MATH3061 - Geometry and Topology 6 credit points Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3001, MATH3006 Description: The aim of the unit is to expand visual/geometric ways of thinking. The geometry section is concerned mainly with transformations of the Euclidean plane (that is, bijections from the plane to itself), with a focus on the study of isometries (proving the classification theorem for transformations which preserve distances between points), symmetries (including the classification of frieze groups) and affine transformations (transformations which map lines to lines). The basic approach is via vectors and matrices, emphasising the interplay between geometry and linear algebra. The study of affine transformations is then extended to the study of collineations in the real projective plane, including collineations which map conics to conics. The topology section considers graphs, surfaces and knots from a combinatorial point of view. Key ideas such as homeomorphism, subdivision, cutting and pasting and the Euler invariant are introduced first for graphs (1-dimensional objects) and then for triangulated surfaces (2-dimensional objects). The classification of surfaces is given in several equivalent forms. The problem of colouring maps on surfaces is interpreted via graphs. The main geometric fact about knots is that every knot bounds a surface in 3-space. This is proven by a simple direct construction, and this fact is used to show that every knot is a sum of prime knots. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam, tutorial tests, assignments. MATH3068 - Analysis 6 credit points Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3008, MATH2007, MATH2907, MATH2962 Description: Analysis grew out of calculus, which leads to the study of limits of functions, sequences and series. The aim of the unit is to present enduring beautiful and practical results that continue to justify and inspire the study of analysis. The unit starts with the foundations of calculus and the real number system. It goes on to study the limiting behaviour of sequences and series of real and complex numbers. This leads naturally to the study of functions defined as limits and to the notion of uniform convergence. Returning to the beginnings of calculus and power series expansions leads to complex variable theory: analytic functions, Taylor expansions and the Cauchy Integral Theorem. Power series are not adequate to solve the problem of representing periodic phenomena such as wave motion. This requires Fourier theory, the expansion of functions as sums of sines and cosines. This unit deals with this theory, Parseval's identity, pointwise convergence theorems and applications. The unit goes on to introduce Bernoulli numbers, Bernoulli polynomials, the Euler MacLaurin formula and applications, the gamma function and the Riemann zeta function. Lastly we return to the foundations of analysis, and study limits from the point of view of topology. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam, tutorial tests, assignments. MATH3075 - Financial Mathematics 6 credit points Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3975, MATH 3015, MATH3933 Description: This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. The lectures in the Normal unit are held concurrently with those of the corresponding Advanced unit. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: Two class quizzes and one 2 hour exam MATH3078 - PDEs and Waves 6 credit points Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3978, MATH3018, MATH3921 Description: This unit of study introduces Sturm-Liouville eigenvalue problems and their role in finding solutions to boundary value problems. Analytical solutions of linear PDEs are found using separation of variables and integral transform methods. Three of the most important equations of mathematical physics - the wave equation, the diffusion (heat) equation and Laplace's equation - are treated, together with a range of applications. There is particular emphasis on wave phenomena, with an introduction to the theory of sound waves and water waves. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam, one lecture quiz Textbooks: Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. 1999. MATH3966 - Modules and Group Representations (Adv) 6 credit points Assumed knowledge: MATH3962 Prerequisites: 12 credit points of Intermediate Mathematics Prohibitions: MATH3906, MATH3907 Description: This unit deals first with generalized linear algebra, in which the field of scalars is replaced by an integral domain. In particular we investigate the structure of modules, which are the analogues of vector spaces in this setting, and which are of fundamental importance in modern pure mathematics. Applications of the theory include the solution over the integers of simultaneous equations with integer coefficients and analysis of the structure of finite abelian groups. In the second half of this unit we focus on linear representations of groups. A group occurs naturally in many contexts as a symmetry group of a set or space. Representation theory provides techniques for analysing these symmetries. The component will deals with the decomposition of representation into simple constituents, the remarkable theory of characters, and orthogonality relations which these characters satisfy. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam, assignments and quizzes MATH3968 - Differential Geometry (Advanced) 6 credit points Assumed knowledge: At least 6 credit points of Advanced Mathematics units of study at Intermediate or Senior level. Prerequisites: 12 credit points of Intermediate Mathematics, including MATH2961 Prohibitions: MATH3903 Description: This unit is an introduction to Differential Geometry, using ideas from calculus of several variables to develop the mathematical theory of geometrical objects such as curves, surfaces and their higher-dimensional analogues. Differential geometry also plays an important part in both classical and modern theoretical physics. The initial aim is to develop geometrical ideas such as curvature in the context of curves and surfaces in space, leading to the famous Gauss-Bonnet formula relating the curvature and topology of a surface. A second aim is to present the calculus of differential forms as the natural setting for the key ideas of vector calculus, along with some applications. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam and 2 assignments MATH3974 - Fluid Dynamics (Advanced) 6 credit points Assumed knowledge: MATH2961, MATH2965 Prerequisites: 12 credit points of Intermediate Mathematics with average grade of at least Credit Prohibitions: MATH3914 Description: This unit of study provides an introduction to fluid dynamics, starting with a description of the governing equations and the simplifications gained by using stream functions or potentials. It develops elementary theorems and tools, including Bernoulli's equation, the role of vorticity, the vorticity equation, Kelvin's circulation theorem, Helmholtz's theorem, and an introduction to the use of tensors. Topics covered include viscous flows, lubrication theory, boundary layers, potential theory, and complex variable methods for 2-D airfoils. The unit concludes with an introduction to hydrodynamic stability theory and the transition to turbulent flow. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam MATH3975 - Financial Mathematics (Advanced) 6 credit points Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions: MATH3933, MATH3015, MATH3075 Description: This unit is an introduction to the mathematical theory of modern finance. Topics include: notion of arbitrage, pricing riskless securities, risky securities, utility theory, fundamental theorems of asset pricing, complete markets, introduction to options, binomial option pricing model, discrete random walks, Brownian motion, derivation of the Black-Scholes option pricing model, extensions and introduction to pricing exotic options, credit derivatives. A strong background in mathematical statistics and partial differential equations is an advantage, but is not essential. Students completing this unit have been highly sought by the finance industry, which continues to need graduates with quantitative skills. Students enrolled in this unit at the Advanced level will be expected to undertake more challenging assessment tasks. The lectures in the Advanced unit are held concurrently with those of the corresponding Normal unit. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: Two class quizzes and one 2 hour exam MATH3978 - PDEs and Waves (Advanced) 6 credit points Assumed knowledge: MATH(2061/2961) and MATH(2065/2965) Prerequisites: 12 credit points of Intermediate Mathematics with at least Credit average Prohibitions: MATH3078, MATH3018, MATH3921 Description: As for MATH3078 PDEs & Waves but with more advanced problem solving and assessment tasks. Some additional topics may be included. Classes: Three 1 hour lectures and one 1 hour tutorial per week. Assessment: One 2 hour exam, one lecture quiz Textbooks: Powers, DL. Boundary Value Problems. Harcourt-Brace 4th Edition. 1999. |