Research / Notes

This page contains notes I have written regarding various research-related topics.

Survey Notes

A Brief Introduction to Mathematical Relativity

These 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.

Part 1: Special Relativity (Original 11/2014; updated 03/2016)
Part 2: General Relativity (Original 08/2016)

Nonlinear Wave Equations

The following are portions of the lecture notes from a postgraduate course on dispersive equations that I co-taught (with Jonathan Ben-Artzi) in 2015. These chapters provide a first introduction to some landmark results on the study of nonlinear wave equations.

Technical Notes

This section contains notes of a more technical nature, such as proofs or explanations of particular theorems or estimates.

Notes on Carleman Estimates

The 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 self-contained 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 self-contained. In particular, it now includes a proof of the sharp Gårding inequality, as well as a few additional microlocal analytic preliminaries.)

Notes on Carleman Estimates (.pdf)
- (Updated 01/2024) Some corrections, added appendix on microlocal analysis background.
- (Original 05/2022)

The Method of Characteristics in PDEs

The following are supplementary notes on topics related to the method of characteristics, used in solving first-order PDEs. These were addenda to a postgraduate PDE module that I taught in 2015 while at Imperial College London.

ODE Theory: C¹-Dependence on Initial Data (.pdf):
- This short note gives a detailed proof of the C¹-dependence of a family of solutions to a system of ODEs on the initial data.
The Cauchy Problem via the Method of Characteristics (.pdf):
- This short note gives a self-contained proof of existence and uniqueness for solutions of fully nonlinear first-order PDE via the method of characteristics.
The Method of Characteristics for Systems and Application to Fully Nonlinear Equations (.pdf):
- This is an alternative presentation of the method of characteristics. In particular, we reformulate fully nonlinear PDEs essentially as a quasilinear system for which all components propagate in the same direction. We show that such systems can be treated using the standard quasilinear method of characteristics.

δ-Distributions in Dispersive Equations

These short notes discuss many of the manipulations involving δ-distributions and pullbacks through δ-distributions 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.

δ-Distributions in Dispersive Equations (.pdf)
- (Original 08/2014)

The Christ-Kiselev Lemma for Dispersive PDE

The Christ-Kiselev Lemma is a maximal-type estimate, which has important applications in dispersive partial differential equations. In particular, this estimate is an important tool in the application of Strichartz estimates.

Christ-Kiselev Lemma (.pdf)
- (Updated 07/2014) More minor corrections.
- (Updated 05/2013) Some minor corrections.
- (Original 02/2013)

Thanks to Yannis Angelopoulos for proofreading and corrections.

Hörmander's Inequality for Wave Equations

These 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.

Hörmander's Inequality (.pdf)
- (Original 10/2012)