General info Edit on GitHub

Financial Mathematics April/May exam Edit on GitHub

Basic introduction to finance, to give students a broad perspective on the topic. Topics covered include: the value of money, interest rates, fair prices, financial markets in discrete and continuous time. It has a review of probability theory, introduction to the basic ideas of Stochastic calculus, and an overview of Actuarial Finance.

Relevant reading available online

• Björk, Tomas. Arbitrage theory in continuous time. 3rd Edition, Oxford University Press 2009

Geometry December exam Edit on GitHub

Differential geometry is the study of geometry using methods of calculus and linear algebra. It has numerous applications in science and mathematics. This course is an introduction to this classical subject in the context of curves and surfaces in Euclidean space.

Honours Algebra December exam Edit on GitHub

It showcases the power of abstraction and brings together several different topics from earlier courses as well as new ideas, in order to present a view on some of the more advanced algebra that is critical for later courses and also in application, as well as of interest in its own right. The course also includes computer algebra, to support and illustrate some of the content.

This is a core course for Honours degrees involving mathematics.

Main topics: Linear Algebra, Rings and Modules, Determinants and Eigenvalues, Inner Product Spaces, and Jordan Normal Form.

Resources

• Cheatsheet from Owen
• Cheatsheet from Will
• Overview Sheet from Sebastian (Source on GitHub)
• Prerequisites: Fundamentals of Pure Mathematics (MATH08064)

Honours Analysis April/May exam Edit on GitHub

This course builds on ideas studied in the analysis portion of Fundamentals of Pure Mathematics (MATH08064). The course introduces uniform convergence and the Reimann integral, and leads onto the abstract concept of a metric space. Finally we look at some applications to solving equations via the fixed-point method, and at the rudiments of Fourier series.

This course also has a “Skills” portion which involves practising explaining mathematics, consider “metacognitive skills” (i.e. stepping back from a problem and consider how the approach is developing), and illustrating your own and others’ mathematical ideas.

This is a core course for Honours degrees involving mathematics.

Main topics: FPM Analysis revision, Cauchy sequences, Bolzano Weierstrass theorem, Uniform convergence of sequences and series of functions, power series, The Reimann integral, uniform convergence, the fundamental theorem of calculus, metric spaces, point set topology, Heine-Borel theorem, Banach contradiction mapping theorem & applications, Fourier series.

Resources

• Cheatsheet from Owen
• Cheatsheet from Will
• Overview Sheet from Sebastian (Source on GitHub)

Relevant reading available online

• W. R. Wade, An Introduction to Analysis

Honours Complex Variables April/May exam Edit on GitHub

This course has a “Skills” portion which involves mathematical reading and writing (i.e. \(\LaTeX\), applicable to writing technical and non technical reports. Students will be required to complete a project researching a topic connected to complex numbers / complex analysis, producing a written report.

This is a core course for Honours degrees involving mathematics.

Main topics: analytic functions, holomorphic functions, Cauchy-Reimann equations, harmonic functions, Moebius transformations, the Reimann sphere, complex integration, series expansions, the residue calculus and its applications.

Resources

• Cheatsheet from Owen
• Cheatsheet from Will
• Overview Sheet from Sebastian (Source on GitHub)

Relevant reading available online

• Bak and Newman, Complex Analysis, 3rd ed. 2010, ISBN 9781441972873
• Priestley, Introduction to Complex Analysis, 2nd edition, ISBN 9780198525622

Honours Differential Equations December exam Edit on GitHub

This course has a “Skills” portion which involves programming in Python, in particular using SciPy ODE solvers. Activities include: plotting phase portraits of 1st order ODEs, exploring dynamical non-linear systems, implementing numerical methods (Euler, Heun, etc), Fourier coefficient computation, plots of 2D functions and animations of PDEs.

This is a core course for Honours degrees involving mathematics.

Main topics: higher order linear equations, Laplace transforms, systems of First Order Linear ODEs, numerical methods, non-linear systems of ODEs, non-linear methods (eg. Lyapunov functions), Fourier Series, use of separation of variables in standard PDEs, Sturm-Liouville Theory.

Resources

• Cheatsheet from Marie (2018/19; TeX file)
• Cheatsheet from Will (2018/19)
• Cheatsheet from Owen (2018/19)

Introduction to Number Theory April/May exam Edit on GitHub

The main theme of this course is the interplay between number theory and rings.

