Arick Shao 邵崇哲

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.

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.)

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