Slides

Here are slides from some of my previous talks.

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Shock Formation for the 2D Compressible Euler Equations in the Presence of Non-zero Vorticity This seminar-style talk concerns my recent joint work with J. Luk in which we proved a sharp stable shock formation result for the 2D compressible Euler equations. The work is significant for three reasons: i) We allow the vorticity to be non-zero at the shock. Thus, we provide the first constructive description of the behavior of vorticity all the way up to the first singularity formed from compression. ii) Our work provides the first proof of shock formation without symmetry assumptions for a hyperbolic system featuring more than one speed of propagation. iii) Our proof relies on a new formulation of the compressible Euler equations that exhibits remarkable structures and is therefore of independent interest.
An Overview of Recent Progress on Shock Formation in Three Spatial Dimensions This is a colloquium-style lecture that provides context and background for the shock-formation results of the next talk.
Shock Formation in Solutions to 3D Wave Equations

This seminar-style talk addresses my recent book on shock formation and also the corresponding survey article, which is joint with Gustav Holzegel, Sergiu Klainerman, and Willie Wong. We showed that Christodoulou's framework for proving shock formation, which was put forth in his remarkable 1000 page monograph, can be extended to apply to several large classes of quasilinear wave equations in 3D. Specifically, for the equations under consideration, we prove that small-data shock formation occurs if and only if Klainerman's null condition fails. We also give a very sharp description of the singularity and the blow-up mechanism.

Stable Big Bang Formation in Solutions to the Einstein-Scalar Field System This talk concerns joint work with Igor Rodnianski. We showed that the well-known FLRW solution to the Einstein-scalar field system is globally stable in the collapsing direction, towards the Big Bang. That is, we proved a stable blow-up result for near-FLRW data, without symmetry assumptions. The same result holds for the Einstein-stiff fluid system (a stiff fluid has a sound speed equal to the speed of light). The most interesting and important aspect of the proof is the new form of L2-type monotonicity that we discovered, which holds in the near-FLRW regime.
Geometric Methods in Hyperbolic PDEs This talk is a brief overview of some of my recent work on global stability problems for various hyperbolic PDEs. The talk focuses on the role that geometry plays in the analysis.
The Stability of the Euler-Einstein System with a Positive Cosmological Constant This talk is an extension of the results addressed in the next talk. The new result is the elimination of the assumption that the fluid is irrotational.
The Stability of the Irrotational Euler-Einstein System with a Positive Cosmological Constant This talk concerns joint work with Igor Rodnianski. We showed that the well-known family of FLRW family of solutions to the Euler-Einstein system is globally future-stable when the cosmological constant is positive, the speed of sound verifies certain assumptions, and the fluid is irrotational. The FLRW solutions lie at the heart of many of the most fundamental predictions of cosmology.
The Global Stability of the Minkowski Spacetime Solution to the Einstein-Nonlinear Electromagnetic System in Wave Coordinates

This talk concerns a global stability result for the trivial solution to the coupled Einstein-nonlinear electromagnetic system in 1 + 3 dimensions. The result is an extension of the well-known Christodoulou-Klainerman proof of the stability of Minkowski spacetime. As in the next talk, the nonlinear electromagnetic matter models are interesting because of their connection to the electromagnetic divergence problem.

The Nonlinear Stability of the Maxwell-Born-Infeld System This talk addresses a global stability result for the 0 solution to the Maxwell-Born-Infeld system of electrodynamics, which is a nonlinear version of Maxwell's equations. This system plays a role in physicists' efforts to fix the electromagnetic divergence problem. This problem is the disappointing fact that the motion of an electron due to its own self-influence is ill-defined in standard Maxwell theory.