# Subject pathways within the Natural Sciences degree (2025/26)

## Mathematics

Mathematics is key to the sciences and to a cross-section of business disciplines. The ongoing drive for economic efficiency, the increasing importance of technology and big data, and new emerging areas such as climate science all mean that mathematics continues to have a significant impact on the world. Demand for mathematics skills comes from all sectors – from business and technology to science, research and development and IT – meaning the career options available are varied and rewarding.

If you decide to take the Mathematics subject pathway as part of your Natural Sciences degree, you'll take a series of optional and compulsory modules within this pathway.

Please note: The modules listed below are indicative of typical options and some of these options may not be available, depending on other modules you have selected already.

### Year 1

**Compulsory module**

**Core Mathematics – 40 credits**

Learn the foundational concepts of function, number and proof, equipping you with the language and skills to tackle your mathematical studies. The module also consolidates basic calculus, extending it to more advanced techniques, such as functions of several variables. These techniques lead to methods for solving simple ordinary differential equations. Linear algebra provides a basis for wide areas of mathematics and this module provides the essential foundation.

### Year 2

For year 2, you’ll take modules from one of the following subject pathways:

**Applied Mathematics****Pure Mathematics****Pure and Applied Mathematics**

**Applied Mathematics pathway**

**Compulsory modules**

**Vector Calculus and Partial Differential Equations – 20 credits**

Vector calculus is the extension of ordinary one-dimensional differential and integral calculus to higher dimensions, and provides the mathematical framework for the study of a wide variety of physical systems, such as fluid mechanics and electromagnetism.

These systems give rise to partial differential equations (PDEs), which can be solved and analysed. you’ll learn to use, among others, techniques introduced in earlier modules as well as being introduced to Fourier methods for PDEs.

**Mathematical Modelling – 20 credits**

Learn analytical and computational techniques for the solution of ordinary and partial differential equations, which describe particle motion in fields, fluids, waves, diffusion and many other phenomena.

**Optional modules**

**Optimisation – 10 credits**

Optimisation, “the quest for the best”, plays a major role in financial and economic theory, such as maximising a company's profits or minimising its production costs. This module develops the theory and practice of maximising or minimising a function of many variables, and thus lays a solid foundation for progression onto more advanced topics, such as dynamic optimisation, which are central to the understanding of realistic economic and financial scenarios.

**Calculus of Variations – 10 credits**

The calculus of variations concerns problems in which one wishes to find the extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics. In this module, you’ll meet the system of differential equations arising from such variational problems: the Euler-Lagrange equations. These equations and the techniques for their solution, will be studied in detail.

**Pure Mathematics pathway**

**Compulsory modules**

**Further Linear Algebra and Discrete Mathematics – 20 credits**

Explore the more abstract ideas of vector spaces and linear transformations, together with introducing the area of discrete mathematics.

**Introduction to Logic – 10 credits**

Discover mathematical logic introducing formal languages that can be used to express mathematical ideas and arguments. It throws light on mathematics itself, because it can be applied to problems in philosophy, linguistics, computer science and other areas.

**Rings and Polynomials – 10 credits**

Rings are one of the fundamental concepts of mathematics, and they play a key role in many areas, including algebraic geometry, number theory, Galois theory and representation theory. The aim of this module is to give an introduction to rings. The emphasis will be on interesting examples of rings and their properties.

**Optional modules**

**Optimisation – 10 credits**

Optimisation, “the quest for the best”, plays a major role in financial and economic theory, such as maximising a company's profits or minimising its production costs. This module develops the theory and practice of maximising or minimising a function of many variables, and thus lays a solid foundation for progression onto more advanced topics, such as dynamic optimisation, which are central to the understanding of realistic economic and financial scenarios.

**Calculus of Variations – 10 credits**

The calculus of variations concerns problems in which one wishes to find the extrema of some quantity over a system that has functional degrees of freedom. Many important problems arise in this way across pure and applied mathematics. In this module, you’ll meet the system of differential equations arising from such variational problems: the Euler-Lagrange equations. These equations and the techniques for their solution, will be studied in detail.

**Pure and Applied Mathematics pathway**

**Compulsory modules**

**Further Linear Algebra and Discrete Mathematics – 20 credits**

Explore the more abstract ideas of vector spaces and linear transformations, together with introducing the area of discrete mathematics.

