London PDE Seminar

$$R_{ a b } - \frac{1}{2} R \cdot g_{ a b } + \Lambda \cdot g_{ a b } = T_{ a b } \text{.}$$

The London PDE Seminar is a London-based seminar featuring speakers at the forefront of mathematical research in partial differential equations.

### Upcoming Seminars

The seminar convenes fortnightly, on approximately every other Friday. Some talks will be held in-person, while others will be held on Zoom.

 Date/Time: 11:15 (UK time), Friday, 27 January 2023 Location: Queen Mary University of London (in-person) —Room: 1.02 iQ (formerly Scape) Building (Google maps) Speaker: Christopher Alexander (UCL) Title: General Relativistic Shock Waves that Exhibit Cosmic Acceleration Abstract: This talk concerns the construction and analysis of a new family of exact general relativistic shock waves. The construction resolves the open problem of determining the expanding waves created behind a shock-wave explosion into a static isothermal sphere with an inverse square density and pressure profile. The construction involves matching two self-similar families of solutions to the perfect fluid Einstein field equations across a spherical shock surface. The matching is accomplished in Schwarzschild coordinates where the shock waves appear one derivative less regular than they actually are. Separately, both families contain singularities, but as matched shock-wave solutions, they are singularity free. This construction is also accompanied by a novel existence proof in the pure radiation case. These shock-wave solutions represent an intriguing new mechanism in General Relativity for exhibiting accelerations in asymptotically Friedmann spacetimes, analogous to the accelerations modelled by the cosmological constant in the Standard Model of Cosmology. However, unlike in the Standard Model, these shock-wave solutions solve the Einstein field equations in the absence of a cosmological constant, opening up the question of whether a purely mathematical mechanism could account for the cosmic acceleration observed today, rather than dark energy. Date/Time: 11:15 (UK time), Friday, 10 February 2023 Location: Queen Mary University of London (in-person) —Room: 1.02 iQ (formerly Scape) Building (Google maps) Speaker: Arthur Touati (IHES) Title: Geometric optics approximation for the Einstein vacuum equations Abstract: In this talk I will present recent work on the rigorous justification of the geometric optics approximation for the Einstein vacuum equations, and its link with the Burnett conjecture in general relativity. I will start by presenting the initial value problem for the Einstein vacuum equations formulated in wave coordinates. Then I will give the state of the art on the Burnett conjecture, focusing on the approaches in U(1) symmetry and double null gauge. I will then present my main result and sketch its proof, highlighting the quasi- and semi-linear challenges. I will conclude my talk by discussing other quadratic wave equations. Date/Time: 11:15 (UK time), Friday, 24 February 2023 Speaker: Hamed Masaood (Imperial) Date/Time: 11:15 (UK time), Friday, 10 March 2023 Speaker: Tobias Barker (Bath) Date/Time: 11:15 (UK time), Friday, 24 March 2023