Mathematics BSc (Hons)

We introduce you to the foundational language and notation that are used throughout mathematics, including mathematical logic, set theory, proofs, number systems, functions, elementary number theory, symmetry groups etc. Through these diverse topics, you will gain a solid understanding of the foundational principles of mathematics, laying the groundwork for more advanced studies in the field. Our emphasis on logical reasoning and mathematical structures will equip you with the skills to construct rigorous mathematical arguments essential for all university-level mathematics. We will also have a series of workshop focusing on how to write mathematics well, how to read and comprehend mathematics, and how to learn mathematics and use feedback effectively. This will help you transition to becoming an university-level mathematician.

Topics may include:

• Introduction to basic language, concepts, methods, and results in Mathematics, aiming to help students transition to university-level mathematics, including learning key independent and group study skills.
• Exploration of mathematical language, logic, and methods of proof, including induction.
• Elementary set theory.
• Functions, including injective, surjective, and bijective functions, as well as inverses.
• Operations, binary operations, equivalence relations and properties of algebraic structures.
• Introduction to groups, permutation groups, and cyclic groups.
• Elementary number theory: congruence, hcf and lcm, Euclidean algorithm, prime numbers, Fundamental Theorem of Arithmetic
• Different types of infinity: countable and uncountable sets.

Differential equations play a pivotal role in modelling numerous mathematical, scientific and engineering problems, stretching across celestial motion dynamics, neuron interactions, cancer progression, bridge stability and financial market trends. This module serves as an introduction to the essential theory and numerical methods used in solving ordinary and partial differential equations (ODEs and PDEs) while exploring their varied applications.

In this module, we will review the essential calculus techniques, including methods of differentiation and integration, necessary to solve ODEs. We will introduce ODEs, see their applications to real-world problems and explore techniques for generating both exact and approximate solutions for ODEs. We will also give a brief introduction to PDEs and their applications.

Topics may include:

• Review of trigonometric functions, hyperbolic functions, limits and differentiation.
• Integration, techniques such as integration by parts, partial fractions, and multiple integration.
• Review of sequences and series, covering convergence and divergence.
• Exploration of complex numbers, covering axiomatic foundations, complex conjugates, loci, polar form, De Moivre's Theorem, and roots.
• Notation and classification of ordinary differential equations.
• Linear ODEs and their applications.
• Selective exploration of non-linear ODEs and their applications.
• Introduction to systems of ODEs.
• Numerical integration: Trapezoidal Rule and Simpson’s Rule.
• Numerical solutions for ODEs: Euler method, using computer code in, for example, MATLAB, or Python.
• Partial differentiation, functions of two variables.
• Brief introduction to PDEs and their applications.

In this module you will have an introduction to both analysis and linear algebra.

Analysis

What is infinity? And how can we talk about a function 'tending to a limit'? Many problems in mathematics can't be solved exactly, but we can find a series of approximations tending to a true solution. However, our intuition can start to go very wrong in these cases as various pathological functions found in the 19th century illustrate. To make sense of this, we need rigorous definitions to enable us to definitively prove results; these can then be applied to accurately solve problems.

Analysis is the rigorous underpinnings and proof of calculus – differentiation and integration. In this module, we begin by laying the foundations for analysis by studying sequences and series. We then use this to explore functions and their continuity and differentiability.

Topics may include:

• Basic properties of the real numbers.
• Sequences and Series.
• Introduction to real valued functions of one variable, including limits.
• Continuous functions and their properties.
• Continuous functions on closed interval: Boundedness Theorem, Extreme Value Theorem, Intermediate Value Theorem.
• Differentiablility and properties of differentiable functions.
• Rolle’s Theorem, Mean Value Theorem.
• L’Hôpital's rule, Taylor’s Theorem.

Linear Algebra

Linear algebra is of fundamental importance throughout science, engineering and computing and it underpins almost all higher mathematics. Its real-world applications are extensive, encompassing error-correcting codes, noise and signal analysis, facial recognition, quantum computer algorithms, search engine ranking, computer game graphics and even providing you personalised suggestions on Netflix.

In this module, we adopt a dual approach, blending theoretical exploration with a computational and practical perspective on linear algebra. On the theoretical side, we will introduce the abstract notion of a vector space and explore linear transformations between them. We will use these theoretical results to tackle various computational questions ranging from solving systems of linear equations to utilising eigenspace structure for quick matrix computation.

