Arick Shao  邵崇哲 
Research / NotesThis page contains notes I have written regarding various researchrelated topics. Survey NotesA Brief Introduction to Mathematical RelativityThese notes, which grew from a short talk given at a postdoc lunch at Imperial College London in 2014, are intended as a brief introduction to the field of mathematical general relativity. The first part discusses the mathematics, in particular the geometry, behind Einstein's theory of special relativity. The second part expands the outlook toward the mathematics behind general relativity.
Nonlinear Wave EquationsThe following are portions of the lecture notes from a postgraduate course on dispersive equations that I cotaught (with Jonathan BenArtzi) in 2015. These chapters provide a first introduction to some landmark results on the study of nonlinear wave equations.
Technical NotesThis section contains notes of a more technical nature, such as proofs or explanations of particular theorems or estimates. Notes on Carleman EstimatesThe following are a further elaboration of some notes I had from a previous reading seminar I ran in 2019. These notes, which are heavily inspired by Nicolas Lerner's monograph on Carleman Equations, give a selfcontained treatment and proof of Hörmander's classical Carleman estimate (from the microlocal point of view) and its application to unique continuation for PDEs. (The newest version of the aims to be entirely selfcontained. In particular, it now includes a proof of the sharp Gårding inequality, as well as a few additional microlocal analytic preliminaries.)
The Method of Characteristics in PDEsThe following are supplementary notes on topics related to the method of characteristics, used in solving firstorder PDEs. These were addenda to a postgraduate PDE module that I taught in 2015 while at Imperial College London.
δDistributions in Dispersive EquationsThese short notes discuss many of the manipulations involving δdistributions and pullbacks through δdistributions found in the study of dispersive equations. The goal is to explain in detail many points that are usually swept under the rug in the standard literature. As an application, we prove a basic bilinear estimate for solutions of the (free) Schrödinger equation.
The ChristKiselev Lemma for Dispersive PDEThe ChristKiselev Lemma is a maximaltype estimate, which has important applications in dispersive partial differential equations. In particular, this estimate is an important tool in the application of Strichartz estimates.
Thanks to Yannis Angelopoulos for proofreading and corrections. Hörmander's Inequality for Wave EquationsThese short notes contains a detailed account of the proof of Hörmander's inequality for wave equations in three spatial dimensions. This estimate played an essential role in establishing small data global existence for nonlinear wave equations satisfying the null condition.
