## Sessões anteriores

### Introduction to the Theory of Shock Waves (IV)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

Planned duration: 6 x 1.5 hours.

Bibliography:

Joel Smoller, “Shock waves and Reaction Diffusion Equations”

Constantine Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”

Alexandre Chorin and Jerrold Marsden, “A Mathematical Introduction to Fluid Mechanics”

Lecture notes of Blake Temple (on his webpage)

### Introduction to the Theory of Shock Waves (III)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

#### Ver também

Notes for this talk

Planned duration: 6 x 1.5 hours.

Bibliography:

Joel Smoller, “Shock waves and Reaction Diffusion Equations”

Constantine Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”

Alexandre Chorin and Jerrold Marsden, “A Mathematical Introduction to Fluid Mechanics”

Lecture notes of Blake Temple (on his webpage)

### Introduction to the Theory of Shock Waves (II)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

#### Ver também

Notes for this talk

Planned duration: 6 x 1.5 hours.

Bibliography:

Joel Smoller, “Shock waves and Reaction Diffusion Equations”

Constantine Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”

Alexandre Chorin and Jerrold Marsden, “A Mathematical Introduction to Fluid Mechanics”

Lecture notes of Blake Temple (on his webpage)

### The nonlinear stability of Schwarzschild

I'll discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.

### Introduction to the Theory of Shock Waves (I)

I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.

#### Ver também

Notes for this talk

Planned duration: 6 x 1.5 hours.

Bibliography:

Joel Smoller, “Shock waves and Reaction Diffusion Equations”

Constantine Dafermos, “Hyperbolic Conservation Laws in Continuum Physics”

Alexandre Chorin and Jerrold Marsden, “A Mathematical Introduction to Fluid Mechanics”

Lecture notes of Blake Temple (on his webpage)

### Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory

I am going to present Blake Temple's and my recent breakthrough regarding optimal metric regularity: We recently derived a set of nonlinear elliptic equations, with differential forms as unknowns, (the "Regularity Transformation equations" or "RT-equations"), and proved existence of solutions. The RT-equations determine whether optimal metric regularity can be achieved in General Relativity. Our existence result applies to connections in and Riemann curvature in $W^{m,p}$, $m\geq1$, $p>n$, and thus yields that such connections can always be smoothed to optimal regularity (one derivative above their curvature) by coordinate transformation. Extending this existence theory to the case of GR shock waves, when the connection is in $L^{\infty}$, is subject of our ongoing research. Our current existence result demonstrates that the method of determining optimal metric regularity by the RT-equations works.

### Stability and instability in spherical symmetry of Reissner-Nordström black holes for the Einstein-Maxwell-Klein-Gordon model

Penrose’s Strong Cosmic Censorship Conjecture is one of the central problems of Mathematical General Relativity. Its proof for the Einstein-Maxwell-Uncharged-Scalar-Field (EMSF) model in spherical symmetry relies on the formation of a Cauchy horizon that is $C^0$ regular but $C^2$ singular for generic Cauchy data. EMSF model however only admits two-ended black holes, unlike its charged analogue that allows for one-ended black holes, relevant to the study of charged gravitational collapse in spherical symmetry. In this talk, I will present my work about spherically symmetric charged and massive scalar fields on black holes. This includes a study of the black hole interior, that relates the behaviour of fields on the event horizon to the formation of a $C^0$ regular and $C^2$ singular Cauchy horizon. I will also mention my more recent work on the black hole exterior stability, for weakly charged massless scalar fields.

### On some stability and instability problems for hard stars in spherical symmetry

I will review Christodoulou's two phase model of relativistic fluids in the context of gravitational collapse, and describe the properties of static solutions to the hard phase with vacuum boundary (stars). We expect small stars to be (orbitally) stable, but limiting configurations with large central density to be unstable, and I will explain some of the underlying heuristics, and related results for these scenarios.

### Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes (II)

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes (I)

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary?

In these talks, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following symmetry extension question: when can a symmetry on the conformal boundary be extended into the interior?

This is joint work with Gustav Holzegel (Imperial College London).

### The quantised Dirac field and the fermionic signature operator (III)

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

### The quantised Dirac field and the fermionic signature operator (II)

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

#### Ver também

Paper by Felix Finster

### The quantised Dirac field and the fermionic signature operator (I)

In this mini course I am going to introduce the Dirac equation (describing fermions) in Minkowski spacetime and explain how to extend the equation to curved spacetimes (Lecture 1). In Lecture 2, I will introduce the canonical quantisation of the Dirac field in Minkowski spacetime and describe the problem of time-dependent external fields in the canonical quantisation formalism. In Lecture 3, I will present a proposal of myself and Felix Finster addressing the problem of time-dependent external fields and spacetimes, based on the fermionic signature operator.