Topics may include:

• Introduction to vectors, Euclidean space and matrices.
• Solving systems of linear equations using matrix techniques.
• Matrices and elementary row operations.
• Generalising Euclidean space over the real numbers to vector spaces over fields.
• Linear dependence and independence, bases, dimension, subspaces.
• Linear maps/transformations between vector spaces.
• Kernel, Image and the Rank-Nullity theorem.
• Matrix determinants and inverses.
• Eigenvalues, eigenvectors and diagonalisation.

Students will be taught through a mixture of lectures, tutorials, computer workshops and through directed self study.

This module is designed with student's employability at the forefront, aiming to develop essential transferable skills which employers look for and value in Mathematics graduates. Programming and computational techniques are all pervasive in today's society, from the next generation of quantum-secure algorithms allowing secure communication with your bank, to using mathematical computer simulations to model real-world phenomena.

The module is split into several different parts. We will provide an in-depth introduction to algorithms and the process of translating these into computer programs, using state-of-the-art software tools, such as Python. This will provide you with a solid foundational understanding to tackle any future computational and programming challenges. You will develop important research and writing skills. You will learn to use LaTeX, an industry-standard typesetting system for produce professional scientific documents. Finally, we will also develop and practice key skills for gaining employment, such as CV building, interview techniques, and giving compelling and engaging presentations.

• Comprehension and understanding of mathematical arguments.
• How to write coherent mathematical arguments.
• Introduction to mathematical and scientific typesetting with the LaTeX package and associated editors.
• Incorporating graphs, figures, tables and bibliographic information in reports and articles, using LaTeX.
• Referencing methodologies in the Mathematical Sciences.
• Introduction to online search tools and searching the library catalogue.
• Introduce algorithms and how to translate them into computer programs.
• Introduce students to computer programming software, for example Python.
• Learn about basic programming concepts including algorithms, loops, conditional statements and functions.  Also learn about more advanced scientific functions.
• Introduction to numerical algorithms from a variety of mathematical areas.
• Working & communicating as part of a scientific team.
• Oral presentation design and delivery.
• CV building, job applications and interview training.

Choose from one of the following modules. Please note you may need GCSE or A-level qualifications in the preferred language to register, although this may not be the case for UO4107 Subsidiary Language for Beginners.

• Subsidiary Language for Beginners
• Chinese: Intermediate Language Development
• French: Intermediate Language Development
• Spanish: Intermediate Language Development
• French: Communication in Practice
• German: Communication in Practice
• Spanish: Communication in Practice

Many optimisation problems in business and industry can be expressed in the form of a linear programming problem and this is even more apparent with increasing reliance on Artificial Intelligence and Machine Learning. Businesses and industry use linear programming to determine what to make in order to maximise their profits, Amazon use it to schedule your parcels for delivery and it is also used widely in Game Theory: you can use it to beat your friends at rock-paper-scissors and other games!

In this module, we will study the theoretical background behind the linear programming methods, learn how to express real-world questions as linear programming problems and solve them by hand and using computer programs. We will also explore some other optimisation methods used in AI and Machine Learning. Topics may include:

• Canonical forms of linear programming problems.
• Theoretical considerations: relevant results from set theory and geometry.
• Integer linear programming.
• Solutions of linear programming models: Simplex and dual simplex methods, Pivot algorithm, and computer-based techniques.
• Degeneracy, cycling, duality.
• Use of a mathematical computer software package, for example, Python, Matlab, Excel etc.
• Application to logistics in transportation and assignment problems, VAM and the Hungarian algorithm.
• Game theory: zero-sum matrix games, multi-phase games.
• Non-linear optimisation techniques in Machine Learning

Stochastic processes serve as essential mathematical models for systems and phenomena exhibiting apparent randomness. Examples encompass diverse scenarios, such as the growth of a bacterial population, fluctuations in electrical current due to thermal noise, or the motion of gas molecules. The applications of stochastic processes span various disciplines, including biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, and financial markets. In order to understand such random behaviour, we introduce and study Markov chains, random walks, Brownian motion and stochastic differential equations. Through these topics, students will not only establish a robust foundation in the principles of stochastic processes but will also gain valuable insights into their diverse applications across numerous domains. The module's goal is to equip students with analytical tools essential for comprehending and modelling complex uncertainties, thereby enhancing their capacity to address real-world challenges in mathematics, statistics, and related fields.

Topics covered include:

• Brief review of Probability Theory via a Measure Theory approach.
• Martingales: Basic definitions, filtrations, stopping times.
• Doob's Martingale inequalities and Convergence Theorem.
• Markov Chains.
• Moment generating functions.
• Characteristic functions.
• Probability generating functions.

Although geometric intuition allows us to understand some simple calculus (differentiation and integration), when things become more complicated we need a more rigorous foundation. Mathematical analysis provides this framework and indispensable to understanding applied techniques such as numerical methods and solving differential equations. By treating the notions of distance and limit formally, this course develops a rigourous theory of convergence, differentiation and integration of real-valued functions.

