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 "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: Matthias Wink (Muenster)

Date: April 11, 2023

Seminar: Topics in general relativity

Title: Einstein metrics on the Ten-Sphere

Abstract: We prove the existence of three non-round, non-isometric Einstein metrics with positive scalar curvature on the sphere S^{10}. Previously, the only even-dimensional spheres known to admit non-round Einstein metrics were S^{6} and S^{8}.

Speaker: Paul Gassiat (Paris-Dauphine)

Date: April 18, 2023

Seminar: PDE colloquium

Title: Long-time behavior of stochastic Hamilton-Jacobi equations

Abstract: Stochastic Hamilton-Jacobi equations appear naturally in a number of contexts, in particular in the level-set formulation of interface motions, when this motion is perturbed by random noise. The "irregular" time dependence of the right-hand side of these equations creates a number of mathematical difficulties. They are part of a class of equations ("fully nonlinear SPDE") which was introduced by Lions and Souganidis at the end of the 90s. Their work showed that one could extend the classical techniques of viscosity solutions to this context. In this talk, after presenting these equations, I will talk about various recent results concerning the long-time behavior of their solutions. Some of these results are only qualitative (convergence towards an equilibrium), but in some cases we can also prove quantitative results on the speed of convergence. We obtain in particular in the case of the movement of a graph by mean curvature that the stochastic term accelerates the convergence. Based on joint works with B. Gess, B. Seeger, P. Souganidis and P.L. Lions.

Speaker: Qian Wang (Oxford)

Date: April 25, 2023

Seminar: Topics in general relativity

Title: Rough solutions of the 3-D compressible Euler equations

Abstract: I will talk about my work on the compressible Euler equations. We prove the local-in-time existence the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $22$. At the opposite extreme, in the incompressible case, i.e. with a constant density, the result is known to hold for $\omega\in H^s$, $s>3/2$ and fails for $s\le 3/2$, see the work of Bourgain-Li. It is thus natural to conjecture that the optimal result should be $(v,\varrho, \omega) \in H^s\times H^s\times H^{s'}$, $s>2, \, s'>\frac{3}{2}$. We view our work as an important step in proving the conjecture. The main difficulty in establishing sharp well-posedness results for general compressible Euler flow is due to the highly nontrivial interaction between the sound waves, governed by quasilinear wave equations, and vorticity which is transported by the flow. To overcome this difficulty, we separate the dispersive part of sound wave from the transported part, and gain regularity significantly by exploiting the nonlinear structure of the system and the geometric structures of the acoustic spacetime.

Speaker: Gustav Holzegel (Muenster)

Date: May 9, 2023

Seminar: Topics in general relativity

Title: Quasilinear wave equations on black hole backgrounds

Abstract: After a brief introduction to the geometric features of black holes, I will discuss recent joint work with Dafermos, Rodnianski and Taylor introducing a new scheme to prove small data global existence results for quasilinear wave equations on black hole backgrounds (see arXiv:2212.14093). A key ingredient is a new physical space estimate at the top order which avoids understanding the detailed structure of trapped geodesics. The relation to the stability problem for black holes will also be discussed.

Speaker: Milos Provci (University of Burgundy)

Date: 16 May, 2023

Seminar: Topics in general relativity

Title: Scattering of Dirac fields in the interior of Kerr-Newman(-anti)-de Sitter black holes

Abstract: The existence of a scattering map from the event horizon to the Cauchy horizon of a black hole from the Kerr-Newman(-anti)-de Sitter (KN(A)dS) family allows for the study of the stability of the Cauchy horizon, and is therefore a necessary first step in understanding the Cosmic Censorship Conjecture. Following work done on Reissner-Nordström (Penrose and Simpson, waves; Chandrasekhar and Hartle, waves; Kehle and Shlapentokh-Rothman, scalar fields; Häfner and Nicolas and Mokdad, Dirac fields), we construct the first scattering theory in the interior of KN(A)dS, of Dirac fields. More particularly, we show existence, uniqueness and asymptotic completeness of scattering data for Dirac fields from the event horizon to the Cauchy horizon. Our approach relies on dynamic scattering: the construction of wave operators for a time-dependent Hamiltonian.

Speaker: Jonas Hirsch (Leipzig)

Date: 16 May, 2023

Seminar: PDE colloquium

Title: On the De Giorgi-Nash-Moser theorem for hypoelliptic operators

Abstract: I would like to present a relative simple approach to show uniform boundedness and a weak Harnack inequality for general hypoelliptic operators where λ ≤ a_{ij} ≤ Λ is uniformly elliptic but merely measurable and the X_i are given smooth vectorfields. Furthermore we assume that they satisfy the Hormander condition i.e. their Lie-Algebra spans R^{n+1}. The novelty is the avoidance of a “general” Sobolev embedding and a “quantitative” Poincare inequality. Somehow our approach shows that one can somehow consider even the classical De Giorgi-Nash- Moser theorem as a “perturbation” of the poisson equation. If time permits I would like to discuss as well how the geometry of the hypoelliptic equations come into play to obtain as a consequence the famous Holder regularity.

Speaker: Mitia Duerinckx (Universite Libre de Bruxelles) CANCELLED

Date: 23 May, 2023

Seminar: PDE colloquium

New date: We have now provisionally fixed a date in the next term: 24th October.

Abstract:

Speaker: Bjoern Bringmann (Princeton) (Time: 14:15)

Date: June 6, 2023

Seminar: PDE colloquium

Title: Invariant Gibbs measures for (1+1)-dimensional wave maps into Lie groups.

Abstract: We consider the wave maps equation for maps from (1+1)-dimensional Minkowski space into a compact Lie group G. The Gibbs measure of this model corresponds to a Brownian motion on the Lie group G, which is a natural object from stochastic differential geometry. Our main result is the invariance of the Gibbs measure under the wave maps equation. The proof combines techniques from differential geometry, partial differential equations, and probability theory.

