59th North British Mathematical Physics Seminar

Abstract: Causal set theory is a framework for quantum gravity where the spacetime continuum is replaced by a locally finite partial ordered set of spacetime events. Given a continuum spacetime manifold, I summarize the construction of the probability measure for the Poisson process called sprinkling that generates causal sets. I present results for the discretization of quantum fields and their equations of motion on a causal set. The outcomes of this project are used to formulate quantum fields on causal sets in the rigorous framework of algebraic quantum field theory. This talk is based on joint work with C. J. Fewster, E. Hawkins, and K. Rejzner.

Abstract: The non-linear Schrödinger (NLS) equation is one of the fundamental and widely studied equations in mathematical physics with broad application in physical problems, such as the non-linear optics and ocean waves. In this talk, we will present some recent results on the semi-discrete matrix NLS-like (DNLS) hierarchy. More precisely, we will construct the Lax pairs for the whole hierarchy via the dressing method and derive the corresponding soliton solutions via suitable choices of Darboux transformations.

Abstract: I will describe joint work with Dietrich Häfner and András Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild or slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. Our proof uses a detailed description of the resolvent of an associated wave equation on symmetric 2-tensors near zero energy.