Geometry is probably the most accessible branch of mathematics, and can provide an easy route to understanding some of the more complex ideas that mathematics can present. This module intends to introduce students to the major geometrical topics taught at undergraduate level, in a manner that is both accessible and rigorous. The course will consider Euclidean geometry and solid geometry, covering topics such as the equation of lines and planes, inner product, cross product, and triple product. Spherical geometry, featuring concepts including great circles, geodesics, and spherical trigonometry, will be explored for its significant applications in fields like astronomy, navigation and understanding the Earth. Additionally, the study will extend to quadratic curves, quadratic surfaces, and planetary motion.  Through this comprehensive study, students will understand the various geometric concepts, laying a foundation that combines accessibility with the necessary mathematical rigour.

Topics in Geometry are drawn from the following:

• Geometry of numbers (including the golden ratio and continued fractions).
• Coordinate geometry: lines, planes and hyperplanes.
• Euclidian geometry.
• Solid geometry: vector products, polyhedra.
• Projective geometry: basic concepts & applications.
• Spherical geometry.
• Use of appropriate software to visualise problems (such as Autograph or Matlab, for example).
• Real-life applications of geometry: examples drawn from a wide range of contexts (e.g. cartography, forensic science, navigation, architecture).

Students have the option to choose to take part in either a:

• Professional work placement
• Language module
• Term abroad

Group theory is the mathematical study of symmetry, a concept which appears throughout mathematics, science and nature. Physicists use group theory to describe the properties of fundamental particles of matter, Chemists use it to investigate crystal structure and it plays a central role in our every-day lives in cryptography. In mathematics, groups are found everywhere, in algebraic objects such as rings, field, vector spaces and matrices, in analysis, and in solutions to equations and differential equations.

This module will study groups and their actions on sets and geometric objects.  Highlights include the Jordan Holder Theorem, which allows us to break down groups into their fundamental building blocks, called the simple groups; and Sylow's theorems which are of fundamental importance to understanding the structure of finite groups. Topics covered may include:

• Groups
• Subgroups
• Cosets, normal subgroups, quotient groups
• Homomorphisms and the Isomorphism Theorems
• The Classification of finite abelian groups
• Group actions and the Orbit-Stabiliser Theorem
• The Jordan Holder Theorem
• Simple Groups
• Sylow Theory
• Solvable Groups

Vector calculus and partial differential equations are indispensable tools in science. These topics are essential to our understanding of electromagnetism and quantum mechanics in physics, and also to the modelling of physical phenomenon such that fluid flow and heat conduction. This course is divided into two halves. The first half will cover topics related to three-dimensional geometry and vector calculus including: vector-valued functions, parametrised curves, line integrals and conservative vector fields, multiple integrals, surface integrals, the theorems of Green and Stokes, and the Divergence Theorem. The second half focusses on the theory of and techniques for solving partial differential equations. Topics covered will include: first order PDEs, classification and techniques for solving linear second order PDEs, Fourier series, the method of separation of variables applied to the heat, wave and Laplace equations, and Fourier transforms.

Numerical analysis is the exploration of algorithms that rely on numerical approximation; this is invaluable for the study of many complicated problems, where an exact solution is very difficult, or even impossible to obtain. Its applications extend across mathematics, physical sciences and engineering, and, in the 21st century, have expanded into life sciences, social sciences, medicine, business, and even the arts. This module serves as an introduction to the fundamental techniques and methodologies for solving mathematical problems using numerical methods and equips students with the skills needed to analyse, design, and implement numerical algorithms effectively across a broad range of mathematical problems.

The topics include:

• Solving nonlinear equations, finding roots, Newton iteration and related methods.
• Introduction to optimisation, optimisation of functions of several variables, with and without constraints.
• Interpolation and approximations: Lagrange interpolation, Hermite interpolations,
• Numerical integration and differentiation: Trapezoidal method, Simpson method.
• Solutions of ordinary differential equations: Euler method, Runge-Kutta method, multistep methods, stability, convergence.
• Solution techniques for partial differential equations including the heat equation.
• Implementation and programming, e.g., Python, MATLAB.
• Error analysis: developing an understanding of the sources of error in numerical computations and methods for analysing and controlling numerical errors.

The project gives the student an opportunity to apply theory learned on the programme and to develop skills of self-discipline, project management and written communication.

Students will negotiate with tutors the precise title and objectives of the project. Students will study the art of mathematical writing and communication. Tutors will provide appropriate levels of support and advice.