Main topics: Binary operations on integers, axioms of a ring, the ring of integers \(\mathbb{Z}\) (primes, units and irreducibles), division algorithm, Euclidean Algorithm, the Fundamental Theorem of Arithmetic, using rings to prove basic results from elementary number theory, integral domains, zero divisors, greatest common divisors, gaussian integers and rings \(\mathbb{Z}[d]\) where \(d\) is some irrational number, units, primes and irreducibles in rings, special focus connections of Gaussian integers and quadratic residues and the Legendre symbol

Resources

• Cheatsheet from Owen

Linear Programming, Modelling, and Solution April/May exam Edit on GitHub

Linear programming (LP) is the fundamental modelling technique in optimal decision-making. This course will introduce the concepts of LP modelling, explore the mathematical properties of general LP problems and study the theory of the simplex algorithm as a solution technique. Theoretical understanding / hand calculation will be assessed via the written examination.

Students will learn to use the Xpress mathematical programming system to create, solve and analyse case studies, then present their work in oral and written form: via a short individual Xpress assignment and a substantial group-based case study.

Numerical Linear Algebra December exam Edit on GitHub

This course explores reliable and computationally efficient numerical techniques for practical linear algebra problems. Traditional linear algebra techniques are usually too computationally intensive to be used for the large matrices encountered in practical applications. Such algorithms will be studied, analysed (and assessed) theoretically, as well as implemented using an advanced programming language (MATLAB? Python?). This course includes significant lab work.

Main topics: solution of linear systems of equations, the solution of least squares problems, finding the eigenvectors and/or eigenvalues of a matrux

Relevant reading available online

• Applied numerical linear algebra, James “Jim” Demmel, SIAM, ISBN: 978-0898713893

Numerical Ordinary Differential Equations and Applications April/May exam Edit on GitHub

Most ordinary differential equations (ODEs) lack solutions that can be given in explicit analytical formulas. Numerical methods for ODEs allow for the computation of approximate solutions and are essential for their quantitative study. In some cases, a numerical method can facilitate qualitative analysis as well, such as probing the long term solution behaviour. As well as studying the theory, the course has a strong emphasis on the practical aspects and implementation of these methods (in Python) to tackle modern applications of ODEs.

Resources

• Cheatsheet from Owen

Relevant reading available online

• D. Griffiths and D. Higham, Numerical Methods for Ordinary Differential Equations, Springer 2010
• A. Iserlies, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2008
• L. N. Trefethen, Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations

Statistical Computing Edit on GitHub

Introduction to programming within the statistical package R. Discussion and investigation of various computer intensive statistical algorithms, and their implementation.

Main topics: basic commands of R, data structures and data manipulation, writing functions and scripts, optimising functions in R, programming statistical techniques and interpreting the results (including bootstrap algorithms)

Relevant reading available online

• Crawley. M. (2013). The R Book (2nd edition). Wiley.
• Venables, W. N. and Ripley, B. D., (2002). Modern Applied Statistics with S (4th edition). Springer.

Statistical Methodology December exam Edit on GitHub

This course provides many of the underlying concepts and theory for Likelihood based statistical analyses.

This course is required for numerous further Year 3-5 courses in statistics.

Main topics: likelihood function, maximum likelihood estimation, likelihood ratio tests, Bayes theorem and posterior distribution, Iterative estimation of the MLE (Fishers’ method of scoring), normal linear models.

Relevant reading available online

• Wood, S. N., Core Statistics, Cambridge University Press, 2015.
• Azzalini, A., Statistical Inference Based on the Likelihood, Chapman & Hall, 1996.
• Held, L. & Bove, D. S., Applied Statistical Inference: Likelihood and Bayes, Springer, 2014.
• Christensen, R. et al., Bayesian Ideas and Data Analysis, An Introduction for Scientists and Statisticians, Chapman & Hall, 2011.
• Weisberg, S., Applied Linear Regression, 2nd Edition, Wiley, 2005.
• Crawley, M. J. The R Book, Wiley, 2013.

Stochastic Modelling December exam Edit on GitHub

An advanced probability course dealing with discrete and continuous time Markov chains. Markov chains has countless applications in many fields raging from finance, operation research and optimization to biology, chemistry and physics. The course covers the fundamental theory, and provides many examples.

Main topics: Probability review, definition of stochastic processes, Markov chains in discrete time, modelling of real-life systems as Markov chains, Poisson processes, Markov processes in continuous time.

Resources

• Notes from Owen

Relevant reading available online

• R. Durrett. Essentials of Stochastic Processes, Springer, 2012.
• V. Kulkarni. Modelling and Analysis of Stochastic Systems, CRC Press, 2010.