Private Homepage | https://imperia.uni-muenster.de/Diffgeo/christophboehm.html |

Project membershipMathematics Münster | B: Spaces and OperatorsC: Models and ApproximationsB1: Smooth, singular and rigid spaces in geometry C4: Geometry-based modelling, approximation, and reduction |

Current Projects | • CRC 1442: Geometry: Deformation and Rigidity - 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. • CRC 1442: Geometry: Deformation and Rigidity - 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. • Dynamical systems and irregular gradient flows The central goal of this project is to study asymptotic properties for gradient flows (GFs) and related dynamical systems. In particular, we intend to establish a priori bounds and related regularity properties for solutions of GFs, we intend to study the behaviour of GFs near unstable critical regions, we intend to derive lower and upper bounds for attracting regions, and we intend to establish convergence speeds towards global attrators. Special attention will be given to GFs with irregularities (discontinuities) in the gradient and for such GFs we also intend to reveal sufficient conditions for existence, uniqueness, and flow properties in dependence of the given potential. We intend to accomplish the above goals by extending techniques and concepts from differential geometry to describe and study attracting and critical regions, by using tools from convex analysis such as subdifferentials and other generalized derivatives, as well as by employing concepts from real algebraic geometry to describe domains of attraction. In particular, we intend to generalize the center-stable manifold theorem from the theory of dynamical systems to the considered non-smooth setting. Beside finite dimensional GFs, we also study GFs in their associated infinite dimensional limits. The considered irregular GFs and related dynamical systems naturally arise, for example, in the context of molecular dynamics (to model the configuration of atoms along temporal evoluation) and machine learning (to model the training process of artificial neural networks).online • EXC 2044 - B1: Smooth, singular and rigid spaces in geometry Many interesting classes of Riemannian manifolds are precompact in the Gromov-Hausdorfftopology. The closure of such a class usually contains singular metric spaces. Understanding thephenomena that occur when passing from the smooth to the singular object is often a first step toprove structure and finiteness results. In some instances one knows or expects to define a smoothRicci flow coming out of the singular objects. If one were to establishe uniqueness of the flow, thedifferentiable stability conjecture would follow. If a dimension drop occurs from the smooth to thesingular object, one often knows or expects that the collapse happens along singular Riemannianfoliations or orbits of isometric group actions.Rigidity aspects of isometric group actions and singular foliations are another focus in this project.For example, we plan to establish rigidity of quasi-isometries of CAT(0) spaces, as well as rigidity oflimits of Type III Ricci flow solutions and of positively curved manifolds with low-dimensional torusactions.We will also investigate area-minimising hypersurfaces by means of a canonical conformal completionof the hypersurface away from its singular set. online • EXC 2044 - C4: Geometry-based modelling, approximation, and reduction In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry. online |

E-Mail | cboehm at uni-muenster dot de |

Phone | +49 251 83-32736 |

FAX | +49 251 83-32711 |

Room | 412 |

Secretary | Sekretariat Huppert Frau Sandra Huppert Telefon +49 251 83-33748 Fax +49 251 83-32711 Zimmer 411 |

Address | Prof. Dr. Christoph Böhm Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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