Year 1
Year 1 introduces a variety of topics useful for the application of mathematics. The foundations are laid for later work in themes developing calculus based techniques with applications modelling the real world and also practical data analysis methodologies. The power of computational methods is introduced, enabling the investigation of more realistic problems, through the use of industry-standard software (such as SAS, Maple and Matlab).
You will have the opportunity to work together with students from other disciplines on case - studies developing team working and professional skills and awareness of the wider professional world.
Core modules
Mathematical Methods and Modelling of Applications
30 credits
This is the first in a spine of 3 core modules progressing through Levels 4 to 6 of the Mathematics BSc which place considerable emphasis on the important topic of calculus and its application to real-world problems. Although the necessary fundamental aspects of calculus, such as that of a limit are introduced, and the continuity and differentiability of functions on the real line explored, the delivery is primarily from an applicable, modelling perspective.
Typically the class sessions are problem-centred in nature where students are first presented with a real-world or other authentic problem as motivation for the solution being sought. A significant proportion of the time in class is spent with students working in small groups where these problems are formulated mathematically, solved (possibly with the use of computing packages) and the results presented in various formats to enhance employability skills. Lecturer input for the modelling methodology (the modelling cycle) and necessary new theory is given at appropriate stages of these formative tasks. In the latter stages of the module the study of ordinary differential equations is commenced which provide the opportunity to model many additional real-world scenarios and also provides essential foundation knowledge for the higher level calculus modules to follow.
On successful completion of the module, students will be able to:
- Solve problems related to the calculus and its foundations;
- Apply the techniques of calculus in a range of real-world modelling scenarios;
- Formulate and solve authentic mathematical models based on simple ordinary differential equations (ODEs);
- Communicate mathematical ideas and arguments in a variety of forms.
Problem Solving and Computational Mathematics
30 credits
This module is taken by all first year undergraduate students undertaking a degree in Mathematics. The module combines computer programming with an introduction to specialist mathematics software in the context of which some fundamentals of linear algebra are explored. Previous experience of programming is not assumed. The module seeks to introduce a foundation for programming that can be built on in subsequent years and that accommodates specialist practice within Mathematics.
Teaching and learning is split between a variety of different units the first of which is in common with module CI4100. As befits a practical discipline like programming, a hands-on approach is used that facilitates self-paced and self-directed learning. This approach continues as students move on to applying algebraic and numerical software to solve mathematical problems that arise in applications. Students are encouraged to engage with, develop and experiment with programs and mathematical problems in a constructivist fashion inspired by bricolage. The intent is to build students' confidence as they progress through the module's topics, and provide a foundation that can be built on so that in later years they can go beyond simple solutions to problems and be ready to engage in fully-fledged mathematical modelling and problem solving.
On successful completion of the module, students will be able to:
- Decompose a mathematical problem or programming task into a set of smaller sub-tasks
- Write programs that demonstrate the appropriate use of variables, arrays of variables, expressions, subroutines, conditional and iterative control flow structures
- Use debugging and problem analysis strategies to find errors and validate problem or model solutions using appropriate tools and techniques
- Apply modern graphical, numerical and algebraic computing techniques to simple mathematical problems or structures
- Use matrices and vectors to represent, analyse and solve simple problems from the real world
- Construct and present rigorous logical arguments (e.g. combining theory and output from mathematical software)
Practical Data Analyst Skills
30 credits
We make the first steps into the analysis of data. We begin by considering what are data, how they are obtained and introduce consideration of aspects of data collection, including designing surveys to obtain the information desired. Then, we look at how to approach data analysis, defining questions and identifying the best techniques to achieve solutions to the problems posed. Some probability concepts are introduced to support the statistical inference methods used as the module progresses. The main objective of the module is to teach practical data analysis skills using a problem-centred approach simulating the practice most commonly encountered in industry and other real-life scenarios, thus improving students' employability. We teach students to work together and ask questions of the data and to find the correct statistical analysis tools to obtain good information and make useful decisions.
