Arick Shao 邵崇哲

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.

## 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:

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:

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