguments. Importantly, the techniques that we will use and the underlying mathematical ideas are universal across projects coming from completely different origin. This particular facet stands as a cornerstone within the RTG, holding significant importance for the participating students. For our students to be able to exploit the synergies between the different projects, they will pass through a structured and rich qualification programme with several specialised courses, regular colloquia and seminars, working groups, and yearly retreats. Moreover, the PhD students will benefit from the lively mathematical community in Münster with a mentoring programme and several interaction and networking activities with other mathematicians and the local industry.
onlineProject members:
Matthias Löwe,
Steffen Dereich,
Caterina Zeppieri,
Zakhar Kabluchko,
Chiranjib Mukherjee,
Christian Seis,
Arnulf Jentzen,
Martin Huesmann,
Anna Gusakova,
Hendrik Weber• CRC 1450 - A05: Targeting immune cell dynamics by longitudinal whole-body imaging and mathematical modelling
onlineProject members:
Benedikt Wirth• CRC 1450 - A06: Improving intravital microscopy of inflammatory cell response by active motion compensation using controlled adaptive optics
We will advance multiphoton fluorescence microscopy by developing a novel optical module comprised of a high-speed deformable mirror that will actively compensate tissue motion during intravital imaging, for instance due to heart beat (8 Hz), breathing (3 Hz, in mm-range) or peristaltic movement of the gut in mice. To control this module in real-time, we will develop and validate mathematical methods that track and predict tissue deformation.
This will allow imaging of inflammatory processes at cellular resolution without mechanical tissue fixation.
onlineProject members:
Benedikt Wirth• CRC 1442 - B01: Curvature and Symmetry
The question of how far geometric properties of a manifold determine its global topology is a classical problem in global differential geometry. Building on recent breakthroughs we investigate this problem for positively curved manifolds with torus symmetry. We also want to complete the classification of positively curved cohomogeneity one manifolds and obtain structure results for the fundamental groups of nonnegatively curved manifolds. Other goals include structure results for singular Riemannian foliations in nonnegative curvature and a differentiable diameter pinching theorem.
onlineProject members:
Burkhard Wilking,
Michael Wiemeler• CRC 1442 - B02: Geometric evolution equations
Hamilton's Ricci flow is a (weakly parabolic) geometric evolution equation, which deforms a given Riemannian metric in its most natural direction. Over the last decades, it has been used to prove several significant conjectures in Riemannian geometry and topology (in dimension three). In this project we focus on Ricci flow in higher dimensions, in particular on heat flow methods, new Ricci flow invariant curvature conditions and the dynamical Alekseevskii conjecture.
onlineProject members:
Burkhard Wilking,
Christoph Böhm• CRC 1442 - D03: Integrability
We investigate blobbed topological recursion for the general Kontsevich matrix model, as well as the behaviour of Baker–Akhiezer spinor kernels for deformations of the spectral curve and for the quartic Kontsevich model. We study relations between spin structures and square roots of Strebel differentials, respectively between topological recursion and free probability. We examine factorisation super-line bundles on infinite-dimensional Grassmannians and motivic characteristic classes for intersection cohomology sheaves of Schubert varieties.
onlineProject members:
Jörg Schürmann,
Raimar Wulkenhaar,
Yifei Zhao• CRC 1442 - B04: Harmonic maps and symmetry
Many important geometric partial differential equations are Euler–Lagrange equations of natural functionals. Amongst the most prominent examples are harmonic and biharmonic maps between Riemannian manifolds (and their generalisations), Einstein manifolds and minimal submanifolds. Since commonly it is extremely difficult to obtain general structure results concerning existence, index and uniqueness, it is natural to examine these partial differential equations under symmetry assumptions.
onlineProject members:
Christoph Böhm,
Anna Siffert• CRC 1442 - B06: Einstein 4-manifolds with two commuting Killing vectors
We will investigate the existence, rigidity and classification of 4-dimensional Lorentzian and Riemannian Einstein metrics with two commuting Killing vectors. Our goal is to address open questions in the study of black hole uniqueness and gravitational instantons. In the Ricci-flat case, the problem reduces to the analysis of axisymmetric harmonic maps from R^3 to the hyperbolic plane. In the case of negative Ricci curvature, a detailed understanding of the conformal boundary value problem for asymptotically hyperbolic Einstein metrics is required.