The module is the basis for much of the work in Statistics and in part the Data Science stream of the Mathematics course. It is foundational for the Data Science degree.
On successful completion of the module, students will be able to:
- characterise the data set in terms of purpose, source timescale and measurement.
- describe and summarise the main features of a dataset by using appropriate tables, diagrams and summary statistics
- construct confidence intervals and conduct hypotheses tests for means and proportions in well-defined, appropriately sized samples and interpret the results;
- enhance employability by using appropriate industry standard software for data manipulation, basic statistical analysis and presentation of data;
- demonstrate awareness of the key principles and practices of survey design and implementation.
Professional Environments 1
30 credits
The goal of the Professional Environments module is to prepare students for professional practice firstly by ensuring they acquire suitable employability assets and secondly by equipping them with an understanding of the role of a professional in society and the role of professional bodies.
While the bulk of the taught programme focuses primarily on domain knowledge, the Professional Environments module focuses on developing key skills (as enumerated in the Programme Specification), personal qualities (eg commercial awareness, reliability and punctuality, understanding the centrality of customers and clients), and professional knowledge including the need to engage with continuing professional development. With such assets, students will generate a CV, an employment portfolio, and a professional online presence.
Being a professional also means understanding the key legal, ethical and societal issues pertinent to the domain, and understanding the need for continuing professional development (CPD) especially when technology develops at such a rapid pace. The module is designed to support different domain areas and to integrate experience from other professions. The subject areas being studied demand a global perspective which encourages the inclusion of our diverse of communities and national practices.
Reflecting the fact that team working is ubiquitous in the modern workplace, a significant proportion of the assessment work on the course is group-work based. There is considerable evidence that group work promotes a much deeper engagement with taught content. It also encourages the development of diverse learning communities. This module will therefore introduce students to best practice in group working covering how to approach group work, how to deal with different types of people, and methods of selecting and managing groups.
Year 2
Year 2, extends the themes introduced in Year 1, refining the integration of mathematical and professional skills to model real-world problems and develop and present solutions. You will continue to build a portfolio of products, showcasing your growing knowledge and skills.
Once Year 2 is successfully completed, you will have the opportunity to take a professional placement year to develop your skills in a real work setting.
Core modules
Applications of Calculus and Linear Systems
30 credits
This is the second in a strand of essentially calculus-based core modules for the BSc in Mathematics, and concepts developed here are used extensively to underpin the knowledge delivered at level 6, including in the capstone project. The module content is designed to build on the work undertaken at level 4 by further developing the students' knowledge and skills necessary to tackle a wide variety of interesting real-world problems. For example, the treatment of ordinary differential equations is extended so that linear systems of these can be considered, both analytically and numerically as befits the application, thus permitting the solution of a much wider range of problems associated with the real-world scenarios. These may be associated with medical applications, industrial processes, environmental hazards and disasters, to name just a few.
The module also considers the topic of multiple integration and vector calculus thus permitting consideration of authentic problems where changes occur in 3-dimensions, such as problems in computer game and animation development. In common with the preceding level 4 calculus module, MA5500 is rooted in the methodology of modelling real-life problems and its delivery is centred on active student participation in tackling interesting and engaging tasks.
On successful completion of the module, students will be able to:
- Evaluate multiple integrals, in different co-ordinate systems.
- Perform vector algebra and calculus, including evaluations of gradient, divergence and curl, and applications of (integral) theorems linking these quantities.
- Solve a variety of ordinary differential equations (ODEs), analytically and numerically as applied to real-world problems through individual and collaborative group work.
- Solve numerically systems of linear and nonlinear equations from real-life problems.
Mathematics of Finance and Investment
30 credits
This module introduces students to basic mathematical models for assessing investments and projects taking place over a period of time. The module goes on to explain how concepts of compound interest and discounting are used to value payments to be made in the future. Compound interest functions are introduced and formulae for regular level or varying payments made for specified periods (annuities certain) are derived. Practical applications are demonstrated by analysing elementary compound interest problems relating to investments such as bonds and ordinary shares. The module provides the basis for the final year modules Financial Portfolios and Derivatives, and Insurance Risk Mathematics.
