Teaching / Postgraduate
Below are the courses that I have taught at the (post)graduate level.
M4P41: Analytic Methods in PDEs
 Position: Instructor
 Location: Imperial College London
 Term: Spring 2016
This is a fourthyear undergraduate course in partial differential equations (PDEs) and the mathematical methods used to study them.
Approximate list of topics covered:
 Review of ODE theory (existence, uniqueness, continuous dependence)
 Method of characteristics (firstorder scalar PDE)
 Analytic PDE, power series solutions (CauchyKovalevskaya and Holmgren theorems)
 Weak derivatives, weak solutions of PDE
 The Laplace and Poisson equations
 The heat equation
 The wave equation
 Mastery content: Existence and uniqueness for the cubic nonlinear Schrödinger equation.
The following supplementary notes covered a couple additional topics that were not fully covered in lecture.
TCC: Dispersive Equations
This is a graduatelevel partial differential equations (PDE) course, a collaborative effort with Jonathan BenArtzi, which introduced the following two areas of study:
 Kinetic theory (JBA): transport equations, classical theory of VlasovPoisson and VlasovMaxwell equations.
 Wave equations (AS): linear waves, classical theory of nonlinear wave equations.
The lectures were broadcast to universities in the TCC network: University of Bath, University of Oxford, University of Bristol, Imperial College London, University of Warwick.
Approximate list of topics covered, by week:
 (AS) Ordinary differential equations, connections to evolutionary PDE
 (JBA) PDE preliminaries (Fourier transforms, Sobolev spaces), linear transport equations (method of characteristics), introduction to kinetic theory
 (JBA) The VlasovPoisson system: local existence and uniqueness
 (JBA) The VlasovPoisson system: global existence
 (AS) Linear wave equations: physical and Fourier representation formulas, energy and dispersive estimates
 (JBA) The VlasovMaxwell system: conditional global existence
 (AS) Nonlinear wave equations: classical local existence and uniqueness
 (AS) Nonlinear wave equations: the vector field method, smalldata global and longtime existence
My contributions to the lecture notes can be found below:
(Much thanks to Vaibhav Jena for proofreading and finding errors within the notes.)
