You will learn about a range of appropriate statistical techniques that are used to analyse time series data. You will be introduced to the different methods that can be used to remove any trend or seasonality that are present in the data and learn how to determine the appropriate time series model for this modified time series. Once the model is chosen, you will learn verification techniques to confirm that you have selected the correct model and then, if required, learn how to forecast future values based on this model.

By the end of the module, you will have developed an awareness of different approaches to analysing time series data and to be able to tailor these techniques based on the initial assessment of the time series data.

Outline Syllabus
On this module, you will cover:
• Differencing methods to remove trends and/or seasonality.
• Diagnostic tools to select appropriate model
• Autoregressive Integrated Moving Average (ARIMA) models
• Model identification methods
• Verification of model
• Seasonal Autoregressive Integrated Moving Average (SARIMA) models and their identification and modelling.

You will achieve proficiency in using appropriate R and or Python statistical packages.

You will learn the various numerical techniques used to solve partial differential equations (PDEs). PDEs are widely used to describe phenomena in the natural world as well as in cultured and manufactured reality. These powerful numerical methods often provide the only means to explore and analyse the PDEs. Various methods will be investigated with emphasis on the underlying ideas and principles of each method. This theoretical understanding will be underpinned by practical implementation of the numerical methods throughout the module. This approach will allow you to develop a well-grounded theoretical base as well as the necessary programming skills to implement solutions in real-life situations.

You will become conversant in the classification of PDEs as well as the stability and convergence of numerical schemes. Using this knowledge as a foundation, you will investigate and appraise state-of-the-art numerical methods. These may include but are not limited to

• Finite difference methods
• Finite element methods
• Finite volume methods
• Spectral methods
• Particle methods

Academic skills when studying away from your home institution can differ due to cultural and language differences in teaching and assessment practices. This module is designed to support your transition in the use and practice of technical language and subject specific skills around assessments and teaching provision in your chosen subject area in the Department of Architecture and Built Environment. The overall aim of this module is to develop your abilities to read and study effectively for academic purposes; to develop your skills in analysing and using source material in seminars and academic writing and to develop your use and application of language and communications skills to a higher level.

The topics you will cover on the module include:

• Understanding assignment briefs and exam questions.
• Developing academic writing skills, including citation, paraphrasing, and summarising.
• Practising ‘critical reading’ and ‘critical writing’.
• Planning and structuring academic assignments (e.g. essays, reports and presentations).
• Avoiding academic misconduct and gaining credit by using academic sources and referencing effectively.
• Listening skills for lectures.
• Speaking in seminar presentations.
• Giving discipline-related academic presentations, experiencing peer observation, and receiving formative feedback.
• Speed reading techniques.
• Discussing ethical issues in research, and analysing results.
• Describing bias and limitations of research.
• Developing self-reflection skills.

You will learn about a range of appropriate statistical techniques that are used to predict and analyse complex systems modelled by random matrices. You will be introduced to the generalisation of probability theory for multivariate calculus, the analysis of the most common ensembles (Gaussian Orthogonal and Unitary Ensembles, the Circular Ensembles) and methods for using these tools efficiently in numerical simulations.

Outline Syllabus
– Review of linear algebra and probability theory
– Numerical techniques to generate and analyse random matrices
– The Circular Unitary Ensemble (CUE): definition, spacing distribution, eigenvalues correlation functions
– The Circular Orthogonal Ensemble (COE)
– The Gaussian Ensembles: unitary, orthogonal, symplectic
– Orthogonal polynomial techniques (large N limit and universality)

Depending on the time the extrema statistics (Tracy-Widom distribution) will be derived as it can be found in numerous applications (combinatorics, biology)

This module will provide you the fundamentals and theoretical underpinnings of the theory of networks, machine learning and their applications, create a solid background to support professional work in relation to a rapidly evolving field of research such as machine learning and artificial intelligence.
You will learn fundamental concepts of graphs theory, representation and quantitative characterisation of networks, statistical mechanics of random networks and their deployment for the realisation of systems which can process information and learn. You will achieve proficiency in relevant computer programming (Python) and suitable packages for network analysis and machine learning.
Topics in the syllabus will include fundamentals of graph theory and elements of neural networks (degree distributions, clustering, shortest paths, portioning, modularity), probability and statistical mechanics of networks, supervised/unsupervised machine learning.

The module covers three broad topics: 1) Ordinary differential equations, 2) eigenvalue problems, and 3) random processes. For each topic, we will explore related techniques and apply them to specific problems.

The syllabus includes:

Ordinary differential equations
Numerical methods (Euler, Runge-Kutta); phase portraits; systems of ordinary differential equations; reflection and transmission; synchronisation (entraining, Adler’s model, Arnold tongues, mutual synchronisation, nonlinear oscillators).

Eigenvalue problems
Fourier transform and series; discrete Fourier transform; asymptotic expansion; dispersion relation in periodic potentials; Bloch theorem.

Random processes
Brownian motion; Langevin equations; Ito and Stratonovich calculus; noise in Fourier space; Wiener-Khinchin theorem; Monte-Carlo integration; metropolis algorithm.

The project requires you to develop and demonstrate the ability to do research and this is primarily demonstrated through your dissertation. Your dissertation will detail a systematic understanding of Mathematical Modelling and its real life application, a critical awareness of knowledge, a critical awareness of current problems and/or new insights into advanced mathematical methods and their importance for professional practice. You will develop a comprehensive understanding of techniques applicable to the research topic you have chosen and advanced scholarship. You will also develop skills for carrying out original research in Mathematical Modelling and practical understanding of how established techniques of research and enquiry are used to create and interpret knowledge in the specific area of interest. You will be trained on how to engage with relevant research articles for your chosen project. You will also learn from seminars in Mathematics that are regularly organised in the department. Importantly, you will learn by 1:1 meetings with your supervisor while working on a topic grounded in the staff research.

Specifically, you will learn how to do the following
1) Conduct a focused literature search of library and web-based materials and critically appraise and analyse the findings.
2) Integrate and/or modify ideas, concepts and theoretical models that have been selectively extracted from scholarly literature.
3) Critically appraise and test the applicability of theoretical models to their researchable topic.
4) Rationalise and defend the key aspects of the work undertaken in the form of a presentation using professional software (e.g. Microsoft PowerPoint, Beamer package in LaTex).
(5) Write an original dissertation in an academically acceptable format, which should be theoretically and methodologically linked, paying particular attention to the integration of the literature review, the methodology and the clear and concise presentation of results and conclusions.

Wave phenomena appear everywhere in nature, from water waves to magnetic materials, from optics to weather forecasts, hence their description and understanding is of fundamental importance both from the theoretical and the applicative points of view.
You will learn the mathematical theory of nonlinear wave motion. Applications include the understanding of shock waves and coherent structures, such as solitons, of some famous integrable partial differential equations that arise from standard modelling processes and the mechanisms of formation and propagation of anomalous waves such as tsunamis and rogue waves. The module is designed to give you a flavour of modern research in this actively developing area of applied mathematics.

Topics will include theory of linear dispersive waves (wave propagation, elements of Fourier analysis, modulated waves), nonlinear hyperbolic waves and integrable nonlinear wave equations and their applications.