The "Topics in General Relativity" seminar is the seminar of Professor Gustav Holzegel's group at Mathematics Münster. It takes place Tuesdays at 12:00 at the Westfälische Wilhelms-Universität, in Room 503 of the department building (Einsteinstr. 62). For further details/to receive e-mails concerning this seminar, please contact Athanasios Chatzikaleas. The page for previous semesters may be found here: Summer 2021-2022, Summer 2022-2023 and Winter 2022-2023.
The "PDE colloquium" takes place Tuesdays at 14:15 at the Westfälische Wilhelms-Universität, in Room SRZ 203 (unless otherwise announced) of the Seminarraumzentrum building (Orléans-Ring 12). If you want to be on the mailing list, please send an email to pde.colloquium@uni-muenster.de. The seminar's webpage for further details may be found here.
Speaker: Milos Provci (University of Muenster)
Date: Tuesday, October 24, 2023 (Time: 12:00)
Seminar: Topics in General Relativity
Title: Double-Null Foliation: A Review
Abstract: I will be going over the double null coordinate construction from Christodoulou's and Aretakis' notes
Speaker: Mahir Hadzic (University College London)
Date: Tuesday, October 24, 2023 (Time: 14:15)
Seminar: PDE colloquium
Title: Construction of naked singularities for the Einstein-Euler system
Abstract: I will review a series of recent works on the construction of imploding singularities for self-gravitating fluids, describing stars. We will briefly explain a rigorous construction of the so-called Larson-Penston (LP) collapse - a Newtonian self-similarly imploding star solution discovered numerically in 1960s. We will then show the existence of its GR analogue, called the relativistic Larson-Penston (RLP) solution, which is a radially symmetric solution of the Einstein-Euler system. We will show that the RLP spacetime can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data. This is based on several joint works with Yan Guo, Juhi Jang, and Matthew Schrecker.
Speaker: Marios-Antonios Apetroaie (University of Muenster)
Date: Tuesday, October 31, 2023
Seminar: Topics in General Relativity
Title: On the Linear (In)stability of Extremal Reissner-Nordström
Abstract: The Reissner-Nordström spacetime, as a solution to the Einstein-Maxwell equations, has been shown to be linearly stable for the full subextremal range, |Q|< M, by Elena Giorgi. We address the aforementioned problem for the extremal case, |Q|=M, which contrary to the subextremal one we show instability results manifesting along the future event horizon of the black hole, \mathcal{H}^+ . In particular, depending on the number of translation invariant derivatives of derived gauge-invariant quantities, there are decay, non-decay, and polynomial blow-up estimates asymptotically along \mathcal{H}^+ . In this presentation, we motivate the main ideas showing that solutions to the generalized Teukolsky system of positive and negative spin satisfy analogous estimates as well. Stronger and unprecedented instabilities are realised for the negative spin solutions, with the extreme curvature component, a , not decaying asymptotically along the event horizon.
Speaker: Annalaura Stingo (Ecole Polytechnique)
Date: Tuesday, November 7, 2023
Seminar: PDE colloquium
Title: Global stability of the Kaluza-Klein spacetime
Abstract: The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems. In this talk I will discuss the problem of the classical global stability of Kaluza-Klein spacetime when d=1. This is a joint work with C. Huneau and Z. Wyatt.
Speaker: Michal Wrochna (Utrecht University)
Date: Tuesday, November 14, 2023 (Time: 12:00)
Seminar: Topics in General Relativity
Title: The Dirichlet-to-Neumann map for the wave equation on anti-de Sitter spacetimes
Abstract: In the setting of the wave equation on asymptotically anti-de Sitter spacetimes, it is possible to define an operator which maps Dirichlet data of solutions to corresponding Neumann data at the conformal boundary. The Dirichlet-to-Neumann map (or in physicists’ terminology, the boundary-to-boundary propagator) is conjectured to determine the bulk metric up to trivial transformations. In this talk I will report on recent progress on this question, based on the proof that the Dirichlet-to-Neumann map is a fractional power for the boundary metric in a suitable sense (based on work in progress with Alberto Enciso and Gunther Uhlmann).
Speaker: Mitia Duerinckx (Universite Libre de Bruxelles)
Date: Tuesday, November 14, 2023 (Time: 14:15)
Seminar: PDE colloquium
Title: Effective description of particle suspensions in fluids
Abstract: This talk will be devoted to the effective large-scale behavior of suspensions of rigid particles in Stokesian fluids. Such systems are ubiquitous both in nature and in practical applications, and are well known to give rise to complex rheological behaviors, in particular to non-Newtonian effects. The main source of complexity is that particles are not point-like and that their interactions via the surrounding fluid flow are thus multibody in nature. Another difficulty lies in the long-range nature of these interactions, which means that renormalization is necessary to express various effective properties. We will address the derivation of so-called Doi models in dilute mean-field regime, the limitations of such effective theories, and the description of effective semi-dilute corrections.
