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 fourth-year 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 (first-order scalar PDE)
- Analytic PDE, power series solutions (Cauchy-Kovalevskaya 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 graduate-level partial differential equations (PDE) course, a collaborative effort with Jonathan Ben-Artzi, which introduced the following two areas of study:
- Kinetic theory (JBA): transport equations, classical theory of Vlasov-Poisson and Vlasov-Maxwell 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 Vlasov-Poisson system: local existence and uniqueness
- (JBA) The Vlasov-Poisson system: global existence
- (AS) Linear wave equations: physical and Fourier representation formulas, energy and dispersive estimates
- (JBA) The Vlasov-Maxwell system: conditional global existence
- (AS) Nonlinear wave equations: classical local existence and uniqueness
- (AS) Nonlinear wave equations: the vector field method, small-data global and long-time existence
My contributions to the lecture notes can be found below:
(Much thanks to Vaibhav Jena for proofreading and finding errors within the notes.)