MEASURE AND INTEGRATION (FALL 2016)
The main text for the course is "Real Analysis" by E. Stein & R. Shakarchi (Princeton University Press). The example sheets are available here
There is also a set of Notes which follow very closely the book. They now contain all material that has been lectured. Please email me for typos and corrections.
Update (14/1/2017): A small section has been added at the end of the notes which is relevant for Mastery Students only.
Update (16/3/2017): Some typos in the Example Sheets (7-9) have been corrected. Solutions to all Sheets will appear here by the end of March.
The first assessed coursework is due November 11th. It consists of Question 1 from Sheet 1, Questions 4+5 from Sheet 2 and Questions 3+4 from Sheet 3. [Returned]
The second assessed coursework is due December 5th. It consists of Questions 1+4 from Sheet 5, Questions 1+3 from Sheet 6 and Question 5 from Sheet 7. [Returned]
FUNCTIONAL ANALYSIS (SPRING 2016)
LECTURENOTES (Notes under construction (last update March 6th 2016).)
PDE (Spring 2013+2014)
PDE examples, ODE review, Picard's theorem, Gronwall's inequality, bootstrap technique
First order PDEs, method of characteristics, Cauchy Problem, Burger's equation, weak solutions (idea + examples)
Cauchy Kovalevskaya Theorem, Holmgren's uniqueness theorem
Proof of Holmgren's theorem. Fritz John's Global Holmgren theorem, domain of dependence, domain of influence, d'Alembert's formula; Week 3 and 4 are in the same pdf.
Distributions I, Fundamental Solution, Laplace's equation, Mean value formulas, Poisson's formula, Maximum Principles
[updated 21/3/2014] Dirichlet problem on a ball, Elliptic Regularity Theory, Existence of Weak Solutions
Boundary Regularity, Fredholm Alternative; Week 6 and 7 are in the same pdf.
[updated 29/3/2014] Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)
[updated 29/3/2014] The Wave equation: Energy Estimate, Domain of Dependence, Fourier Synthesis, Kirchhoff's formula, Duhamel formula, existence and uniqueness for perturbations of the wave operator
Revision Sheet (updated 21/3/2014)
The Mastery Question will be a question combining different techniques that you have seen. It's a good idea to review the section(s) on the wave equation.
ESI SummER SCHOOL Vienna 2014
Quick Intro to Lorentzian Geometry (Edinburgh 2017)
Holzegel G, Dafermos M, Rodnianski I, A scattering theory construction of dynamical vacuum black holes, Journal of Differential Geometry, ISSN:1945-743X
Holzegel G, Schmelzer T, Warnick C, Ricci Flow of Biaxial Bianchi IX Metrics, Classical and Quantum Gravity, ISSN:1361-6382
Holzegel G, 2016, Conservation laws and flux bounds for gravitational perturbations of the Schwarzschild metric, Classical and Quantum Gravity, Vol:33, ISSN:0264-9381
et al., 2016, Small-data shock formation in solutions to 3D quasilinear wave equations: An overview, Journal of Hyperbolic Differential Equations, Vol:13, ISSN:0219-8916, Pages:1-105
Holzegel G, Shao A, 2016, Unique Continuation from Infinity in Asymptotically Anti-de Sitter Spacetimes, Communications in Mathematical Physics, Vol:347, ISSN:0010-3616, Pages:723-775