Imperial College London

DrGustavHolzegel

Faculty of Natural SciencesDepartment of Mathematics

Reader in Pure Mathematics
 
 
 
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Contact

 

g.holzegel CV

 
 
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Location

 

625Huxley BuildingSouth Kensington Campus

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Summary

 

Summary

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

Sheet1, Sheet2, Sheet3Sheet4Sheet5

Sheet6Sheet7Sheet8

There is also a set of Notes which follow very closely the book.

Notes have been updated up to and including lectures on Monday 28/11/16. 

Some typos in Sheet 7 have been corrected.

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.

Sol1sol2Sol3

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.

FUNCTIONAL ANALYSIS (SPRING 2016)

LECTURENOTES (Notes under construction (last update March 6th 2016).)

Addendum: Baire category theorem

Revision Sheet (last updated 22/3/16)

PDE (Spring 2013+2014)

Week1.pdf

PDE examples, ODE review, Picard's theorem, Gronwall's inequality, bootstrap technique

Week2.pdf

First order PDEs, method of characteristics, Cauchy Problem, Burger's equation, weak solutions (idea + examples)

Week3.pdf 

Cauchy Kovalevskaya Theorem, Holmgren's uniqueness theorem 

Week4.pdf

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.

Week5.pdf

Distributions I, Fundamental Solution, Laplace's equation, Mean value formulas, Poisson's formula, Maximum Principles

Week6.pdf

[updated 21/3/2014] Dirichlet problem on a ball, Elliptic Regularity Theory, Existence of Weak Solutions

Week7.pdf

Boundary Regularity, Fredholm Alternative; Week 6 and 7 are in the same pdf.

Week8+9.pdf 

[updated 29/3/2014] Basic Fourier-Analysis, Schroedinger equation, Non-linear Schroedinger, Heat equation (fundamental solution, boundary initial value problem, maximum principle)

Week9+10.pdf 

[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

 

More Information

Lectures 1-5 

 

WORKSHOP ON HYPERBOLIC PDE -- SEPT 30TH--OCT 2ND 2015

Publications

Journals

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

Holzegel G, Klainerman S, Speck J, 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

More Publications