On successful completion of the module, students will be able to:
- Distinguish between interest rates expressed in different time periods and derive the relationships between them;
- Evaluate the present value and the accumulated value of a given cash flow series;
- Define and derive compound interest functions including annuities certain;
- Construct a schedule of loan repayments and evaluate the price of, or yield from a bond using the concept of an equation of value;
- Apply discounted cash flow and equation of value techniques in investment project appraisal;
- Demonstrate an understanding of the term structure of interest rates and evaluate spot rates, forward rates, duration, convexity and immunisation.
Modelling Real World Data with Statistics
30 credits
This module develops and builds on the concepts of probability and statistics introduced in the module Practical Data Analyst Skills . It is a core module for students taking Mathematics, and Data Science degrees.
Probability underpins aspects of statistics and we need a sound grounding in those topics that are directly applicable to many real world applications of the subject. We also need to be able to apply these probability distributions to real world data in order to obtain more information. In addition, we will be looking at how data from experiments and related studies are analysed and how we can make useful sense of data. We also study some of the general linear models which give us understanding how various factors influence output data.
On successful completion of the module, students will be able to:
- solve problems in applied probability, using discrete and continuous random variables and probability distributions
- estimate the parameters of certain probability distributions using appropriate techniques of statistical inference
- choose and apply the appropriate techniques of General Linear Models to obtain useful information from experimental and other related data sets and test the efficacy of these models
- develop and apply statistical and other software to analyse data, and communicate the results of the analysis
Professional Environments 2
30 credits
Following a project-based pedagogic approach, students will undertake a major inter-disciplinary team-work project drawn from a list of authentic industrial problems. Achieving the goals of the project will require students, firstly, to apply the various development methodologies they have acquired on their course and, secondly, to develop professional skills in project management and team working.
While the bulk of the taught programme focuses primarily on the learning of domain knowledge, the goal of the Professional Environments 2 module is to prepare students for professional practice in their respective domains. They will develop the necessary project management and team-working skills, and, by working as a team on an authentic industrial project, they will gain a high degree of familiarity with the typical requirements capture, design, and development methodologies relevant to their discipline. With the focus on making real-world artefacts, the students will integrate their work into an employment focused portfolio.
Being a professional practitioner also mean critically assessing both goals and solutions from legal, ethical and societal perspectives as well as addressing security and safety concerns. Students are also encouraged to consider their continuing professional development needs and to engage with their professional bodies. To encourage career management skills and promote employability after graduation, students are expected to integrate the artefacts they produce and reflective practice narratives into their employability portfolios and personal development plans.
The module is designed to support different domain areas and to integrate experience from other professions. The subject areas being studied demand a global perspective which encourages the inclusion of our diverse of communities and national practices.
Optional sandwich year
Core modules
Industrial Placement
60 credits
This module is an essential course programme component for students on the sandwich route of an honours degree "with professional placement". It is a key element in providing an extended period in industry gaining real world employability skills. Students are supported both before and through their placement by the SEC Placement team. Students that successfully complete their placement year will graduate with a 4 year sandwich degree.
Final year
Final year completes the calculus based modelling journey with the study of partial differential equations and nonlinear systems (areas of mathematics that are applicable to many real-world problems). Everyone undertakes a major project (independent study) as the culmination of the theme integrating mathematical and professional skills in preparation for future employment. In addition,
you may select specialist option modules from different areas of mathematics and statistics, such as modelling financial investments, optimisation or modern applications in the analysis of 'Big Data'.
Core modules
Advanced Mathematical Methods and Models
30 credits
This module is core to the Mathematics BSc and it completes the theme of modules at lower levels which concentrate on calculus and differential equations. There are two main topics which advance the earlier material, namely partial differential equations (PDEs) and nonlinear systems. Whereas the ordinary differential equations (ODEs) you have studied at lower levels can handle only a single independent variable, PDEs accept several variables such as situations where a quantity being measured or predicted varies both with position and time. The enriched range of real-world scenarios that can be modelled mathematically then include traffic flows, heat conduction and vibrations in bodies, electrical properties in transmission lines, fluid dynamics, acoustics and option pricing in the banking industry.