Speaker: Marcel Oliver (Katholische Universität Eichstätt - Ingolstadt)
Date: Tuesday, November 21, 2023
Seminar: PDE colloquium
Title: Backscatter parameterizations for eddy permitting ocean models
Abstract: Ocean models at eddy-permitting resolution are generally overdissipative, damping the intensity of the mesoscale eddy field. To reduce overdissipation, backscatter parameterizations introduce an artifical energy pathway from small to large scales to close the energy cycle. In this talk, I will review the underlying issue, compare several possible deterministic, stochastic, and data-driven approaches, and present mathematical results regarding well-poseness and stability of the underlying parameterized PDE.
Speaker: Lili He (Johns Hopkins University) (Time: 12:00-13:00)
Date: Wednesday, November 22, 2023 (please note the special date and time)
Seminar: Topics in General Relativity
Title: The linear stability of weakly charged and slowly rotating Kerr-Newman family of charged black holes.
Abstract: I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.
Speaker: Istvan Kadar (University of Cambridge)
Date: Tuesday, November 28, 2023 (Time: 12:00)
Seminar: Topics in General Relativity
Title: Stability for nonlinear wave equations: The role of asymptotics
Abstract: In their pioneering works, Christodoulou and Klainerman proved the sufficiency of the null condition for stability of the trivial solution for systems of wave equations in Minkowski space. Subsequently, during their study of the Einstein equations in harmonic gauge, Lindblad and Rodnianski introduced the weak null condition, sparking extensive research on stability problems. However, the sufficiency of this condition remained a puzzle. In this talk, I will to tell you a tale about the late time behaviour of waves in asymptotically flat backgrounds and how this relates to stability for systems of wave equations. In particular, I'll present a system satisfying the weak null condition, but which has unstable zero solution.
Speaker: Angeliki Menegaki (Imperial College London)
Date: Tuesday, November 28, 2023 (Time: 14:15)
Seminar: PDE colloquium
Title: L^2-stability for the 4-waves kinetic equation around the Rayleigh-Jeans equilibrium
Abstract: We consider the four-waves spatial homogeneous kinetic equation arising in weak wave turbulence theory. In this talk I will present some new results on the existence and long-time behaviour of solutions around the Rayleigh-Jeans thermodynamic equilibrium solutions. In particular, I will present an $L^2$ stability of mild solutions on the whole frequency space for initial data close to Rayleigh-Jeans when assuming radial solutions for the equation, as well as a stability of the same kind without the radial solution-assumption but after introducing a cut-off on the frequencies. Parts of this talk are joint works with Miguel Escobedo (UPV/EHU).
Speaker: Tommaso Rosati (University of Warwick)
Date: Tuesday, December 5, 2023
Seminar: PDE colloquium
Title: Lower bounds to Lyapunov exponents for stochastic PDEs
Abstract: Inspired by problems from fluid dynamics, we introduce an approach to obtain lower bounds for Lyapunov exponents of stochastic PDEs. Our proof relies on the introduction of a novel Lyapunov functional for the projective process associated to the equation, based on the study of dynamics of the energy median and on a notion of non-degeneracy of the noise that leads to high-frequency stochastic instability. Joint work with Martin Hairer.
Speaker: Rita Teixeira da Costa (Cambridge and Princeton)
Date: Tuesday, December 12, 2023
Seminar: Topics in General Relativity
Title: The Teukolsky equation on Kerr in the full subextremal range |a| smaller than M
Abstract: The Teukolsky equation is one of the fundamental equations governing linear gravitational perturbations of the Kerr black hole family as solutions to the vacuum Einstein equations. We show that solutions arising from suitably regular initial data remain uniformly bounded in the energy space without derivative loss, and satisfy a suitable “integrated local energy decay” statement. A corollary of our work is that such solutions in fact decay inverse polynomially in time. Our proof holds for the entire subextremal range of Kerr black hole parameters, |a| smaller than M. This is joint work with Yakov Shlapentokh-Rothman (Toronto).
Speaker: Rita Teixeira da Costa (Cambridge and Princeton)
Date: Thursday, December 14, 2023 (please note the special date)
Seminar: Colloquium Wilhelm Killing
Title: On the stability of rotating black holes.
Abstract: In General Relativity, spacetime is a Lorentzian manifold whose curvature is sourced by the matter content through nonlinear partial differential equations -- the Einstein equations. Due to their nonlinear character, even in the absence of matter, there are solutions with nontrivial geometry, such as black holes. A long standing conjecture in the field posits that stationary black holes should be stable as solutions to the Einstein vacuum equation. A remarkable result of Dafermos, Holzegel, Rodnianski and Taylor (2021) established the conjecture for spherically symmetric black holes. This talk aims to give an overview of the state of the conjecture for rotating black holes. The talk begins with a gentle introduction to General Relativity and black holes. We will then describe the key insights from Geometry and Analysis that have allowed for progress on this problem. Some of the results presented are joint work with Yakov Shlapentokh-Rothman (Toronto) and Marc Casals (Leipzig/UCD/CBPF).
