``An educated person in Nottingham is as precious and rare a find as jam in a wartime doughnut'' (Grahame Greene)



Hyperbolic Geometry (MA 448)

I am taught this course for 4th year Mathematics students at Warwick University.

The lectures (in term 2) are: Tuesday 11-12 (B3.02); Thursday 1-2 (B3.02); Friday 11-12 (B3.02).

Below are scans of my hand written lecture notes (added/corrected as the course progresses).
lectures 1-3 (Overview, History and Euclid's 5th postulate, Poincare upper half plane, lengths of curves and distance, boundary of hyperbolic space, Mobius maps and isometries)

lectures 4-12 (Geodesics, Explicit form for metric,Hyperbolic trianges, Cosine and Sine laws, Gauss-Bonnet, Cross ratio, Poincare disk model, lengths of circles and areas of balls)

lectures 13-18 (Tilings and tessellations, triangle groups, cross rations and distance, three dimensional hyperbolic space, more cross ratios, volumes of ideal tetrahedra)

lectures 19-23 (Fuchsian groups, Criteria for groups to be Fuchsian, Orbits, Prperly discontinuous actions, Fundamental Domains, Dirichlet Fundamental Domains)
lectures 14-27 (Elliptic cycles, side pairings, Poincare's Theorem, genus and signature, Limit sets)

Below are copies of the example sheets and solution sheets (to be corrected as the course progresses).


Example Sheet 1 Solution Sheet 1
Example Sheet 2 Solution Sheet 2
Example Sheet 3 Solution Sheet 3
Example Sheet 4 Solution Sheet 4
Example Sheet 5 Solution Sheet 5
Example Sheet 6 Solution Sheet 6
Example Sheet 7 Solution Sheet 7
Example Sheet 8 Solution Sheet 8



Ergodic Theory (MA 427)

The lectures (in term 2) are: Tuesday 16-17 (MS.04); Wednesday 12-13 (B3.03); Thursday 14-15 (B3.03)

I taught this course for 4th year Mathematics students at Warwick University. 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;

Solution sheet 1;
Solution sheet 2;
Solution sheet 3;
Solution sheet 4;



Complex Analysis (MA 3B8)

I taught 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).

There are typed lecture notes for the course, with hand drawn pictures.

Below are scans of my hand written lecture notes (often corrected as the course progressed)
lecture Notes 1 (Overview)
lecture Notes 2 (Mobius maps)
lecture Notes 3 (Applications of Mobius maps) and an article by Sarnak on Circle packings
lecture Notes 4 (Analyticity, Cauchy-Riemann equations)
lecture Notes 5 (Application:Non-acculumulation of zeros)
lecture Notes 6 (Integrals, Theorems of Gousart and Cauchy) (Correction)
lecture Notes 7 (Applications of Theorems of Gousart and Cauchy)
lecture Notes 8 (Properties of Analytic functions:Arg and Log)
lecture Notes 9 (Location of zeros: Argument Principle and Rouche's Theorem)
lecture Notes 10 (Weierstrauss-Hurwicz theorem, Liouville theorem)
lecture Notes 11 (Montel's Theorem)
lecture Notes 12 (Riemann Mapping Theorem)
lecture Notes 13 (Schwarz-Christoffel Theorem, Schwarz Reflection Principle, Osgood Caratheodory Theorem)
lecture Notes 14 (Harmonic functions)

Example and Solution sheets:
Example sheet 1
Example sheet 2
Example sheet 3
Example sheet 4
Example sheet 5
Example sheet 6

Solution sheet 1
Solution sheet 2 (TA's solutions)
Solution sheet 3 (TA's solutions)
Solution sheet 5 (TA's solutions)
Solution sheet 6 (TA's solutions)
Solution sheet 7 (TA's solutions)



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