Teaching

Major modules I have been lecturer for over the years:

Foundations

Exploring the foundations of undergraduate mathematics from introducing set theory, idea of proof to equivalence relations and basic group theory:

• Proof by induction
• Euclid’s algorithm
• Sets, set notation and set operations
• Symbolic logic, logical connectives, quantifiers and the relation between logic and forms of proof
• Functions, domains, targets, compositions and inverses
• Equivalence relations
• Infinite sets and cardinality
• The construction of number systems (including complex numbers) from set theory
• Modular arithmetic and permutations as examples of groups
• Introductory abstract groups theory: axioms, subgroups, cosets and Lagrange’s theorem

Differential Equations

Introduction to differential equations from trivial first order ODEs through second order linear ODEs to systems of ODEs:
1. FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS
• Trivial first order
• Existence and uniqueness
• First order linear equations with examples
• Substitution methods
• Direction Fields
• Autonomous first order ODEs

2. SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS
• General homogeneous equations
• Linear second order equations with constant coefficients
• Mass/spring systems
• Inhomogeneous linear second order equations
• Mass/spring systems with forcing

3. INTRODUCTION TO DIFFERENCE EQUATIONS
• Motivation (numerical methods)
• First order homogeneous linear difference equations
• Second order linear equations
• First order autonomous nonlinear equations

4. SYSTEMS OF FIRST ORDER LINEAR ODES
• In general
• Coupled 2×2 linear systems with constant coefficients
• Phase portraits and change of variable
• Functions of two variables and linearisation

5. DISCUSSION OF FURTHER TOPICS (non-examinable)
• Three dimensional systems, chaos, nonautonomous second order differential equations

3D Geometry and Motion

Line integrals and multi-variate calculus.

• Curves and line integrals in 2 and 3 dimensions Curves and their parameterisation; distinction between the two; sketching simple curves; conic sections; derivation of Cartesian equation of ellipse and hyperbola from their geometric definition; polar coordinates on R2; polar form of conic sections; unit tangent, curvature and principal normal; line integral of a function (i. e. integration with respect to arc length).
• Integration Integration over rectangles; interpretation in case of two variables
  as signed volume between graph and the xy plane; calculation of the multidimensional integral by a repeated integral (i. e. loose
  statement of Fubini’s theorem); change of order of integration; integration
  over domains bounded by graphs; change of variables formula; Jacobian matrix and Jacobian determinant; average of a function; mass, centre of mass, centroid.
• Surfaces and Surface Integrals Simple examples, including plane, sphere, cylinder, torus, surfaces of revolution, graph of a function of two variables; regular parameterisation of a surface; equation of tangent plane and normal to a surface; integration with respect to element of area (including flux integrals) on a surface in R3. Examples.
• Introduction to particle mechanics The concept of force; Newton’s Laws and calculation of particle paths when the force is given as a function of time; constant gravitational acceleration; motion of projectiles; angular momentum and its geometric interpretation; moment/torque of a force. Motion under central forces. Conservation of angular momentum and its geometric interpretation; velocity and speed in polar coordinates; Newton’s Law of Gravitation. Gravitational potential energy. Planetary and satellite orbits; Kepler’s laws.

Modelling Nature’s Non-linearity

General introduction to techniques and applications in dynamical systems through examples in the natural world. 100% assessed through two assignments and a mini-project actually starting to build a mathematical model of a topical real world problem.

• Evolutionary Game Theory Hawk-Dove and Hawk-Dove-Bully-Retaliator games used to introduce simple Game Theory techniques. Evolutionary Stable Strategies, corresponding dynamical systems, fixed points and dynamics on symplices. Essentially how population dynamics can be explained through how individuals interact with each other.
• Nonlinear Oscillations Overview of linear oscillations (by mechanical and electrical examples), self excited oscillations (including Van Der Pol oscillator), Duffings Equation including forcing and non-forcing cases, vector fields, extended phase space, Poincare Maps, stability, Homoclinic Tangles (leading to chaos). Epidemics SEIR equations, discussion of contact rates and attractors for measles and chicken-pox, are such epidemics periodic or chaotic? Kermack-McKendrick model as a more analytical example.
• Dynamical Systems Discrete and continuous dynamical systems, fixed points and periodic points/orbits and stabilities. Examples include Hawk-Dove and Henon map. Invariant sets, attractors, Strange Attractors, structural stability.
• Sensitive Dependence Lyapunov Characteristic Exponents, Lyapunov Spectrum, chaotic attractors, practical method for computing LCE. Discussion on implications of chaos, and ways to ’control’ chaotic systems.
• Bifurcations and Catastrophes How small changes to parameters can lead to large (’catastrophic’) changes to a physical system. Euler Strut, cusp catastrophes, decision making (intelligent vs. non-intelligent), the Canonical Cusp Catastrophe, catastrophe set, singularities, the Spruce-Budworm and brief mention of the seven elementary catastrophes. Hopf bifurcation (introduced by wavy rolls in convection), linearisation and eigenvalues at a Hopf bifurcation, Hopf theorems (a Hopf bifurcation is a process which forms periodic orbits).
• Coupled Oscillators Fireflies as an example of coupled integrate and fire models, linearly coupled relaxation oscillators, introduction to patterns observed in symmetrically coupled oscillators through the symmetries of animal gaits. A discussion of how symmetry affects ’steady-state’solutions.
• Fractals A rather basic introduction to Fractals through the Middle Third Cantor Set, Koch Curve, Julia Sets, spaces of Fractals, chaotic repellors, iterated function systems, dimension (counting and Hausdorff dimensions).