Information website: Colloquium Wilhelm Killing
Speaker: Jan Sbierski (University of Edinburgh)
Date: Monday, December 18, 2023 (please note the special date)
Seminar: Oberseminar Differential Geometry
Title: On the uniqueness problem for extensions of Lorentzian manifolds
Abstract:
Information website: Oberseminar Differential Geometry
Speaker: Frederico Cornalba (University of Bath)
Date: Tuesday, December 19, 2023
Seminar: PDE colloquium
Title:High-order discretisation and variance reduction for the Dean-Kawasaki model from Fluctuating Hydrodynamics
Abstract: Large-scale systems of interacting particles are ubiquitous in several relevant applications (in fact, the term particle may stand for actual particles, individuals, animals, parameters in neural networks, etc?). In many circumstances, it may be convenient to describe such particle systems on the continuum scale, i.e., to view their empirical density as the solution of suitable stochastic PDEs. This way of looking at particle systems has given rise to the theory nowadays referred to as Fluctuating Hydrodynamics (FH). In this talk, I plan to (1) give a brief overview of the theory of FH, and (2) discuss high-order numerical discretisations and variance reduction techniques for the Dean-Kawasaki model, a prototype model of FH (based on joint works with Julian Fischer, Jonas Ingmanns, Claudia Raithel)
Speaker: Hamed Masaood (Imperial College London)
Date: Tuesday, January 9, 2024 (Time: 12:00)
Seminar: Topics in General Relativity
Title: A Scattering Theory for Linearised Gravity on the Exterior of the Schwarzschild Black Hole
Abstract: I will talk about the scattering problem in general relativity, and present a construction of a scattering theory for the linearised Einstein equations in a double null gauge against a Schwarzschild background. The construction begins by dealing with the gauge invariant components of the linearised system via the spin $\pm2$ Teukolsky equations and the Teukolsky--Starobinsky identities; this was the subject of the paper arXiv:2007.13658. In this talk I will discuss how this theory can then be extended to the full system. Key to this step is the identification of suitable asymptotic gauge conditions on scattering data. Here, a Bondi-adapted double null gauge is shown to provide the necessary gauge rigidity, in a manner that enables the identification of a Hilbert-space isomorphism between finite energy scattering data and a suitable space of finite energy Cauchy data. In particular, the scattering theory we construct enables us to show that, in the global scattering problem, the linear memory effects at past and future null infinity are related by an antipodal map on the 2-sphere.
Speaker: Pawel Duch (Adam Mickiewicz University, Poznan)
Date: Tuesday, January 9, 2024 (Time: 14:15)
Seminar: PDE colloquium
Title: Flow equation approach to singular stochastic PDEs
Abstract: The macroscopic or mesoscopic behavior of many physical systems interacting with a random environment can be described in terms of non-linear stochastic partial differential equations. Such equations are usually very singular and cannot be solved using standard tools from PDE theory. In the talk, I will present a new technique of solving singular stochastic PDEs based on the renormalization group flow equation. The technique is applicable to a large class of semi-linear parabolic or elliptic SPDEs with fractional Laplacian and covers equations in the whole subcritical regime. A nice feature of the method is that it avoids the algebraic and combinatorial problems arising in different approaches.
Speaker: Vindula Kumaranayake (University of Muenster)
Date: Tuesday, January 16, 2024
Seminar: Topics in General Relativity
Title: Generic blow-up results for the wave equation in the interior of a Schwarzschild black hole.
Abstract: I will present an analysis of the aforementioned paper authored by Grigorios Fournodavlos and Jan Sbierski in 2019. Smooth Solutions to the Wave Equation in the Interior of the Schwarzschild Black Hole (Region II) were studied by the authors. They derive a fully asymptotic expansion centered around r=0, wherein each solution is distinguished by a principal logarithmic term and a bounded second order term. The paper identifies an open set of initial data, leading to a solution exhibiting logarithmic blow-up on the entire hypersurface {r=0}.The analysis incorporates weighted energy estimates, eliminating the need for symmetry assumptions. Despite this, the monotonicity of the spherically symmetric part of the solution is considered a pivotal argument in their study.
Speaker: Béatrice Bonga (Radboud University)
Date: Tuesday, January 30, 2024
Seminar: Topics in General Relativity
Title: Non-linearities at black hole horizons
Abstract: The gravitational waves emitted by a perturbed black hole ringing down are well described by damped sinusoids, whose frequencies are those of quasinormal modes. Typically, first-order black hole perturbation theory is used to calculate these frequencies. Recently, it was shown that second-order frequencies are necessary to model the gravitational-wave signal observed by a distant observer in binary black hole merger simulations. In this talk, I will show that the horizon of a newly formed black hole after the head-on collision of two black holes also shows evidence of non-linear modes.