- Second order elliptic PDEs
- -De Giorgi's theorem on $C^\alpha$ regularity, Harnack principles, Krylov-Safonov estimate.
- Advanced PDE 1
- -This is a first year course for PhD students for the local CDT.
Syllabus and notes.
- Real Analysis
- - Covering theorems, maximal functions, Lebesgue's differentiation theorem, singular integrals,
$L^p$ estimates, generalised functions.
- Calculus of variations and differential geometry
- - Theory of curves in $\mathbb R^2$ and $\mathbb R^3$. Hypersurfaces in $\mathbb R^3$.
- Combinatorics and graph theory
- - Basic counting laws, pigeon hole principle, recurrence relations, elements of graph theory.
- Metric spaces
- - Compactness, completeness, contraction and the fixed point theorem.
- Linear Analysis
- - Hilbert and Banach spaces, spectral theorem, Schwartz classes and distributions.
- Mathematics for Engineers
- - Basic ordinary differential equations and applications.