The models of systems of ODEs studied at Level 5 make the assumption that all the equations are linear; in the other major topic of this module that restriction is removed and so again a larger range of models can be created which can represent such scenarios as interacting populations, chemical reactions, electrical circuits, mechanical and control systems. Of particular importance is the predicted stability of systems arising in the models produced. In both the cases of PDEs and nonlinear systems the analytical solutions may be impossible to find, but you will be introduced to tools to aid in their analysis including approximations afforded by industry standard computing packages. As with the earlier modules in this core strand, the exciting applications of these techniques to authentic scenarios makes the module ideal for a problem-centred, active learning environment where significant student participation is the norm.
On successful completion of the module, students will be able to:
- solve appropriate partial differential equations analytically by separation of variables, the method of characteristics or integral transform techniques
- formulate mathematical models of real-world problems using partial differential equations and critically evaluate their suitability by reference to the assumptions on which they are constructed
- employ finite difference methods for solving PDEs and understand limitations of numerical methods
- use appropriate techniques such as phase-plane analysis, aided when appropriate by mathematical software, to extract qualitative information from real-world systems
Mathematics Individual Project
30 credits
This module provides the opportunity for you to showcase your accumulated knowledge and practical skills gained throughout the programme, including development of an end product which showcases your skills portfolio and may be a useful discussion tool for interviews and graduate employment. The project will often draw from several different areas of your course, highlighting connections (and often interdependence) between the different skills acquired thus giving you experience of mathematics in practice. The project aims to further develop vital skills in areas of research, time and project management, and presentation as well as in technical areas. You can choose from a varied range of proposed project titles or work with academic staff to develop a title of your own within a particular field of interest and on completion of the project you will have gained expertise in your chosen topic area.
Optional modules
Financial Portfolios and Derivatives
30 credits
This module is on the Financial Modelling Guided Option Route within the Mathematics field. It serves as an introduction to financial markets, the mathematics of modern portfolio theory, the stochastic models of risky assets and the theory of pricing contracts based on these assets. The first part of this module introduces the main theories and techniques of modern pricing models and portfolio management. The second part of the module exhibits the basic features of financial derivatives (internationally traded financial contracts that depend on the values of underlying assets such as stocks or bonds). These instruments are defined, their payoffs and the markets in which they are traded are considered, and the importance of valuing these instruments in the absence of arbitrage is discussed. The topics covered in this module provide students with a thorough understanding of the characteristics and mechanics of financial markets and therefore enhance the employability of graduates wishing to pursue a career in trading, investment banking or risk management.
On successful completion of the module, students will be able to:
- Employ appropriate measures to select, analyse and optimise the returns of portfolios and other investments,
- Describe various models of asset returns and equilibrium and perform calculations using these models,
- Discuss the various forms of the efficient market hypothesis and their limitations,
- Formulate models of the securities markets using discrete and continuous-time stochastic processes and partial differential equations,
- Use and extend these models to obtain the value of options and other contingent claims on assets.
Insurance Risk Mathematics
30 credits
The module provides a grounding in mathematical techniques which can be used for pricing and valuing life insurance and pension products, with examples drawn from current professional practice. Mathematical techniques used to model and value cashflows which depend on death, survival or other uncertain risks are explained. The module goes on to define simple assurance and annuity contracts and develop practical methods of evaluating cash flows arising from the contracts. This module provides students with an insight into the methods used by a professional in the insurance industry as well as many other sectors where risk modelling is needed.
On successful completion of the module, students will be able to:
- Construct and analyse statistical distributions for risk modelling;
- Formulate the model of lifetime or failure time as a random variable and evaluate survival probabilities and rates of mortality;
- Compute estimates of mortality rates and hazard rates based on a range of modelling assumptions.
