Teaching

Second order elliptic PDEs: -De Giorgi's theorem on $C^\alpha$ regularity, Harnack principles, Krylov-Safonov estimate. Notes.

Advanced PDE 1: -This is a first year course for PhD students for the local CDT. Syllabus and notes.

Syllabus
1) Examples of elliptic equations, maximum principles (strong weak), Hopf's Lemma, comparison principle.
2) Classical solutions, Bernstein's gradient estimate, applications.
3) Schauder estimates.
4) Approximation by smooth functions, Sobolev spaces, embeddings, traces.
5) Weak solutions, Lax-Milgram.
6) Interior regularity, Boundary regularity.
7) Parabolic equations, main examples, maximum principle.
8) Parabolic setting and Sobolev spaces.
9) Energy estimates
10) Global in time solutions for nonlinear parabolic equations with small initial data.

Prerequisites
1) Rigorous multivariable calculus (continuity, differentiability, chain rule, integration)
2) Metric spaces, Banach spaces, Hilbert space, weak/strong convergence
3) Vector calculus, Green's formula, (normal, tangent/vectors, parametrisation of surfaces and curves.)

We propose 4 (2+2) sets of homework

The suggested text book is L.C. Evans, Partial Differential Equations, AMS Graduate Studies in Mathematics.

Some material will be from elsewhere.

Notes-1

Notes-2

Notes-3 (Schauder estimates)

Notes-4 (Weak solutions)

Homework 1

Homework 2

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 $\mathbb R^3$. Notes.

Part 1, Part 2.

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. Course page.

Mathematics for Engineers: - Basic ordinary differential equations and applications.