onlineProject members:
Hans-Joachim Hein,
Gustav Holzegel• Mathematical analysis of bubble rings in ideal fluids
In this project, the evolution of toroidal bubble vortices is to be investigated. Bubble vortices are special vortices that occur in two-phase fluids. A typical and fascinating example is an air bubble ring in water created by dolphins or beluga whales. The underlying mathematical model is given by the two-phase Euler equations with surface tension. One major goal is a thorough mathematical construction of steady rings that move without changing their shape, and of perturbations of these. Such traveling waves are known for the classical Euler equations, but their existence is unknown for surface tension dependent models. Of particular interest is the role of the surface tension for the shape of the ring, which will be investigated. A second objective of this project is to understand how the effect of surface tension can be exploited to rigorously justify certain nonlinear motion laws of one or more interacting bubble rings. The understanding of such motion laws for the classical Euler equations is poor, and it is expected that the regularising effect of surface tension helps to mathematically tame the problem.
onlineProject members:
Christian Seis• Overcoming the curse of dimensionality through nonlinear stochastic algorithms: Nonlinear Monte Carlo type methods for high-dimensional approximation problems
In many relevant real-world problems it is of fundamental importance to approximately compute evaluations of high-dimensional functions. Standard deterministic approximation methods often suffer in this context from the so-called curse of dimensionality in the sense that the number of computational operations of the approximation method grows at least exponentially in the problem dimension. It is the key objective of the ERC-funded MONTECARLO project to employ multilevel Monte Carlo and stochastic gradient descent type methods to design and analyse algorithms which provably overcome the curse of dimensionality in the numerical approximation of several high-dimensional functions; these include solutions of certain stochastic optimal control problems of some nonlinear partial differential equations and of certain supervised learning problems.
onlineProject members:
Arnulf Jentzen• Global Estimates for non-linear stochastic PDEs
Semi-linear stochastic partial differential equations: global solutions’ behaviours
Partial differential equations are fundamental to describing processes in which one variable is dependent on two or more others – most situations in real life. Stochastic partial differential equations (SPDEs) describe physical systems subject to random effects. In the description of scaling limits of interacting particle systems and in quantum field theories analysis, the randomness is due to fluctuations related to noise terms on all length scales. The presence of a non-linear term can lead to divergencies. Funded by the European Research Council, the GE4SPDE project will describe the global behaviour of solutions of some of the most prominent examples of semi-linear SPDEs, building on the systematic treatment of the renormalisation procedure used to deal with these divergencies.
onlineProject members:
Hendrik Weber• Chemical biology of membrane transport proteins
The Topical Program deals with the so-called membrane transport proteins on a molecular level. The scientists work with methods from chemical biology and transfer their findings from basic research to physiological processes in animals and humans. In a workshop, various experts from the University of Münster will be brought together to discuss the topic in more detail and to further develop the concept of the Topical Program into a Collaborative Research Centre. A successful Collaborative Research Centre in this field will be a pillar for a chemical-medical Cluster of Excellence in the next but one application period.
The Topical Program follows on the Research Training Group "Chemical Biology of Ion Channels (Chembion)" funded by the German Research Foundation (DFG). Since the end of 2019, the research training group has been dedicated to the development, synthesis and modification of small organic ion channel modulators and their application in the field of molecular, cellular and systemic ion channel (patho)physiology.
onlineProject members:
Angela Stevens• Interdisziplinäres Lehrprogramm zu maschinellem Lernen und künstlicher Intelligenz
The aim of the project is to establish and test a graduated university-wide teaching programme on machine learning (ML) and artificial intelligence (AI). AI is taught as an interdisciplinary cross-sectional topic that has diverse application possibilities in basic research as well as in economy and society, but consequently also raises social, ethical and ecological challenges.
The modular teaching program is designed to enable students to build up their AI knowledge, apply it independently and transfer it directly to various application areas. The courses take place in a broad interdisciplinary context, i.e., students from different departments take the courses together and work together on projects.
onlineProject members:
Xiaoyi Jiang,
Christian Engwer• Mathematical Research Data Initiative - TA2: Scientific Computing Driven by the needs and requirements of mathematical research as well as scientific disciplines using mathematics, the NFDI-consortium MaRDI (Mathematical Research Data Initiative) will develop and establish standards and services for mathematical research data.
Mathematical research data ranges from databases of special functions and mathematical objects, aspects of scientific computing such as models and algorithms to statistical analysis of data with uncertainties. It is also widely used in other scientific disciplines due to the cross-disciplinary nature of mathematical methods. online
Project members: Mario Ohlberger, Stephan Rave