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