| Private Homepage | https://www.uni-muenster.de/Diffgeo/christophboehm.html |
| Selected Publications | • Böhm, Christoph Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces. Inventiones Mathematicae Vol. 134 (1), 1998, pp 145--176 online • Böhm, Christoph On the long time behavior of homogeneous Ricci flows. Commentarii Mathematici Helvetici Vol. 90 (3), 2015, pp 543-571 online • Böhm, Christoph; Buttsworth, Timothy; Clarke, Brian Scalar curvature along Ebin geodesics. Journal für die reine und angewandte Mathematik Vol. 813, 2024 online • Böhm, Christoph; Lafuente, Ramiro Non-compact Einstein manifolds with symmetry. Journal of the American Mathematical Society Vol. 36 (3), 2023 online • Böhm, Christoph; Lafuente, Ramiro; Simon, Miles Optimal Curvature Estimates for Homogeneous Ricci Flows. International Mathematics Research Notices Vol. 2019 (14), 2019 online • Böhm, Christoph; Lafuente, Ramiro Immortal homogeneous Ricci flows. Inventiones Mathematicae Vol. 212, 2018 online • Böhm, Christoph; Lafuente, Ramiro A. The Ricci flow on solvmanifolds of real type. Advances in Mathematics Vol. 353, 2019 online • Böhm, Christoph; Lafuente, Ramiro A. Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. Geometry and Topology Vol. 26, 2022 online • Böhm, Christoph, Wilking, Burkhard Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geometric And Functional Analysis Vol. 17 (3), 2007, pp 665-681 online • Böhm, Christoph; Wilking, Burkhard Manifolds with positive curvature operators are space forms. Annals of Mathematics Vol. 167 (3), 2008 online |
| Topics in Mathematics Münster | T4: Groups and actions T5: Curvature, shape, and global analysis T6: Singularities and PDEs T10: Deep learning and surrogate methods |
| Current Publications | • Böhm, Christoph; Buttsworth, Timothy; Clarke, Brian Scalar curvature along Ebin geodesics. Journal für die reine und angewandte Mathematik Vol. 813, 2024 online • Böhm, Christoph; Lafuente, Ramiro Non-compact Einstein manifolds with symmetry. Journal of the American Mathematical Society Vol. 36 (3), 2023 online • Böhm, Christoph; Lafuente, Ramiro A. Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds. Geometry and Topology Vol. 26, 2022 online • Böhm, Christoph; Lafuente, Ramiro; Simon, Miles Optimal Curvature Estimates for Homogeneous Ricci Flows. International Mathematics Research Notices Vol. 2019 (14), 2019 online • Böhm, Christoph; Lafuente, Ramiro A. The Ricci flow on solvmanifolds of real type. Advances in Mathematics Vol. 353, 2019 online • Böhm, Christoph; Lafuente, Ramiro Immortal homogeneous Ricci flows. Inventiones Mathematicae Vol. 212, 2018 online |
| Current Projects | • EXC 2044 - T04: Groups and actions The study of symmetry and space through the medium of groups and their actions has long been a central theme in modern mathematics, indeed one that cuts across a wide spectrum of research within the Cluster. There are two main constellations of activity in the Cluster that coalesce around groups and dynamics as basic objects of study. Much of this research focuses on aspects of groups and dynamics grounded in measure and topology in their most abstract sense, treating infinite discrete groups as geometric or combinatorial objects and employing tools from functional analysis, probability, and combinatorics. Other research examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry. online • EXC 2044 - T05: Curvature, shape and global analysis Riemannian manifolds or geodesic metric spaces of finite or infinite dimension occur in many areas of mathematics. We are interested in the interplay between their local geometry and global topological and analytical properties, which in general are strongly intertwined. For instance, it is well known that certain positivity assumptions on the curvature tensor (a local geometric object) imply topological obstructions of the underlying manifold. online • EXC 2044 - T06: Singularities and PDEs Our goal is to utilise and further develop the theory of non-linear PDEs to understand singular phenomena arising in geometry and in the description of the physical world. Particular emphasis is put on the interplay of geometry and partial differential equations and also on the connection with theoretical physics. The concrete research projects range from problems originating in geometric analysis such as understanding the type of singularities developing along a sequence of four-dimensional Einstein manifolds, to problems in evolutionary PDEs, such as the Einstein equations of general relativity or the Euler equations of fluid mechanics, where one would like to understand the formation and dynamics (in time) of singularities. online • EXC 2044 - T10: Deep learning and surrogate methods In this topic we will advance the fundamental mathematical understanding of artificial neural networks, e.g., through the design and rigorous analysis of stochastic gradient descent methods for their training. Combining data-driven machine learning approaches with model order reduction methods, we will develop fully certified multi-fidelity modelling frameworks for parameterised PDEs, design and study higher-order deep learning-based approximation schemes for parametric SPDEs and construct cost-optimal multi-fidelity surrogate methods for PDE-constrained optimisation and inverse problems. online • 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. • 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. • 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 | 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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