PDE (Spring 2018)
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
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
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.
Update (15/5/2017): Correction in Example Sheet 9, Question 2.
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).)
ESI SummER SCHOOL Vienna 2014
Quick Intro to Lorentzian Geometry (Edinburgh 2017)
Holzegel G, Shao A, 2017, Unique continuation from infinity in asymptotically anti-de Sitter spacetimes II: Non-static boundaries, Communications in Partial Differential Equations, Vol:42, ISSN:0360-5302, Pages:1871-1922
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
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