# Analysis of Functions (Part II D-course)

The analysis of functions has its roots in the rigorous study of the equations of mathematical physics, and is now a key part of modern mathematics. This course builds on the Part II courses Linear Analysis and Probability & Measure, applying the theory of integration and the tools of functional analysis to explore such topics as Lebesgue and Sobolev spaces, the Fourier transform and the generalised derivative. These topics are important and interesting in themselves, but the emphasis is on their use in other areas of mathematics (for instance in the representation of functions and in partial differential equations), rather than their maximal generalisation. You can get an idea of the flavour of the course by browsing the Analysis by Lieb & Loss (Springer).

Prerequisites: A good understanding of the methods and results in the IB courses Linear Algebra, Analysis II and Metric and Topological Spaces is essential. Part II Linear Analysis and integration theory from Part II Probability and Measure are essential.

Syllabus 2016

Chapter 1 (Integration of Functions) 2018

Chapter 2 (Vector spaces of Functions) 2018

Chapter 3 (Fourier decomposition of functions)

Chapter 4 (Generalised derivative of functions)

Mock exam 2017

Exam 2017

Example sheet 1 2018

Example sheet 2 2018

Example sheet 3

# Mathematical Topics in Kinetic Theory (Part III course)

### — Part III, michaelmas terms 2010 & 2011 —

Course description (2011-2012)

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6 (soon!)

Example sheet 1 (chapter 2)

Example sheet 2 (chapter 3)

Example sheet 3 (chapter 4)

Example sheet 4 (chapter 5)

Exam paper 2011

Exam paper 2012

Bibliography

Some links:

Some elements of history of Kinetic Theory (up to 2001) from the webpage of Dave Levermore.

Attack of the Boltzmann Brains (Jim Kakalios)

Sources: Some parts of these lecture notes owe a lot to two sets of hand-written lecture notes on kinetic equations written by Laure Saint-Raymond and Laurent Desvillettes, as well as to the lecture notes available online of G. Allaire and F. Golse on transport and diffusion processes. They also inspire from the classical books of R. Glassey, of C. Cercignani, and of C. Cercignani – R. Illner – M. Pulvirenti, as well as from various recent research papers. Many thanks to Otis Chodosh, who typewrote a first latex version of these notes.

# Analysis of Partial Differential Equations

### — Part III & CCA graduate course, michaelmas term 2016 —

Course description

Examinable syllabus

Lectured with Mihalis Dafermos

Example classes taught by Georgios Moschidis

Example sheet 1

Example sheet 2

Example sheet 3

Example sheet 4

This course also forms part of the first year programme of the CCA, where it is supplemented with extra projects of a more advanced nature. Midterm assignments for CCA students are available here. (Midterm presentations took place on Monday 7 November 2016.) The four longer assignments for CCA students are given below:

### — Part III & CCA graduate course, michaelmas term 2014 —

Course description

Example classes taught by Harsha Hutridurga

Wikipedia commons

Chapter 1 (updated 2014): From ODEs to PDEs

Example sheet of chapter 1 (updated 2014)

Chapter 2 (updated 2014): The Cauchy-Kovalevskaya theorem

Example sheet of chapter 2 (updated 2014)

Chapter 3 (update 2014 in progress): Ellipticity

Example sheet of chapter 3 (updated 2014)

Chapter 4: Hyperbolicity

Example sheet of chapter 4 (updated 2014)

Midterm assignements 2013

Midterm assignements 2014

Mock Exam 2014

Exam 2014

Additional material:

Remarks on the prehistory of Sobolev spaces, J. Naumann

Notes on Sobolev spaces, W. Wong

Final assignements 2012-2013:

Existence and Regularity of Solutions to the Minimal Surface Equation (Tom Begley, Henry Jackson, James Mathews, Parousia Rockstroh)

Existence and Uniqueness for the Vacuum Einstein Equations (Rob Hocking, Tim Talbot, Marcus Webb)

The Cauchy problem for the Boltzmann equation (Alexander Bastounis, Thomas Holding, Vittoria Silvestri)

Heat Flow Methods in Geometry and Topology (Helge Dietert, Kim Moore, Parousia Rockstroh, Giles Shaw)

Final assignements 2013-2014:

An Overview of the Cauchy Problem for the Vlasov-Poisson Equation (Ben Jennings, Adam B Kashlak, Davide Piazzoli)

Existence and Continuation for Euler Equations (David Driver and Harold Lim)

Gaussian beams and the propagation of singularities (Dominic Dold, Karen Habermann, Ellen Powell)

Leray–Schauder Existence Theory for Quasilinear Elliptic Equations (Sam Forster, Eavan Gleeson, Franca Hoffmann)

Sources:

Some significant parts of these lecture notes are borrowed from some lecture notes previously written by Mihalis Dafermos for this CCA course, and from some lecture notes I wrote for a similar course I gave in Paris-Dauphine 2006-2009. The lecture notes are also strongly inspired by the books of C. Evans, E. Lieb & M. Loss, W. Strauss, F. John, by the review paper Partial differential equations of S. Klainerman, and by various lecture notes available online by L. Boutet de Monvel, B. Driver, B. Helffer, G. Tsogtgerel and C. Villani.