``An educated person in Nottingham is as precious and rare a find as jam in a
wartime doughnut'' (Grahame Greene)
Ergodic Theory (MA 427)
I am teaching this course for 4th year Mathematics students at
Warwick University.
The lectures (in term 2) are: Tuesday 16-17 (MS.04); Wednesday 12-13 (B3.03);
Thursday 14-15 (B3.03).
An important prerequisite is measure theory.
Here is an informal summary of
those topics it would be useful to know:
Summary of measure theory
Below are scans of my hand written lecture notes (often corrected as the
course progressed)
lecture 1 (Overview)
lecture 2 (Invariant measures, Examples, Existance)
lecture 3 (Poincare Recurrence Theorem,
Multiple Recurrence, Geodesic and horocycle flows)
lecture 4 (Ergodic measures, Examples, Existance)
lecture 5 (Kac's Theorem, von Neumann Ergodic Theorem)
lecture 6 (Birkhoff Ergodic Theorem)
lecture 7 (Subadditive ergodic theorem)
lecture 8 (Ergodicity of flows and Hopf
method)
lecture 9 (Spectral Theory)
lecture 9.5 (Entropy)
lecture 10 (SMB theorem. OW theorem)
lecture 11 (Markov operators and
Dunford-Schwartz-Hopf Ergodic Theorem)
lecture 12 (Pressure and equilibrium states)
Below are additional references
Paper of Kaznelson-Weiss: Short proof of
ergodic theorem (L^1 functions and invariant measures)
Appendix of Parry
book: Proof of spectral theorems
Below are typed lecture notes (often corrected as the
course progressed)
lecture Notes: First part
lecture Notes: Second part
lecture Notes: All the notes
Example and Solution sheets:
Example sheet 1;
Example sheet 2;
Example sheet 3;
Example sheet 4;
Example sheet 5; ... Probably to be revised
Example sheet 6;
Example sheet 7;
Example sheet 8;
Complex Analysis (MA 3B8)
I am teaching this course for 3rd year Mathematics students at
Warwick University.
The lectures (in term 2) are: Monday 9-10 (H.0.51); Wednesday 9-10 (MS.01);
Friday 10-11 (H0.51).
Below are scans of my hand written lecture notes (often corrected as the
course progressed)
lecture 1 (Overview)
lecture 2 (Mobius maps)
lecture 3 (Applications of Mobius maps) and
an article by Sarnak on
Circle packings
lecture 4 (Analyticity, Cauchy-Riemann equations)
lecture 5 (Application:Non-acculumulation
of zeros)
lecture 6 (Integrals, Theorems of Gousart
and Cauchy)
lecture 7 (Applications of Theorems of Gousart
and Cauchy)
Example and Solution sheets:
Example sheet 1
Example sheet 2
Example sheet 3
Analysis I (MA131)
I taught this course for 1st year Mathematics students.
Here are copies of two previous exams:
Exam 1,
Exam 2
Here are copies of:
workbook 1, assignment book
1
workbook 2, assignment book
2
workbook 3, assignment book
3
workbook 4, assignment book
4
workbook 5, assignment book
5
workbook 6, assignment book
6
workbook 7 assignment book
7
Dynamical Systems (MA428)
This was a 4th year course I taught, whose main prerequisite was Metric Spaces.
Below are scans of my personal lecture notes
(often corrected as the course progressed):
lecture 1 (The introductory lecture)
lecture 2 (Rotations, minimality,transitivity, leading digit
of 2^n)
lecture 3 (Period 3 implies chaos)
lecture 4 (Rotations numbers)- plus a note on lifts.
lecture 5 (Minimal homeomorphisms and Conjugacy)
lecture 6 (Denjoy's Theorem)
lecture 7 (Expanding maps, transitivity [again], mixing, periodic points)
lecture 8 (Sensitive dependence, Quadratic maps)
lecture 9 (Subshifts of finite type, mixing, periodic points)
lecture 10 (Aside: Seach Engines)
lecture 11 (Two sided shifts, minimality
[again], van der Waerden's theorem) - plus a typed version
lecture 12 (Coding expanding maps of the circle using shifts)
lecture 13 (Coding Smale horsehoes, toral
automorphisms and solenoids)
lecture 14 (Toral automorphisms [again],
periodic points, the Shadowing property) plus a
note on lyapunov exponents
lecture 15 (Structural stability)
lecture 16 (Topological entropy, spanning sets
and seperating sets)
lecture 17 (Examples of Topological entropy:
shift maps, toral automorphisms, billiards)
lecture 18 (Rational Maps, the Julia set,
the Mandelbrot set)
Here are also some Example sheets (and Solution sheets):
Example sheet 0.5
and
Solution sheet 0.5
Example sheet 1 and
Solution sheet 1
Example sheet 2 and
Solution sheet 2
Example sheet 3
and
Solution sheet 3
Example sheet 3.5
Example sheet 4
Example sheet 5
I gave
a talk in Durham (January 2009) to encourage students to study
smooth ergodic theory .
In October 2009 I gave two 10 minute presentations to 4th year undergraduates
about
analysls courses at Warwick
and applying to do a PhD at Warwick