You can expect to study a range of introductory courses in your first year, covering key mathematics topics such as abstract algebra, calculus, complex numbers, differential equations, geometry, number theory, probability and statistics. You’ll then move on to more advanced study, and will need to choose from a range of elective courses. Popular mathematics topics include:
Complex analysis involves investigating the functions of complex numbers – numbers which can be expressed in a form which allows for the combination of real and imaginary numbers. Complex analysis is useful in many branches of mathematics, including algebraic geometry, number theory and applied mathematics, so it is an essential starting point for the further study of mathematics. You’ll learn about the analytic functions of complex variables, complex functions and differentiation of complex functions, how complex variables can be applied to the real world and cover the many theorems surrounding complex functions such as Cauchy’s theorem, Morera’s theorem, Rouché’s theorem, Cauchy-Riemann equations and the Riemann sphere to name a few.
Discrete mathematics involves mathematical structures that are fundamentally discrete (with finite, distinct, separate values) rather than continuous. This includes topics such as integers, graphs, trees, sets, chromatic numbers, recurrence relations and mathematical logic. Discrete mathematics usually involves examining the interrelations between probability and combinatorics. You’ll also learn about the complexity of algorithms, how to use algorithmic thinking in problem solving, algorithmic applications of random processes, asymptotic analysis, finite calculus and partitions. You’ll learn how discrete mathematics is applied to other topics within mathematics, and you’ll also look into broader academic fields such as computer science.
Mechanics is concerned with the study of forces that act on bodies and any resultant motion that they experience. Advanced study of mechanics involves quantum mechanics and relativity, covering topics such as electromagnetism, the Schrödinger equation, the Dirac equation and its transformation properties, the Klein-Gordon equation, pair production, Gamma matrix algebra, equivalence transformations and negative energy states. You’ll also look at how relativistic quantum mechanics can be used to explain physical phenomena such as spin, the gyromagnetic ratios of the electron and the fine structures of the hydrogen atom. You could also study statistical mechanics, which covers topics such as inference, multivariate complex systems, state variables, fluctuations, equilibrium systems, transport models, dynamical ordering and phase transitions, and emergent behavior in non-equilibrium systems.
Measure theory originates from real analysis and is used in many areas of mathematics such as geometry, probability theory, dynamical systems and functional analysis. It is concerned with notions of length, area or volume, with a measure within a set being a systematic way to assign a number to a subset of that set. You’ll look at the definition of a measurable space, additive measures, construction of measures, measurable functions, integrals with respect to a measure, differentiability of monotone functions, k-dimensional measures in n-dimensional space, Lebesgue-Stieltjes measure and Lebesgue measure. Theorems you will cover include Lusin’s theorem, Egoroff’s theorem, Fatou’s lemma, monotone convergence theorem, dominated convergence theorem, Fubini’s theorem, Radon-Nikodym theorem, Riesz representation theorem and divergence theorem.
The mathematical concept of ‘fractals’ is difficult to formally define, even for mathematicians! Fractals are geometric forms that display self-similar patterns on all scales of magnification, making them look the same when seen from near as from far. Fractal geometry looks at the mathematical theory behind fractals, the definition and properties of Hausdorff dimensioning and iterated function systems. You’ll gain intimacy with forms such as the middle third Cantor set, the Mandelbrot set and the von Koch snowflake curve.
Useful for students interested in engineering and aerospace, fluid dynamics addresses fluid phenomena of various scales from a mathematical viewpoint. You’ll apply mathematics topics such as ordinary and partial differential equations, basic mechanics and multivariable calculus, and will learn about governing equations, how to deduce the equations of motion from conservation laws (mass, momentum, energy), vorticity, dimensional analysis, scale-invariant solutions, universal turbulence spectra, gravity and rotation in atmospheric and oceanic dynamics, equations of motion such as boundary layer equations, flow kinematics, classical and simple laminar flows and flow instabilities. You’ll cover Euler’s equation, Navier-Stokes equation, Bernoulli’s equation, Kelvin’s circulation theorem, Taylor-Proudman theorem, Reynold’s number, Rayleigh number, Ekman number and Prandtl’s boundary layer theory.
Other mathematics topics you can choose from include: algorithms, applied mathematics, calculus, commutative algebra, computational mathematics, computer game technology, cryptography, differential equations, financial mathematics, financial modelling, functional analysis, geometry, knot theory, linear algebra, linear equations, mathematical biology, mathematical modelling, matrix analysis, multivariable calculus, number theory, numerical analysis, probability, pure mathematics, qualitative theory, real analysis, set theory, statistics, theoretical physics, topology and vectors.