We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

We will discuss shock reflection phenomena, mathematical formulation of shock reflection problem, structures of  shock reflection configurations, and von Neumann conjectures on transition between regular and Mach reflections. Then we will describe the results on existence and properties of regular reflection solutions for potential flow equation. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear  elliptic equation in self-similar coordinates, where the reflected shock is the free boundary, and ellipticity degenerates near a part of a fixed boundary. We will discuss the techniques and methods used in the study of such free boundary problems.

 

I will discuss the free compact boson CFT thrown out of equilibrium by marginal deformations, modeled by quenching or periodically driving the compactification radius of the free boson between two different values. All the dynamics will be shown to be crucially dependent on the ratio of the compactification radii via the Zamolodchikov distance in the space of marginal deformations. I will present various exact analytic results for the Loschmidt echo and the time evolution of energy density for both the quench and the periodic drive. Finally, I will present a non-perturbative computation of the  Rényi divergence, an information-theoretic distance measure, between two marginally deformed thermal density matrices.

The talk will be based on the recent preprint: arXiv:2206.11287

In General Relativity, an impulsive gravitational wave is a localized and singular solution of the 

Einstein equations modeling the spacetime distortions created by a strongly gravitating source.
I will present a comprehensive theory allowing for ternary interactions of such impulsive gravitational waves in translation-symmetry, offering the first examples of such an interaction.  

The proof combines new techniques from harmonic analysis, Lorentzian geometry, and hyperbolic PDEs that are helpful to treat highly anisotropic low-regularity questions beyond the considered problem.  

This is joint work with Jonathan Luk.

Motivated by the physical relevance of many Hodge Laplace-type PDEs from the finite element exterior calculus, we analyse the Hodge Laplacian boundary value problem arising from the strain space in the linear elasticity complex, an exact sequence of function spaces naturally arising in several areas of continuum mechanics. We propose a discretisation based on the adaptation of discontinuous Galerkin FEM for the incompatibility operator $\mathrm{inc} := \mathrm{rot}\circ\mathrm{rot}$, using the symmetric-tensor-valued Regge finite element to discretise  the strain field; via the 'Regge calculus', this element has already been successfully applied to discretise another metric tensor, namely that arising in general relativity. Of central interest is the characterisation of the associated Sobolev space $H(\mathrm{inc};\mathbb{R}^{d\times d}_{\mathrm{sym}})$. Building on the pioneering work of van Goethem and coauthors, we also discuss promising connections between functional analysis of the $\mathrm{inc}$ operator and Kröner's theory of intrinsic elasticity in the presence of defects.

This is based on ongoing work with Dr Kaibo Hu.

Regularization of linear least squares problems is necessary in a variety of contexts. However, the optimal regularization parameter is usually unknown a priori and is often to be determined in an ad hoc manner, which may involve solving the problem for multiple regularization parameters. In this talk, we will discuss three randomized algorithms, building on the sketch-and-precondition framework in randomized numerical linear algebra (RNLA), to efficiently solve this set of problems. In particular, we consider preconditioners for a set of Tikhonov regularization problems to be solved iteratively. The first algorithm is a Cholesky-based algorithm employing a single sketch for multiple parameters; the second algorithm is SVD-based and improves the computational complexity by requiring a single decomposition of the sketch for multiple parameters. Finally, we introduce an algorithm capable of exploiting low-rank structure (specifically, low statistical dimension), requiring a single sketch and a single decomposition to compute multiple preconditioners with low-rank structure. This algorithm avoids the Gram matrix, resulting in improved stability as compared to related work.

Time-optimal control can be used to improve driving efficiency for autonomous
vehicles and it enables us explore vehicle and driver behaviour in extreme
situations. Due to the computational cost and limited scope of classical
optimal control methods we have seen new interest in applying reinforcement
learning algorithms to autonomous driving tasks.
In this talk we present methods for translating time-optimal vehicle control
problems into reinforcement learning environments. For this translation we
construct a sequence of environments, starting from the closest representation
of our optimisation problem, and gradually improve the environments reward
signal and feature quality. The trained agents we obtain are able to generalise
across different race tracks and obtain near optimal solutions, which can then
be used to speed up the solution of classical time-optimal control problems.