Speaker: Gianni Dal Maso (SISSA, Trieste)

Date: 6 June, 2023

Seminar: PDE colloquium

Title: New results on homogenisation of free discontinuity problems.

Abstract: We study deterministic and stochastic homogenisation problems for free discontinuity functionals under new hypotheses on the surface energies. The results are based on a compactness theorem with respect to Gamma-convergence, on the characterisation of the integrands of the Gamma-limit by means of limits of minimum values of some auxiliary minimum problems on small cubes, and on the subadditive ergodic theorem for the stochastic part.

Speaker: Roberta Bianchini (IAC, Rome)

Date: 13 June, 2023

Seminar: PDE colloquium

Title: Mathematical analysis of stably stratified fluids

Abstract: We will be interested in the analysis of a system of PDEs modeling continuously stratified fluids under the influence of gravity. The system is obtained by a linearization of the equations of incompressible non-homogeneous fluids around a background density profile that increases with depth (spectrally stable density profile). I will present some mathematical problems related to well/ill-posedness, stability and long-time dynamics.

Speaker: Stefan Czimek (Leipzig)

Date: 20 June, 2023

Seminar: Topics in general relativity

Title: Obstruction-free gluing for the Einstein equations

Abstract: We present a new approach to the gluing problem in General Relativity, that is, the problem of matching two solutions of the Einstein equations along a spacelike or characteristic (null) hypersurface. In contrast to previous constructions, the new perspective actively utilizes the nonlinearity of the constraint equations. As a result, we are able to remove the 10- dimensional spaces of obstructions to gluing present in the literature. As application, we show that any asymptotically flat spacelike initial data set can be glued to Schwarzschild initial data of sufficiently large mass. This is joint work with I. Rodnianski.

Speaker: Markus Tempelmayr (Muenster)

Date: 20 June, 2023

Seminar: PDE colloquium

Title: Some recent progress on quasilinear SPDEs

Abstract: We give an overview of the solution theory for singular SPDEs in case of a quasi-linear equation, following the recent approach of Otto, Sauer, Smith and Weber. The basic idea is to parametrize the model, which captures the local solution behaviour, by partial derivatives w.r.t. the nonlinearity. This allows for an efficient bookkeeping and an inductive construction of the model. Malliavin calculus plays a prominent role, on the one hand to obtain convergence of renormalized models based on a spectral gap assumption of the driving noise, on the other hand to obtain a characterization of models establishing a universality-type result. Based on joint work with Pablo Linares, Felix Otto and Pavlos Tsatsoulis.

Speaker: Yakov Shlapentokh-Rothman (Toronto)

Date: 27 June, 2023

Seminar: PDE colloquium

Title: Massive Scalar Fields on Black Hole Spacetimes

Abstract: After a brief discussion of the motivation for the study of massive scalar fields in general relativity and in relation to black holes, I will recall some old results concerning situations where one expects instabilities for massive scalar fields and then discuss a new result concerning situations where one can prove decay for the scalar field. This final decay result is joint with Federico Pasqualotto and Maxime Van de Moortel.

Speaker: Max Berning (Muenster)

Date: 4 July, 2023

Seminar: Topics in general relativity

Title: Improved region of existence for the characteristic initial value problem

Abstract: I will present a recent result about an improved region for local existence of vacuum-solutions to the Characteristic Initial Value Problem in General Relativity by Johnathan Luk (2018). I'll start by going through the general setup such as the double-null foliation, the harmonic gauge and talk about the structure of the equations that result from this choice of coordinate system. I'll also briefly recall the Theorem of Rendall, which is the result that is improved here and which acts as a basis for the work from Luk. In the end I'll talk about the difficulties that are present when trying to improve the region of existence and outline the structure as well as the main ideas of the proof.

Speaker: Giorgios Mavrogiannis (Rutgers)

Date: 11 July, 2023

Seminar: Topics in general relativity

Title: Relatively non-degenerate estimates on Kerr de Sitter spacetimes

Abstract: We will start discussing a new method of how to prove exponential decay for the solutions of the wave equation on a Schwarzschild de Sitter black hole spacetime by exploiting a novel "relatively non-degenerate" estimate. This estimate does not degenerate at trapping. The main ingredient in proving this estimate is to commute with a novel vector field that "sees" trapping. Then, we will discuss a natural generalization of the vector field commutation in Schwarzschild de Sitter to the entire subextremal Kerr de Sitter black hole spacetime, by commuting with a pseudodifferential operator. There are more technicalities because of the elaborate nature of trapping. Time permitting we will discuss how to use this black box estimate to prove stability and exponential decay for the solutions of a quasilinear wave equation on Kerr de Sitter.

Speaker: Arick Shao (Queen Mary)

Date: 18 July, 2023

Seminar: Topics in general relativity

Title: Bulk-boundary correspondence for vacuum asymptotically Anti-de Sitter spacetimes

Abstract: The AdS/CFT conjecture in physics posits the existence of a correspondence between gravitational theories in asymptotically Anti-de Sitter (aAdS) spacetimes and field theories on their conformal boundary. In this presentation, we prove a rigorous mathematical statement toward this conjecture in the classical relativistic setting. In particular, we show there is a one-to- one correspondence between aAdS solutions of the Einstein-vacuum equations and a suitable space of data on the conformal boundary (consisting of the boundary metric and the boundary stress-energy tensor), provided the boundary satisfies a geometric condition. We also discuss applications of this result to symmetry extension, as well as its connection to unique continuation problems. This is joint work with Gustav Holzegel, and builds upon joint works with Athanasios Chatzikaleas, Simon Guisset, and Alex McGill.