Winter School: Wave equations and Fourier integral operators on models of an expanding universe

Time

January 15-17, 2020.

Place

The mathematics department at the Universtity of Potsdam, Campus Golm, House 14, Room 0.45.

Meeting Point

We meet on January 15 either in front of the mathematics department at 12:00 and go together for lunch in the mensa, or at the first talk at 13:30 (see below for the schedule).

Organizers

Klaus Kröncke, Hamburg

Jan Metzger, Potsdam

Oliver Petersen, Hamburg

Prerequisites

The topic of the winter school is a recent advanced development in the theory of wave equations on curved backgrounds (for more details, see the abstract below). A basic understanding of Fourier integral operators, or at least pseudo-differentialoperators, is necessary. The participants will be PhD students and Postdocs.

Format

The winter school will consist of 8 talks, each 75 min long + discussions. The talks will be given by participants of the winter school.

Questions

For questions concerning the talks please write to Oliver Petersen (Hamburg)

For questions regarding local organization please contact Sylke Pfeiffer, Jan Metzger or Philip Thonke

Schedule

Wednesday, 15.01.2020

13:30-15:00, Talk 1: Introduction

Speaker: Oliver Petersen

Main reference: [Vas10, Sec. 1]

 

15:00-16:00: Coffee break

 

16:00-17:30, Talk 2: 0-geometry and propagation of 0-singularities

Speaker: Klaus Kröncke

Main references: [MM87], [Vas10, Sec. 2]

 

Thursday, 16.01.2020

10:00-11:30, Talk 3: Local solvability near ∂X

Speaker: Felix Lubbe

Main reference: [Vas10, Sec. 3]

 

11:30-13:00: Lunch

 

13:00-14:30, Talk 4: Conormal regularity

Speaker: Alexander Friedrich

Main reference: [Vas10, Sec. 4]

 

14:30 - 15:30: Coffee break

 

15:30-17:00, Talk 5: Global solvability of the Cauchy problem

Speaker: Philip Thonke

Main reference: [Vas10, Sec. 5-6]

 

17:30 - 18:00: Discussion session

 

19:00 - : Conference dinner

Restaurant Loft, Brandenburger Str. 30-31, 14467 Potsdam (Potsdam City Center)

We will leave together after the discussion session to take Bus 606 at 18:24 from Golm to stop Luisenplatz-Süd. From there the restaurant is a 5 minute walk.

Friday, 17.01.2020

10:00-11:30, Talk 6: The scattering operator I

Speaker: Orville Damaschke

Main reference: [Vas10, Sec. 7]

 

11:30-13:00: Lunch

 

13:00-14:30, Talk 7: The scattering operator II

Speaker: Jørgen Lye

Main reference: [Vas10, Sec. 7]

 

14:30-15:30: Coffee break

 

15:30-17:00, Talk 8: The scattering operator III

Speaker: Jan Metzger

Main reference: [Vas10, Sec. 7]

Abstract

Mathematical general relativity is based on the theory of wave equations on curved spaces. In fact, even the gravitational field itself satisfies a non-linear wave equation, known as Einstein’s equation. Einstein’s equation is, in general, very difficult to solve and one therefore studies the wave equations on special models of the universe. The most simple model is the Minkowski space, which is the model of the universe in special relativity. This model is, however, not “expanding” and therefore not a good model for our universe (which is known to expand). The second most simple model for the universe is the so-called de Sitter space, which is believed to be a more accurate, since it is expanding. Fortunately, it is much easier to predict (estimate) solutions to wave equations on de Sitter space, than on Minkowski space. Moreover, there are natural generalizations of the de Sitter space including models for black holes. One of the most remarkable recent results in mathematical general relativity is the proof that such blackhole models are stable, by work of Hintz and Vasy [HV18].

In this winter school, we study wave equations on de Sitter space and the generalizations, including black holes. Though the proof of stability, mentioned above, lies beyond the scope of this winter school, the topics we will discuss are the natural first steps in this direction. We mainly will focus on a paper by Vasy [Vas10], where linear wave equations on asymptotically de Sitter spaces (generalizing de Sitter space) are studied in detail. Many of the non-technical ideas and basic structures in [HV18 ]are already present in [Vas10]. Vasy uses a microlocal analysis point of view,which gives a modern treatment of a classical problem. The methods include the construction and analysis of so-called Fourier integral operators (generalizing pseudo-differential operators), which will be studied in detail. From the time when the paper [Vas10] was published, a sequence of papers including [Vas13], [MSBV14], and [HV15], was published, building up for the final stability result in [HV18]. Even though most of the focus will be on [Vas10], the winter school will include a brief overview of all these papers and in particular an outline of the approach towards [HV18].
 

References

[HV15] Peter Hintz and András Vasy, Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes, Anal. PDE8(2015), no. 8,1807–1890.2

[HV18] Peter Hintz and András Vasy, The global non-linear stability of the Kerr–de Sitter family of blackholes, Acta Math.220(2018), no. 1, 1–206.

[MSBV14] Richard Melrose, Antônio Sá Barreto, and András Vasy, Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space, Comm. Partial Differential Equations39(2014), no. 3, 512–529.

[MM87] R. R. Mazzeo and R. B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct.Anal.75(1987), no. 2, 260–310.

[Vas10] András Vasy, The wave equation on asymptotically de Sitter-like spaces, Adv.Math.223(2010), no. 1, 49–97.

[Vas13] András Vasy, Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitterspaces (with an appendix by Semyon Dyatlov), Invent. Math.194(2013), no. 2,381–513.