The "Topics in General Relativity" seminar is the seminar of the Holzegel group at Mathematics Münster. It takes place every Tuesday at 12:00 at the Westfälische Wilhelms-Universität. For further details/to receive e-mails concerning this seminar, feel free to contact Allen Juntao Fang, the current organiser of the seminar. The page for previous semesters may be found here: Summer 2022, Winter 2022-2023, Summer 2023, and Winter 2023-2024 .

The "PDE colloquium" takes places 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.

Topics in General Relativity**Marica Minucci (2nd April 2024, room MA 503)***Title: Conformal geodesics and the evolution of spacetimes with positive Cosmological constant*Abstract: Conformal geodesics were introduced into general relativity by Friedrich and Schmidt in 1987. Their motivation was to construct coordinates well suited for the local study of conformal boundaries. The conformal Gaussian coordinates are geometrically defined coordinates generated by timelike conformal geodesics. These coordinates, a conformal connection and an orthonormal frame solution to the conformal geodesic equations give rise to a conformal Gaussian gauge. In this talk, I will discuss a strategy based on this gauge choice and the conformal Einstein field equations to study the evolution of de Sitter-like and sub-extremal Schwarschild-de Sitter spacetimes as in arXiv:2103.08919 and arXiv:2302.04004.

Topics in General Relativity**Christiane Klein (16th April 2024, room MA 503)***Title: Quantum effects on the strong cosmic censorship conjecture*Abstract: The strong cosmic censorship conjecture states that the inner horizons of black holes should become singular to avoid travel into spacetime regions where determinism breaks down. In this talk, I present results on the influence of quantum effects on the strong cosmic censorship conjecture in asymptotically de Sitter black hole spacetimes. I will discuss the universality of these effects and present numerical results also showing how the quantum effects may influence the charge and angular momentum of the black hole. This is based on joint work with Peter Hintz, Jochen Zahn, Stefan Hollands, Marc Casals and Mojgan Soltani.

**Volker Schlue (16th April 2024, room SRZ 205)**PDE Colloquim

*Title: The scattering problem for classical wave equations with sources*Abstract: For the classical wave equation the map from initial data to Friedlander's radiation field is an isometry in energy spaces. I will explain the limitations of classical scattering theory, in the setting of slowly decaying initial data, or slowly decaying forcing terms, which entail the failure of Huygens principle. I will then describe my recent joint work with Hans Lindblad (Johns Hopkins) on the construction of scattering solutions in this context, and highlight the role of homogeneous functions in the interior and exterior of the light cone

Topics in General Relativity**Pascal Millet (30th April 2024, room MA 503)***Title: Leading-order term expansion for the Teukolsky equation on subextremal Kerr black holes / Oberseminar Topics in General Relativity*Abstract: The study of wave propagation on black hole spacetimes has been an intense field of research in the past decades. This interest has been driven by the stability problem for black holes and by questions related to scattering theory. On Kerr black holes, the analysis of Maxwell's equations and the equations of linearized gravity, can be simplified by introducing the Teukolsky equation, which offers the advantage of being scalar in nature. After explaining this reduction, I will present a result providing the large time leading-order term for initially localized and regular solutions of the Teukolsky equation, valid for the full subextremal range of black hole parameters and for all spins. I will explain how such a development follows naturally from the precise analysis of the resolvent operator on the real axis. Recent advances in microlocal analysis are used to establish the existence and mapping properties of the resolvent.

**Théophile Dolmaire (30th April 2024, room SRZ 205)**PDE Colloquim

*Title: Inelastic collapse of three particles in dimension d ≥ 2*Abstract: The Boltzmann equation can be derived rigorously from a system of elastic hard spheres (Lanford?s theorem, [11], [8]). Kinetic theory may also be fruitfully used to model large systems of particles that interact inelastically (sand, snow, interstellar dust, see [3], [10], [4]). Such materials are known as granular media. The theory enables for instance to explain the onset of inhomogeneities, as well as to quantify the decay of the temperature (Haff?s law, [9], [3]). In this case, the derivation of the inelastic Boltzmann equation is still open, mainly due to the complicated dynamics of the particles. In particular, it is still unknown if the dynamics of such particle systems is well-posed. One major difficulty comes from the phenomenon of inelastic collapse. A system of particles is said to experience an inelastic collapse when infinitely many collisions take place in finite time. It is known that inelastic collapse may take place for systems of only three particles [12]. We studied systems of three particles, in dimension d ? 2. Assuming that the restitution coefficient r is constant, we obtained general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff [13], and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision. Another important question is the following: since inelastic collapse can take place, is it possible to continue the dynamics of the particles anyway? We report also partial results in this direction. Considering on the other hand another law of collision, prescribing that a fixed quantity of kinetic energy is lost during each collision, we obtained results on systems of an arbitrary number of particles interacting according to this law, that look a priori contradictory. Namely, we proved that the flow of such a system of particles conserves the measure in the phase space, whereas the kinetic energy is not conserved. From these results, we deduce an Alexander?s theorem [1] for such systems of particles: for almost every initial datum, the dynamics of such systems is globally well-posed. To the best of our knowledge, this is the first result of global well-posedness concerning the dynamics of systems of inelastic particles. The results are taken from [5], [7], [6], obtained in collaboration with Juan J. L. Vel´azquez(Universitaet Bonn)

