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Teaching
I was awarded a
UC teaching award at the April 2010 graduation ceremony.
A brief write-up is
here.
2012 courses
- Econ321 Mathematical
techniques in Microeconomics (Semester 1). This course concerns several of the principal
mathematical tools that are now in standard use in microeconomic theory. The course builds
upon the mathematics offered at stage 1, and upon the particular usages of mathematics in Intermediate
Microeconomics. Above all, the objective is to equip students with the necessary
toolkit to successfully tackle higher study in microeconomic theory, although an
underlying objective is to get students to see the importance of some of the
mathematical modelling that was used in intermediate microeconomics. Throughout,
each of the mathematical techniques that is introduced will be supplemented with
concrete examples of their use in microeconomics. I will teach half of this course,
and Assoc Prof Richard Watts (Dept of Economics and Finance) teaches the other half.
- Math103
Mathematics 1B (Semester 2). MATH103 consolidates concepts from MATH102 and introduces
more advanced ideas in calculus and linear algebra. It includes applications of
this mathematics to applied problems. It also incorporates some study of
probability. After passing MATH103 you will be able to enrol in any mathematics
course at the 200 level.
- Math201 Mathematics
2 (Semester 1). This course deals with techniques in multivariable
calculus and linear algebra and interesting applications in many areas of
science, commerce and engineering. It is required for all Math majors, and is
a foundation course for students who want mathematics beyond first year to
support other subjects. Topics covered: Rank and fundamental subspaces
of a matrix; Eigenvalues and eigenvectors; systems of linear differential
equations; introduction to vector calculus (parametrized curves, tangent
vectors, line integrals and work); differentiation of multivariable functions (with geometric understanding); constrained and unconstrained optimization in two or more variables; multivariable Taylor expansions; iterated integrals; polar coordinates; Green's theorem in the plane.
- Math363 Dynamical
sytems (Semester 2).
Dynamical systems is the study of global, long-term behaviour of mathematical systems whose state evolves with time. Most of the systems studied arise from differential equations models of an applied problem from Physics, Biology, Economics, Chemistry, Engineering..... The aim of this course is to understand asymptotic behaviour using a combination of geometric reasoning, intelligent approximations, computer assistance and mathematical insight. This will be accomplished without grinding out the solutions of special classes of differential equations.
Topics covered: Overview of dynamics;
flows on the line; one and two parameter bifurcations; 2d linear systems; phase plane for nonlinear 2d systems;
limit cycles; Hopf bifurcations; applications; introduction to chaos in 3d flows.
- Math420 Hilbert spaces (Semester 1). The theory of Hilbert spaces is
fundamental in many areas of modern mathematical analysis, having a clear and easy-to-grasp geometric
structure, just like Euclidean spaces. However, unlike Euclidean spaces, Hilbert spaces may be
infinite dimensional. The course will be self-contained, introducing important spaces (especially
L2(m)), operators on them, and basic spectral theory.
Applications in dynamical systems (von Neumann's ergodic theorem) and quantum mechanics
will be included as time permits. Prior exposure to MATH 343 would be an asset, but is not mandatory.
Recently taught courses
Undergraduate
- Math102 Mathematics 1A (2010)
- Math103 Mathematics 1B (2011)
- Math201 Mathematics 2 (2011)
- Emth204/Math264 Multivariate calculus and differential equations (2008, 2009)
- Math342 Applications of complex variables (2008, 2009)
- Math363 Dynamical systems (2008, 2009, 2011)
Honours (beginning graduate)
- Math402 Dynamical systems 2
(2011). You may have heard of Lorenz's famous question: "Does the flap of a butterfly's
wings in Brazil set off a tornado in Texas?". The question was meant to
portray one of the main concepts in chaos theory (sensitive dependence
to initial conditions). This course will cover the mathematics behind
chaos theory from two points of view: topological and probabilistic. We will see
that deterministic dynamics (like solutions to nonlinear differential equations)
can be thought of in a similar way to purely random systems.
Topics covered: Devaney's definition of chaos; topological horsehoes;
topological entropy; subshifts of finite type; invariant probability measures;
metric entropy and the variational principle; ergodicity; Lyapunov exponents
and Ruelle's inequality. Please note: Math401 is NOT a prerequisite for this course.
- Math418 Measure and integration (2009). This course covers
the essence of Lebesgue's theory of integration. The main idea is that "the integral of a function"
is determined by the distribution of function values, so one does not need to proceed via repetitive
subdivision of the domain (as required by "Riemann sums"). The Lebesgue integral is flexible (you can
weight different parts of the domain with a "measure" of your choice), convenient (there are useful
limit theorems which don't hold for Riemann integrals), and general (multi-dimensional, discrete and
topologically awkward domains use the same theory). The construction is also very natural if you are
used to thinking about probability. (If you're are not used to probability, MATH418 is a good place
to start.) In this course you'll learn some really great theorems (Birkhoff's Ergodic Theorem is a
personal favourite of the lecturer), as well as getting good preparation for further courses.
Topics covered: Riemann integrability: examples, failure and annoying properties.
Sigma-algebras, measure, simple functions and the Lebesgue integral.
Fatou's lemma, monotone and dominated convergence theorems. Lp spaces.
Construction of measures including Lebesgue and product measures. Measurable dynamical systems: invariant
measures, maximal and Birkhoff ergodic theorems, applications including the strong law of large numbers.
- Math420 Hilbert spaces (2010).
Other courses taught
- First year Calculus (Waikato, NZ and UVic, Canada)
- First year Algebra and General/Management mathematics (Waikato, NZ)
- Second year Analysis, Algebra, Discrete mathematics (Waikato, NZ)
- Third year Numerical analysis and Analysis (Waikato, NZ)
- Third year/honours Metric spaces and Optimisation (Waikato, NZ)
- Honours/Masters Dynamical systems (Waikato, NZ and UCL, UK)
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