Courses
Mathematical General Relativity
Lecturer: Prof. Aretakis
Dates: June 12, 13, 14, 15
Course description: This course will start with a basic introduction to Lorentzian geometry and General Relativity that will cover Special Relativity, causality theory, null hypersurfaces, trapped surfaces, and black holes. We will then consider linear and nonlinear wave equations, with and without symmetry assumptions, as model problems for the stability problem in the context of the Cauchy problem for the Einstein equations. We will then review the celebrated stability of Minkowski space by Christodoulou and Klainerman and discuss various applications such as 1) the memory effect, 2) formation of trapped surfaces, 3) stability of black holes.
Suggested background reading: Chapters 1-4 of General Relativity by Wald
Recent Advances in Classical and Relativistic Fluids
Lecturer: Prof. Disconzi
Dates: June 18, 19, 20, 21
Course description: We will discuss some recent results in the mathematical theory of classical and relativistic fluids. For classical fluids, the focus will be on the free-boundary Euler equations with surface tension, with emphasis on the compressible setting. Topics will include a priori estimates, local well-posedness, and the incompressible limit. For relativistic fluids, we will begin with a discussion of the importance of the topic in the physics literature, contrasting it with the lack of rigorous mathematical results in the case of viscous fluids. We will then present results concerning the local well-posedness of certain equations describing relativistic fluids with viscosity, including their coupling to Einstein's equations. If time allows, we will make brief comments on the problem of relativistic free-boundary fluids.
Suggested background reading:
- Chapter 1 of Vorticity and Incompressible Flow by Majda and Bertozzi
- Sections 3.1-3.10 of Relativistic Hydrodynamics by Rezzolla and Zanotti
Solitons, Bubbling, and Blowup for Semilinear PDEs
Lecturer: Prof. Lawrie
Dates: June 18, 19, 20, 22
Course description: Solitons (coherent solitary waves) are the building blocks of both global-in-time dynamics and singularity formation for critical semilinear dispersive PDEs. In the case of a globally defined solution, the soliton resolution conjecture asserts that as a solution evolves, it decomposes into a finite number of asymptotically decoupled solitons plus a remainder exhibiting linear dynamics. In the case of a solution that develops a singularity by concentrating mass or energy (bubbling), solitons often play the role of a universal blow-up profile -- zooming in on the solution near the singularity, the shape of a soliton comes into view. In this course we'll discuss these phenomena in the context of a few model geometric PDEs such as the wave maps equation and the related harmonic map heat flow. The focus will be on developing fundamental technique, especially concentration-compactness methods and modulation theoretic analysis.
Suggested background reading: Chapter 2 of Invariant Manifolds and Dispersive Hamiltonian Evolution Equations by Nakanishi and W. Schlag
The Formation of Singularities in General Relativity
Lecturer: Prof. Speck
Dates: June 11, 12, 14, 15
Course description: We will discuss stable blowup results for solutions to quasilinear wave equations, starting first with some model problems, and then proceeding to Einstein's equations. For Einstein's equations, the celebrated Hawking-Penrose theorems show that a large, open set of initial conditions lead to geodesically incomplete solutions. However, these theorems are "soft" in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is due to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness due to lack of information for how to continue the solution uniquely (this is roughly known as the formation of a Cauchy horizon). In the main part of the course, we will discuss some recent results that, for the first time, eliminate the ambiguity for an open set of initial conditions. The results show in particular that the famous Friedmann-Lemaître-Robertson-Walker (FLRW) solution to the Einstein-scalar field system, which plays a fundamental role in cosmology, is dynamically stable near its Big Bang singularity. In particular, perturbations of the FLRW initial conditions lead to a solution that dynamically develops (in the past) a Big Bang singularity, where the spacetime curvature blows up. Physically, this corresponds to "predicting" (under appropriate assumptions) that a Big Bang happened in the past. At the end of the course, we will discuss some important open problems in the field.
Suggested background reading:
- Chapters 1-4 of General Relativity by Wald
- Sections 6.3-6.4 of Lectures on Nonlinear Hyperbolic Differential Equations by Hörmander