Core modules
Our degree programme consists of core and optional modules. In core modules, you will study essential topics in algebra, analysis and applied mathematics. Optional modules cover the entire range of mathematical sciences, including algebra, combinatorics, number theory, geometry, topology, pure and applied analysis, differential equations, and applications to physical, biological and data sciences. There are core modules in the first and second years of study.
The third and fourth years comprise optional modules, plus the fourth-year project. At Warwick, our wide range of options enables you to explore in depth your love of mathematics, while the flexible system allows you to explore other subjects you enjoy outside of mathematics (as much as 25% in each year can be in non-maths modules).
Year One
Foundations
This module will bridge the gap between school and university mathematics, taking you from a calculation and method-based approach to a deeper understanding of mathematics based on proofs. In this module, you will learn the art of mathematical proofs through understanding key methods of proof and creating your own.
In addition to proofs, the module will also introduce you to essential university mathematics, including set theory, logic, number theory, algorithms and cryptography.
Read more about the Foundations moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2022/23 year of study).
Analysis I/II
Analysis is the rigorous study of calculus. In this module, there will be a considerable emphasis throughout on the need to lay out mathematical arguments 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, say, differentiate and integrate a function, to the point where you can develop your own rigorous proofs of calculus results that you may have taken for granted. 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.
Read more about these modules, including the methods of teaching and assessment (content applies to 2023/24 year of study):
Mathematical Methods and Modelling I and II
Mathematical Methods and Modelling I introduces you to the fundamentals of mathematical modelling, before discussing and analysing difference and differential equations in 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 Mathematical Methods and Modelling II, 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.
Read more about these modules, including the methods of teaching and assessment (content applies to 2023/24 year of study):
Algebra I and II
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 methods of calculating them. You will learn to define and calculate eigenvalues and eigenvectors of a linear map or a matrix. You will gain a solid understanding of matrices and vector spaces for later modules to build on.
Read more about these modules, including the methods of teaching and assessment (content applies to 2023/24 year of study):
Mathematics by Computer
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.
Read more about the Mathematics by Computer moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Introduction to Probability
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 and famous probability theorems.
Year Two
Methods of Mathematical Modelling III
You will study a number of key concepts in mathematical modelling: (i) Optimisation (including critical points in multi-dimensions, linear programming, least squares, regression, convexity, steepest descent algorithms, optimisation with constraints, neural network); (ii) The Fast Fourier Transform (including its application to signal processing and audio and video compression) (iii) Hilbert Spaces (including orthogonal functions and their use in approximation problems).
Read more about the Methods of Mathematical Modelling III moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Algebra III
This course focuses on developing your understanding and application of the theories of groups and rings, improving your ability to manipulate them and extending the results from Year One algebra. You will learn how to prove the isomorphism theorems for groups in general, and analogously, for rings. You will also encounter the Orbit-Stabiliser Theorem, the Chinese Remainder Theorem, and Gauss’s theorem on unique factorisation in polynomial rings. You will study applications in number theory, geometry and combinatorics.
Read more about the Algebra III moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Norms, Metrics and Topologies
Roughly speaking, a metric space is any set provided with a sensible notion of the “distance” between points. The ways in which distance is measured and the sets involved may be very diverse. For example, the set could be the sphere, and we could measure distance either along great circles or along straight lines through the globe; or the set could be New York and we could measure distance “as the crow flies” or by counting blocks. This module examines how the important concepts introduced in first-year Mathematical Analysis, such as convergence of sequences and continuity of functions, can be extended to general metric spaces. Applying these ideas, we will be able to prove some powerful and important results, used in many parts of mathematics.
Read more about the Norms, Metrics and Topologies moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Mathematical Analysis III
In the first half of this module, you will investigate some applications of year one analysis: integrals of limits and series; differentiation under an integral sign; a first look at Fourier series. In the second half you will study analysis of complex functions of a complex variable: contour integration and Cauchy’s theorem, and its application to Taylor and Laurent series and the evaluation of real integrals.
Read more about the Analysis III moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Scientific Communication
You will undertake independent research on a mathematical topic with guidance and feedback from your Personal Tutor. You will investigate mathematics that may not be covered in the core curriculum. You will then communicate your research in a scientific report and an oral presentation.
Read more about the Scientific Communication moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Multilinear Algebra
In this module, you will develop and continue your study of linear algebra: the Jordan normal form for matrices; functions of matrices; symmetric and quadratic forms; tensors; bilinear forms; dual spaces.
Read more about the Multilinear Algebra is moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Multivariable Analysis
This module introduces the concept of continuity and differentiability for multivariable functions, generalising the concepts studied in previous Analysis and Mathematical Methods modules. You will study multivariable generalisations of the derivative, the inverse-function and implicit-function theorems. You will revisit the divergence and Stokes’ theorems from the point of view of multivariable analysis, and study solutions of partial differential equations.
Read more about the Multivariable Analysis moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Year Three
There are no core modules. Instead you will select from an extensive range of optional modules in both mathematics and a range of other subjects from departments across the university. You will be able to take up to 50% (BSc) or 25% (MMath) of your options in subjects other than mathematics should you wish to do so.
Year Four
Research Project
The primary aim of the Research Project is to give you experience of mathematics as it is being pursued close to the frontiers of research, not just as a spectator sport but as an engaging, evolving activity in which you can play a part. You will communicate your findings in a written scholarly report.
Read more about the Research Project moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
or
Maths-in-Action (MiA-Projects)
The MiA projects are primarily aimed at those who seek to further develop their skills in public speaking and writing. The project involves understanding deeply how mathematics underpins a particular topic in the modern world and then communicating this understanding in the form of a presentation to the general public, a written popular science article, and a written scholarly report at the MMath level.
Read more about the Maths-in-Action (MiA-Projects) moduleLink opens in a new window, including the methods of teaching and assessment (content applies to 2023/24 year of study).
Optional modules
Optional modules can vary from year to year. Example optional modules may include:
- Mathematics: Number Theory, Galois Theory, Fractal Geometry, Topology, Partial Differential Equations, Curves and Surfaces, Fluid Mechanics, Machine Learning
- Statistics: Mathematical Finance; Brownian Motion; Medical Statistics; Designed Experiments
- Computer Science: Complexity of Algorithms; Computer Graphics
- Physics: Introduction to Astronomy; Introduction to Particle Physics; Quantum Phenomena; Nuclear Physics; Stars and Galaxies
- Economics: Mathematical Economics
- Other: Introduction to Secondary School Teaching; Climate Change; Language Options (at all levels)
