> Rua's homepage


2018 courses

  • Ench298 Chemical Engineering Mathematics (Semester 1). ENCH298 covers linear algebra, numerical solution of algebraic and differential equations, numerical integration, multivariable calculus, fourier series and Laplace transform techniques for applications in Chemical Engineering.
  • Math102 Mathematics 1A (Semester 1). The work in MATH102 is based around the fundamental ideas in linear algebra and calculus which students started studying at school. MATH102 can be taken as a standalone course or as the first part of a core mathematics programme.
  • Math202 Differential equations (Semester 2). This core second year course covers geometric, analytic and numerical techniques for ordinary differential equations. As well as the classical theory for linear equations (reduction of order, variation of parameters, eigenvector solutions, power series, Laplace transforms and Fourier series) geometric ideas for nonlinear equations are introduced. The numerical solution of DEs via MATLAB is integrated into the course, and a weekly computer lab is required.
  • Math401 Dynamical systems 1 (Semester 2). Dynamical systems is a rapidly developing branch of Mathematics with growing applications in diverse fields from traditional areas of applied mathematics to numerical analysis, biological systems, economic models and medicine. This course will teach theory, techniques and applications of systems of nonlinear equations. In all cases, we are concerned with dynamics: the time evolution of spatial structure. We will cover the mathematics behind chaos theory, and learn techniques for analysing nonlinear systems. Since it is usually difficult or impossible to write down an exact solution to systems of nonlinear equations, the emphasis will be on qualitative techniques for classifying and understanding the behaviour of nonlinear systems. Both main types of dynamical system will be studied: discrete systems, consisting of an iterated map; and continuous systems, arising from nonlinear differential equations. The natural relationships between discrete and continuous time systems will be emphasised too. Probabilistic aspects will be discussed, and transfer operators will be used to study transport. This course is independent of Math363 Dynamical systems, although previous enrolment there is desirable. Topics covered: maps and flows as dynamical systems (including return maps); behaviour of one-dimensional maps, including period-doubling. Hyperbolicity, linearisation and stability of fixed points and period orbits. Invariant manifolds and phase portrait analysis; centre manifolds and local bifurcations; global bifurcations. Probability, transport and mixing; Ulam's method. Topological dynamics, symbolic dynamics and chaos.

Previously taught courses at UC


  • Ench298 Chemical engineering mathematics (2017)
  • Emth119 Engineering mathematics 1B (2013, 2014, 2015, 2016, 2017)
  • Math102 Mathematics 1A (2010)
  • Math103 Mathematics 1B (2011,2012)
  • Math201 Mathematics 2 (2011,2012,2013)
  • Math202 Differential equations (2016, 2017)
  • Emth204/Math264 Multivariate calculus and differential equations (2008, 2009)
  • Math342 Applications of complex variables (2008, 2009)
  • Math363 Dynamical systems (2008, 2009, 2011, 2012, 2013, 2014, 2015, 2016)
  • Econ321 Mathematical techniques in Microeconomics (2012, 2013).

Honours (beginning graduate)

  • Math401 Dynamical systems 1 (2013-2017)
  • 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, 2012). 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.

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)

> Rua's homepage