# Herr Prof. Dr. Gustav Holzegel, Mathematisches Institut

Member of Mathematics MünsterInvestigator in Mathematics Münster

Investigator in Mathematics Münster

Private Homepage | https://www.uni-muenster.de/IVV5WS/WebHop/user/gholzege/ |

Project membershipMathematics Münster | B: Spaces and OperatorsC: Models and ApproximationsB1: Smooth, singular and rigid spaces in geometry C1: Evolution and asymptotics C4: Geometry-based modelling, approximation, and reduction |

Current Publications | • Graf, O; Holzegel, G Mode stability results for the Teukolsky equations on Kerr-anti-de Sitter spacetimes. Classical and Quantum Gravity Vol. 40 (4), 2023 online• Holzegel, G; Kauffman, C The wave equation on subextremal Kerr spacetimes with small non-decaying first order terms. , 2023 online• Holzegel, G; Shao, A The bulk-boundary correspondence for the Einstein equations in asymptotically anti-de Sitter spacetimes. Archive for Rational Mechanics and Analysis Vol. 247, 2023 online• Dafermos, M; Holzegel, G; Rodnianski, I; Taylor, M Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes. , 2022 online• Dafermos, M; Holzegel, G; Rodnianski, I; Taylor, M The non-linear stability of the Schwarzschild family of black holes. , 2021 online• Holzegel, G; Luk, J; Smulevici, J; Warnick, C Asymptotic properties of linear field equations in anti-de Sitter space. Communications in Mathematical Physics Vol. 374, 2020, pp 1125-1178 online• Holzegel, G; Kauffman, C A note on the wave equation on black hole spacetimes with small non-decaying first order terms. , 2020 online• Dafermos, M; Holzegel, G; Rodnianski, I Boundedness and decay for the Teukolsky equation on Kerr spacetimes I: the case |a|≪M. Annals of PDE Vol. 5 (2), 2019, pp Paper No. 2, 118 online• Dafermos, M; Holzegel, G; Rodnianski, I The linear stability of the Schwarzschild solution to gravitational perturbations. Acta Mathematica Vol. 222 (1), 2019 online |

Current Projects | • CRC 1442: Geometry: Deformation and Rigidity - 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. • 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 - C1: Evolution and asymptotics In this unit, we will use generalisations of optimal transport metrics to develop gradient flow descriptions of (cross)-diffusion-reaction systems, rigorously analyse their pattern forming properties, and develop corresponding efficient numerical schemes. Related transport-type- and hyperbolic systems will be compared with respect to their pattern-forming behaviour, especially when mass is conserved. Bifurcations and the effects of noise perturbations will be explored.Moreover, we aim to understand defect structures, their stability and their interactions. Examples are the evolution of fractures in brittle materials and of vortices in fluids. Our analysis will explore the underlying geometry of defect dynamics such as gradient descents or Hamiltonian structures. Also, we will further develop continuum mechanics and asymptotic descriptions for multiple bodies which deform, divide, move, and dynamically attach to each other in order to better describe the bio-mechanics of growing and dividing soft tissues. Finally, we are interested in the asymptotic analysis of various random structures as the size or the dimension of the structure goes to infinity. More specifically, we shall consider random polytopes and random trees.For random polytopes we would like to compute the expected number of faces in all dimensions, the expected (intrinsic) volume, and absorption probabilities, as well as higher moments and limit distributions for these quantities. 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 | gholzege@uni-muenster.de |

Phone | +49 251 83-33743 |

Room | 519 |

Secretary | Sekretariat Holzegel Frau Anke Pietsch Telefon +49 251 83-33901 Zimmer 306 Das Sekretariat ist montags bis donnerstags von 08:00 bis 13:00 geöffnet. |

Address | Herr Prof. Dr. Gustav Holzegel Mathematisches Institut Fachbereich Mathematik und Informatik der Universität Münster Einsteinstrasse 62 48149 Münster Deutschland |

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