Topics in General Relativity**Gabriele Benomio (7th May 2024, room MA 503)***Title: A new gauge for gravitational perturbations of Kerr spacetimes*Abstract: I will present a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range. Central to the framework is a new formulation of nonlinear gravitational perturbations of Kerr in a geometric gauge tailored to the outgoing principal null geodesics of Kerr. The main features of the framework will be illustrated in the context of the linearised theory, which serves as a fundamental building block in nonlinear applications.

**Elena Salguero (7th May 2024, room SRZ 205)**PDE Colloquim

*Title: Global solutions of the two-phase gravity-Stokes system*Abstract: The gravity-Stokes system serves as a fundamental model for understanding the dynamics of incompressible fluids in certain regimes. We focus on the scenario where two fluids of different densities interact in a two-dimensional region without mixing. The density difference together with the gravity influence induce the dynamics of the two fluids and hence the evolution of the free interface arising between them. Through a contour dynamics approach, we address questions such as the existence and uniqueness of global solutions for this system and their asymptotic behavior, making emphasis on the properties of the free boundary.

Topics in General Relativity**Mariem Mohammed (14th May 2024, room MA 503)***Title: Conformal methods and asymptotic charges on vacuum spacetimes*Abstract: One of the common approaches in the studies of the asymptotic structure of spacetimes involves the use of conformal transformations. These transformations allow us to study the behaviour of the gravitational field ‘at infinity’ using local differential geometry by mapping points at infinite distances in one manifold to finite distances in another. In recent years, the subject of asymptotic symmetries/charges has acquired renewed interest, in part due to its relation with soft theorems, the gravitational memory effect and the information loss paradox. During my talk, I will focus on the role of Friedrich’s formulation of spatial infinity in evaluating asymptotic charges near spatial infinity and the implications of our analysis in the recent articles: arXiv:2311.07294 and arXiv:2112.03890

Topics in General Relativity**Sharmila Gunasekaran (28th May 2024, room MA 503)***Title: Rigidity of near horizon geometries*Abstract: Extreme black holes possess event horizons at zero temperature, referred to as degenerate Killing horizons. These horizons are exclusively delineated by a specific limiting procedure, defining a near-horizon geometry or, more broadly, a quasi-Einstein equation which governs their properties. Solutions to this equation manifest as triples (M, g, X), where M represents a closed manifold (the horizon), g denotes a Riemannian metric, and X is a 1-form. The talk will be a overview of these concepts and relevant results which characterize solutions to the quasi-Einstein equation. This is joint work with Eric Bahuaud, Hari Kunduri, and Eric Woolgar.

PDE Colloquim**Timothy Crin-Barat (28th May 2024, room SRZ 205)***Title: Hyperbolic approximation of the Navier-Stokes-Fourier system: hypocoercivity and hybrid Besov spaces*Abstract: We investigate the global well-posedness of partially dissipative hyperbolic systems and their associated relaxation limits. As we shall see, these systems can be interpreted as hyperbolic approximations of parabolic systems and provide an element of response to the infinite speed of propagation paradox arising in viscous fluid mechanics. To demonstrate this, we study a hyperbolic approximation of the multi-dimensional compressible Navier-Stokes-Fourier system and establish its hyperbolic-parabolic strong relaxation limit. For this purpose, we use and present techniques from the hypocoercivity theory and precise frequency decomposition of the solutions via the Littlewood-Paley theory.

PDE Colloquim**Francis Nier (2nd July 2024, room SRZ 205)***Title: Persistent homology and small eigenvalues of Witten and Bismut's hypoelliptic Laplacian*Abstract: After the two historical descriptions by Einstein and Langevin of Brownian motion, the now well known generators acting on p-forms, are on one side the Witten Laplacian (Einstein) and on the other side Bismut's hypoelliptic Laplacian (Langevin). The accurate computation of exponentially small eigenvalues has many applications, in particular for the design of effective molecular dynamics algorithms. In the case of the Witten Laplacian, I will present the result obtained a few years ago with D. Le Peutrec and C. Viterbo, which makes the connection between the various exponential scales of small eigenvalues and the bar code of persistent homology. This provides a natural topological extension of the well known Arrhenius law in the scalar case, for general potential functions not assumed to be Morse. I will also present the more recent result obtained with X. Sang and F. White, which provides the same determination of the different spectral exponential scales in terms of the persistent homology bar code, in the double asymptotic regime of large friction and small temperature for Bismut's hypoelliptic Laplacian.

Topics in General Relativity**Jingbo Wan (9th July 2024, room MA 503)***Title: TBD*Abstract: TBD

**Simon Guisset (11th June 2024, room MA 503)**

*Title: TBD*

Abstract: TBD