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Mathematics Münster

B: Spaces and Operators
C: Models and Approximations

B1: Smooth, singular and rigid spaces in geometry
C1: Evolution and asymptotics
C4: Geometry-based modelling, approximation, and reduction
Current PublicationsGraf, 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; Kauffman, C A note on the wave equation on black hole spacetimes with small non-decaying first order terms. , 2020 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
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 ProjectsThe Black Hole Stability Problem and the Analysis of asymptotically anti-de Sitter spacetimes The present proposal is concerned with the analysis of the Einstein equations of general relativity, a non-linear system of geometric partial differential equations describing phenomena from the bending of light to the dynamics of black holes. The theory has recently been confirmed in a spectacular fashion with the detection of gravitational waves.The main objective of the proposal is to consolidate my research group based at Imperial College by developing novel mathematical techniques that will fundamentally advance our understanding of the Einstein equations. Here the proposal builds on mathematical progress in the last decade resulting from achievements in the fields of partial differential equations, differential geometry, microlocal analysis and theoretical physics.The Black Hole Stability ProblemA major open problem in general relativity is to prove the non-linear stability of the Kerr family of black hole solutions. Recent advances in the problem of linear stability made by the PI and collaborators open the door to finally address a complete resolution of the stability problem. In this proposal we will describe what non-linear techniques will need to be developed in addition to achieve this goal. A successful resolution of this program would conclude an almost 50-year-old problem.The Analysis of asymptotically anti-de Sitter (aAdS) spacetimesWe propose to prove the stability of pure AdS if so-called dissipative boundary conditions are imposed at the boundary. This result would align with the well-known stability results for the other maximally-symmetric solutions of the Einstein equations, Minkowski space and de Sitter space.As a second -- related -- theme we propose to formulate and prove a unique continuation principle for the full non-linear Einstein equations on aAdS spacetimes. This goal will be achieved by advancing techniques that have recently been developed by the PI and collaborators for non-linear wave equations on aAdS spacetimes. 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 - 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
Phone+49 251 83-33743
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.
AddressHerr Prof. Dr. Gustav Holzegel
Mathematisches Institut
Fachbereich Mathematik und Informatik der Universität Münster
Einsteinstrasse 62
48149 Münster
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