#### Ver também

Notes for this talk

### Boundedness of energy for the Wake Klein-Gordon model

We consider the global-in-time existence theory for the Wave-Klein-Gordon model for the Einstein-Klein-Gordon equations introduced by LeFloch and Ma. By using the hyperboloidal foliation method, we prove that a hierarchy of weighted energies of the solutions remain (essentially) bounded for all times.

### Perturbations of the asymptotic region of the Schwarzschild-de Sitter spacetime

Although the study of the Cauchy problem in General Relativity started in the decade of 1950 with the work of Foures-Bruhat, addressing the problem of global non-linear stability of solutions to the Einstein field equations is in general a hard problem. The first non-linear global non-linear stability result in General Relativity was obtained for the de Sitter spacetime by H. Friedrich in the decade of 1980. In this talk the main tool used in the above result is introduced: a conformal (regular) representation of the Einstein field equations — the so-called conformal Einstein field equations (CEFE). Then, the conformal structure of the Schwarzschild-de Sitter spacetime is analysed using the extended conformal Einstein field equations (XCEFE). To this end, initial data for an asymptotic initial value problem for the Schwarzschild-de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild-de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild-de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild-de Sitter spacetime in the asymptotic region. Finally, we'll show that small enough perturbations of asymptotic initial data for the Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild-de Sitter spacetime.

This seminar is joint with CENTRA, and will take place on the Physics Department (seminar room, 2nd floor).

### Test fields cannot destroy extremal black holes

We prove that (possibly charged) test fields satisfying the null energy condition at the event horizon cannot overspin/overcharge extremal Kerr-Newman or Kerr-Newman-anti de Sitter black holes, that is, the weak cosmic censorship conjecture cannot be violated in the test field approximation. The argument relies on black hole thermodynamics (without assuming cosmic censorship), and does not depend on the precise nature of the fields. We also discuss generalizations of this result to other extremal black holes.

This seminar is joint with CENTRA.

### The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

The classical singularity theorems of General Relativity show that a Lorentzian manifold with a $C^2$-metric that satisfies physically reasonable conditions cannot be geodesically complete. One of the questions left unanswered by the classical singularity theorems is whether one could extend such a spacetime with a lower regularity Lorentzian metric such that the extension still satisfies these physically reasonable conditions and does no longer contain any incomplete causal geodesics. In other words, the question is if a lower differentiability of the metric is sufficient for the theorems to hold. The natural differentiability class to consider here is $C^{1,1}$. This regularity corresponds, via the field equations, to a finite jump in the matter variables, a situation that is not a priori regarded as singular from the viewpoint of physics and it is the minimal condition that ensures unique solvability of the geodesic equations. Recent progress in low-regularity Lorentzian geometry has allowed one to tackle this question and show that, in fact, the classical singularity theorems of Penrose, Hawking, and Hawking-Penrose remain valid for $C^{1,1}$-metrics. In this talk I will focus on the Hawking-Penrose theorem, being the most recent and in a sense most general of the aforementioned results, and some of the methods from low regularity causality and comparison geometry that were employed in its proof. This is joint work with J. D. E. Grant, M. Kunzinger and R. Steinbauer.

### Wave equations with initial data on compact Cauchy horizons

I will present a new energy estimate for wave equations close to compact non-degenerate Cauchy horizons. The estimate allows one to conclude several existence and uniqueness results for wave equations with initial data on the Cauchy horizon. This generalizes classical results that were proven under the assumption of either analyticity or symmetry of the spacetime or closedness of the generators. In particular, the results are useful in understanding the geometry of vacuum spacetimes with a compact non-degenerate Cauchy horizon (without making any extra assumptions). This problem is closely related to the strong cosmic censorship conjecture.

### On the Cosmic No-Hair Conjecture in $\mathbb{T}^2$-symmetric non-linear scalar field spacetimes

At late times, cosmological spacetimes solving Einstein's field equations, at least when assuming a positive cosmological constant, are conjectured to isotropise and appear like the de Sitter spacetime to late time observers. This is the statement of the Cosmic No-Hair conjecture. In this talk, I consider Einstein's non-linear scalar field equations and spacetimes with $\mathbb{T}^2$-symmetry. I present results on future global existence of such solutions and discuss the conjecture in the setting of a constant potential.

This talk is based on arXiv:1712.01801.