- Define various assurance and annuity contracts and develop formulae for the expected values and variances of payments under the contracts;
- Explain and derive premiums and reserves of assurance and annuity contracts and apply a profit test to the product;
- Predict expected future cash flows for various assurance and annuity contracts.
Optimisation Techniques and Applications
30 credits
This module can be taken as an option module by students studying on the BSc Mathematics degree course. The module introduces students to a variety of Optimisation Techniques and their applications. The module consists of two distinct but interrelated parts. In the optimisation section the ideas of using calculus to find stationary points of functions (of one or two variables), introduced in earlier modules are generalised and extended to cases where the functions are constrained (by both equations and inequalities). A variety of calculus based methods for finding optima is considered and their appropriateness for different situations and applications is discussed. Whilst, in the operational research section, the basic concepts and ideas of Mathematical Programming are introduced. The section goes on to explain how to apply operational research techniques such as network models, location models, inventory models and heuristics to real life problem solving issues.
The module shows how industrial problems of optimisation may be written in mathematical form. The module also introduces the simplex algorithm and its variants and demonstrates how such problems may be solved via these methods. Numerical software is employed to develop the students' practical skills to solve optimisation problems and to verify solutions from theoretical analysis. The module provides a depth of detail that sufficiently prepares students for further study and research into more advanced techniques while the exciting applications of these techniques to authentic scenarios makes the module ideal for a problem-centred, active learning environment where significant student participation is the norm.
On successful completion of the module, students will be able to:
- find analytically the extrema of functions of two or more variables, with and without constraints,
- apply appropriate numerical methods to solve unconstrained and constrained optimisation problems, and evaluate the relative merits and limitations of the methods;
- apply the above methods in a range of application areas;
- use several types of operational research methodology to formulate models, solve them, interpret and define the limitations of solutions found by the above methodology;
- identify the basic principles of mathematical programming methods and display a deep understanding of several mathematical programming algorithms for linear programming;
- develop models for the solution of real life problems by optimization techniques using software to find, verify and interpret solutions.
Practical Applications of Advanced Statistics
30 credits
This module is designed to introduce students to further developments of statistical modelling methodologies introduced at Level 5. The module will be taught in a very practical way using an example driven approach to present applications of the theory, and subsequently interpretation and communication of the outcomes. Students will also be introduced to the applications of advanced models in real life scenarios including within the Business and Health fields where demand for such skills is consistently high. During the module students will gain practical experience of how to determine and apply appropriate statistical methodologies and how to interpret, present and contextualize the findings of such analyses to the standard expected in a professional setting. They will also learn about the processes involved in such applications such as the full cycle of clinical trial analysis and the practical implementation of forecasting methods in business. Throughout students will be instructed in appropriate statistical software for carrying out such analyses and in the effective communication of their results, hence enhancing employability potential.
On successful completion of the module, students will be able to:
- demonstrate understanding of selected Generalised Linear Models and when it is appropriate to use them;
- choose and apply a statistical methodology appropriate to a given data analysis problem;
- identify and analyse data obtained from clinical studies, and interpret and report the results of such analyses;
- select and apply appropriate forecasting techniques for data analysis and critically assess the validity of the modelling results for time series data from the computer output; and
- design, implement and produce solutions using appropriate modern computational software for the statistical techniques learned.
Artificial Intelligence and Machine Learning
30 credits
This module is an elective (option) module for the BSc Mathematics programme. It builds upon the foundations of Data Analysis & Modelling and computing skills developed in earlier modules. This module aims to introduce the study of artificial intelligence with applications in research-informed topics such as language modelling, speech recognition or pattern recognition in "big data" applications.
It introduces both "traditional" (logic-based) and "modern" (eg neural networks, including "Deep Learning", decision tree-based and probabilistic) "machine learning" approaches to artificial intelligence, and includes some case studies of modern practical applications. These are important mathematical and statistical concepts that are essential attributes for employable data scientists, mathematicians and statisticians in the modern, data-driven world.
