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Expositions

This page contains a number of notes and articles—intended for various audiences and pitched at various levels—that I have written over the years.

Foundational Questions

You're a mathematician? What do you actually study?

I occasionally get asked this. Though mathematicians know well what they do, it remains a tough question to answer. The dilemma is that the question has either a five-second answer (which is too short to be helpful) or a fifteen-minute answer (which tries the audience's patience).

The problem is that a wide gulf has developed between what is taught in mathematics curricula and what mathematicians actually think about, so that "mathematician" has become one of the most misunderstood professions. Unfortunately, neither mathematicians themselves nor the education system have been so effective at bridging that gap. Nowadays, when university students progress from lower-level courses (e.g., calculus, linear algebra) into the rigorous advanced courses (e.g., real analysis, abstract algebra), they basically must learn a whole new field and an entirely new way of thinking.

The following articles are an attempt to give "fifteen-minute answers" to the question—"What do mathematicians actually study?"—and to other tangentially related questions. Of course, this comes with a disclaimer: what is written is merely my own views and is not definitive, and other mathematicians may have different perspectives.

• What Do Mathematicians Do? (Added 07/2014; updated 04/2017)
• Why Prove Things? (Added 07/2014; updated 04/2017)
• Why Study Abstract Things? (Added 07/2014; updated 04/2017)
• What Are the Foundations of Mathematics? (Added 08/2017)

Outreach Articles

Below are some additional expository articles on miscellaneous topics.

• 1 + 2 + 3 + 4 + 5 + ... = WTF? (Added 05/2014)

Teaching Material

The following notes are intended for undergraduate and postgraduate students, containing material of difficulty comparable to taught university mathematics curricula.

Undergraduate Lecture Notes

Below are the full lecture notes from the 2nd-year undergraduate course MTH5113: Introduction to Differential Geometry (2022/23), at Queen Mary University of London.

• Lecture Notes, Introduction to Differential Geometry (.pdf) (updated 2023)
  • This module combines the study of curves and surfaces with vector calculus.

Below are the full lecture notes from the 2nd-year undergraduate course MTH5109: Geometry II, Knots and Surfaces (2017/18), at Queen Mary University of London.

• Lecture Notes, Geometry II, Knots and Surfaces (.pdf) (updated 2017)
  • The module combines the study of curves and surfaces with a bit of knot theory.

The following is the first chapter of unfinished lecture notes from the undergraduate course MAT334: Complex Analysis, at University of Toronto.

• A Basic Guide to Complex Variables (.pdf) (updated 09/2013)
  • This gives a very short introduction to the complex number system.

Postgraduate Lecture Notes

The following are chapters of lecture notes from a PhD-level course Dispersive Equations, which I co-taught (with Jonathan Ben-Artzi in 2015) while at Imperial College London.

• ODEs and Connections to Evolution Equations (.pdf) (updated 2015)
  • The chapter provides a rigorous study of ODEs as warm-up and motivation for dispersive and wave equations.
• Linear Wave Equations (.pdf) (updated 2015)
  • The chapter covers basic properties of linear wave equations, from both the physical and Fourier perspectives.
• Nonlinear Wave Equations: Classical Existence and Uniqueness (.pdf) (updated 2015)
  • The chapter covers local well-posedness results for some nonlinear wave equations.
• Nonlinear Wave Equations: Vector Field Methods, Global and Long-time Existence (.pdf) (updated 2015)
  • The chapter gives an informal survey of small-data global and long-time well-posedness results for nonlinear wave equations.

The following are supplementary short notes from the postgraduate course M4P41: Analytic Methods in PDEs, at Imperial College London,

• ODE Theory: \(C^1\)-Dependence on Initial Data (.pdf) (updated 2016)
  • This short note proves the \(C^1\)-dependence of a family of solutions to a system of ODEs on the initial data.
• The Cauchy Problem via the Method of Characteristics (.pdf) (updated 2016)
  • 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) (updated 2016)
  • 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. Such systems can then be treated using the standard quasilinear method of characteristics.

Miscellaneous Notes

The following are supplementary notes for learning control theory of ODEs.

• The Eigenvalue Placement Problem (original 02/2025)
  • These short notes give a rigorous mathematical treatment on the eigenvalue placement problem for linear systems of ODE with constant coefficients.

Research Material

The following notes cover more specialized research topics, and are mostly intended for PhD students and beyond.

Survey Notes

The following notes, which grew from a short talk given at a postdoc lunch at Imperial College London in 2014, are intended as a very brief introduction to the field of mathematical general relativity.

• A Brief Introduction to Mathematical Relativity—Part 1: Special Relativity (Original 11/2014; updated 03/2016)
  • The first part discusses the mathematics, in particular the geometry, behind Einstein's theory of special relativity.
• A Brief Introduction to Mathematical Relativity—Part 2: General Relativity (Original 08/2016)
  • The second part expands the outlook toward the mathematics behind general relativity.

Advanced Topics

The following notes give an entirely 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. These are heavily inspired by Nicolas Lerner's monograph on Carleman Equations and arose from a reading seminar I ran in 2019. The latest version also includes an appendix with microlocal analytic preliminaries and a proof of the sharp Gårding inequality.

• Notes on Carleman Estimates (.pdf) (Original 05/2022; updated 01/2024)

Miscellaneous Topics

The following short notes discuss manipulations involving δ-distributions and pullbacks through δ-distributions that are found in dispersive PDEs, explaining in detail some 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 following short notes provide a proof of the Christ-Kiselev Lemma—a maximal-type estimate with important applications in dispersive partial differential equations, in particular for Strichartz estimates.

• The Christ-Kiselev Lemma (.pdf) (Original 02/2013; updated 07/2014)
  • Thanks to Yannis Angelopoulos for proofreading and corrections.

These short notes contains a detailed account of the proof of Hörmander's inequality for wave equations in three spatial dimensions. Though now outdated, this estimate previously played a key role in proving small data global existence for nonlinear wave equations satisfying the null condition.

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