• Home
• Introduction
• Research
  • Publications
  • Notes
  • Presentations
  • Miscellaneous
• Teaching
  • Undergraduate
  • Postgraduate
  • Miscellaneous
• Other Interests
• About Me

• Teaching
• • Undergraduate
• • Postgraduate
• • Miscellaneous

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:

1. Review of ODE theory (existence, uniqueness, continuous dependence)
2. Method of characteristics (first-order scalar PDE)
3. Analytic PDE, power series solutions (Cauchy-Kovalevskaya and Holmgren theorems)
4. Weak derivatives, weak solutions of PDE
5. The Laplace and Poisson equations
6. The heat equation
7. The wave equation
8. 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.

• 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.

TCC: Dispersive Equations

• Position: Co-instructor
• Location: Taught Course Centre (Imperial College London)
• Term: Fall 2015

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:

1. (AS) Ordinary differential equations, connections to evolutionary PDE
2. (JBA) PDE preliminaries (Fourier transforms, Sobolev spaces), linear transport equations (method of characteristics), introduction to kinetic theory
3. (JBA) The Vlasov-Poisson system: local existence and uniqueness
4. (JBA) The Vlasov-Poisson system: global existence
5. (AS) Linear wave equations: physical and Fourier representation formulas, energy and dispersive estimates
6. (JBA) The Vlasov-Maxwell system: conditional global existence
7. (AS) Nonlinear wave equations: classical local existence and uniqueness
8. (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:

• Week 1: ODEs and Connections to Evolution Equations (.pdf)
• Week 5: Linear Wave Equations (.pdf)
• Week 7: Nonlinear Wave Equations: Classical Existence and Uniqueness (.pdf)
• Week 8: Nonlinear Wave Equations: Vector Field Methods, Global and Long-time Existence (.pdf)

(Much thanks to Vaibhav Jena for proofreading and finding errors within the notes.)