It is in its proofs that the strength and richness of mathematics are to be found. University mathematics introduces progressively more abstract ideas and structures and demands more in the way of proof until most of your time is occupied with understanding proofs and creating your own. Learning to deal with abstraction and with proofs takes time. This module will bridge the gap between school and university mathematics, taking you from concrete techniques where the emphasis is on the calculation, and gradually moving towards abstraction and proof.

This module also looks at algorithms and operational complexity, including cryptographic keys and RSA.

Analysis is the rigorous study of calculus. In this module, there will be a considerable emphasis throughout on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers, and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. The module will allow you to deal carefully with limits and infinite summations, approximations to pi and e, and the Taylor series. The module ends with the construction of the integral and the Fundamental Theorem of Calculus.

This first half of this module will introduce you to abstract algebra, covering group theory and ring theory, making you familiar with symmetry groups and groups of permutations and matrices, subgroups and Lagrange’s theorem. You will understand the abstract definition of a group, and become familiar with the basic types of examples, including number systems, polynomials, and matrices. You will be able to calculate the unit groups of the integers modulo n.

The second half concerns linear algebra, and addresses simultaneous linear equations. You will learn about the properties of vector spaces, linear mappings and their representation by matrices. Applications include solving simultaneous linear equations, properties of vectors and matrices, properties of determinants and ways of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or matrix. You will have an understanding of matrices and vector spaces for later modules to build on.

This module contains a Python mini-course and an introduction to the Latex scientific document preparation package. It will involve a group project, involving computation, and students will develop their research skills, including planning and use of library and internet resources, and their presentation skills including a video presentation.

Methods of Mathematical Modelling 1 introduces you to the fundamentals of mathematical modelling and scaling analysis, before discussing and analysing difference and differential equation models in the context of physics, chemistry, engineering as well as the life and social sciences. This will require the basic theory of ordinary differential equations (ODEs), the cornerstone of all applied mathematics. ODE theory later proves invaluable in branches of pure mathematics, such as geometry and topology. You will be introduced to simple differential and difference equations, methods for obtaining their solutions and numerical approximation.

In the second term for Methods of Mathematical Modelling 2, you will study the differential geometry of curves, calculus of functions of several variables, multi-dimensional integrals, calculus of vector functions of several variables (divergence and circulation), and their uses in line and surface integrals.

This module takes you further in your exploration of probability and random outcomes. Starting with examples of discrete and continuous probability spaces, you will learn methods of counting (inclusion-exclusion formula and multinomial coefficients), and examine theoretical topics including independence of events and conditional probabilities. You will study random variables and their probability distribution functions. Finally, you will study variance and co-variance, including Chebyshev’s and Cauchy-Schwarz inequalities. The module ends with a discussion of the celebrated Central Limit Theorem.

• Introduction to Probability
• Statistical Laboratory

• Classical Mechanics and Special Relativity
• Electricity and Magnetism
• Astronomy, Quantum Phenomena

• Discrete Mathematics and its Applications 2

• Mind and Reality
• Reason, Argument and Analysis
• Logic I:Introduction to Symbolic Logic

• Introduction to Quantitative Economics

• Mathematical Programming I

The Language Centre offers academic modules in Arabic, Chinese, French, German, Italian, Japanese, Korean, Portuguese, Russian and Spanish at a wide range of levels.