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)

Chapter 2 (Vector spaces of functions)

Chapter 3 (Fourier decomposition of functions)

Chapter 4 (Generalised derivative of functions) (draft)

Mock exam (three “section II” questions resp. in Papers 1, 3, 4)

Example sheet 1

Example sheet 2

Example sheet 3

Birth of a Theorem… Villani on the BBC

Birth of a TheoremCédric Villani was on BBC 2 on the occasion of the publication of the book “Birth of a Theorem” (translation of his book “Théorème vivant” describing the making of our theorem).
Here is the link (Villani starts at 37:45):
http://www.bbc.co.uk/iplayer/episode/b055v6pf/newsnight-11032015
Also, BBC RADIO 4 has a series of programs with Cedric Villani this week:
1. The Mathematical Mind with Cedric Villani: http://www.bbc.co.uk/programmes/b054pj6y
2. Book of the week (Birth of a Theorem by Cedric Villani):
Episode 1: http://www.bbc.co.uk/programmes/b054pmgr
Episode 2: http://www.bbc.co.uk/programmes/b054qc1m
Episode 3: http://www.bbc.co.uk/programmes/b054qd81
Episode 4: http://www.bbc.co.uk/programmes/b054t3s4
Episode 5: http://www.bbc.co.uk/programmes/b054tl6z