**Vector Calculus and Partial Differential Equations – 20 credits**

Vector calculus is the extension of ordinary one-dimensional differential and integral calculus to higher dimensions, and provides the mathematical framework for the study of a wide variety of physical systems, such as fluid mechanics and electromagnetism.

These systems give rise to partial differential equations (PDEs), which can be solved and analysed. Students will learn to use, among others, techniques introduced in earlier modules as well as being introduced to Fourier methods for PDEs.

**Optional modules**

**Introduction to Logic – 10 credits**

Discover mathematical logic introducing formal languages that can be used to express mathematical ideas and arguments. It throws light on mathematics itself, because it can be applied to problems in philosophy, linguistics, computer science and other areas.

**Mathematical Modelling – 20 credits**

Learn analytical and computational techniques for the solution of ordinary and partial differential equations, which describe particle motion in fields, fluids, waves, diffusion and many other phenomena.

### Year 3

For year 3, you’ll continue on the same pathway you chose in year 2:

**Applied Mathematics****Pure Mathematics****Pure and Applied Mathematics**

**Applied Mathematics pathway**

**Compulsory module**

**Methods of Applied Mathematics – 20 credits**

Develop techniques to solve ordinary and partial differential equations arising in mathematical physics. For the important case of second-order PDEs, we distinguish between elliptic equations (e.g., Laplace's equation), parabolic equations (e.g., heat equation) and hyperbolic equations (e.g., wave equation), and physically interpret the solutions. When there is not an exact solution in closed form, approximate solutions (so-called perturbation expansions) can be constructed if there is a small or large parameter.

**Optional modules**

**Entropy and Quantum Mechanics – 20 credits**

The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. Further, at the microscopic atomic-scale particles behave like waves, with dynamics that is known only statistically. So, why is it that the materials around us behave in predictable and regular ways? One reason is that random behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. A second reason is that the mathematics of quantum mechanics provides incredibly accurate predictions at the atomic scale. This module studies calculations involving both entropy and quantum mechanics, as applied to the matter that makes up our world.

**Fluid Dynamics – 20 credits**

Fluid dynamics is the science that describes the motion of materials that flow. It constitutes a significant mathematical challenge with important implications in an enormous range of fields in science and engineering, including aerodynamics, astrophysics, climate modelling, and physiology. This module sets out the fundamental concepts of fluid dynamics, for both inviscid and viscous flows. It includes a formal mathematical description of fluid flow and the derivation of the governing equations, using techniques from vector calculus. Solutions of the governing equations are derived for a range of simple flows, giving you a feel for how fluids behave and experience in modelling everyday phenomena.

**Mathematical Biology – 20 credits**

Mathematics is increasingly important in biological and medical research. This module aims to introduce you to some areas of mathematical biology and medicine, using tools from applied mathematics.

**Nonlinear Dynamical Systems and Chaos – 20 credits**

Many applications, ranging from biology to physics and engineering, are described by nonlinear dynamical systems, in which a change in output is not proportional to a change in input. Nonlinear dynamical systems can exhibit sudden changes in behaviour as parameters are varied and even unpredictable, chaotic dynamics. This module will provide you with the mathematical tools to analyse nonlinear dynamical systems, including identifying bifurcations and chaotic dynamics.

**Pure Mathematics pathway**

**Compulsory module**

**Groups and Symmetry – 20 credits**

Group theory is the mathematical theory of symmetry. Groups arise naturally in pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials. Groups have beautiful applications to counting problems, answering questions like: "How many ways are there to colour the faces of a cube with m colours, up to rotation of the cube?

**Optional modules**

**Graph Theory and Combinatorics – 20 credits**

Graph theory is one of the primary subjects in discrete mathematics. It arises wherever networks as seen in computers or transportation are found, and it has applications to fields diverse as chemistry, computing, linguistics, navigation and more. More generally, combinatorics concerns finding patterns in discrete mathematical structures, often with the goal of counting the occurrences of such patterns. This module provides a foundation in graph theory and combinatorics.

**Proof and Computation – 20 credits**

The main goal of this module is to prove Gödel's First Incompleteness Theorem (1931) which shows that, if any reasonable formal theory has strong enough axioms, there are statements which it can neither prove nor refute. This module will also provide background to the impact of Gödel's Theorem on the modern world and the way it sets an agenda for further research.

**Numbers and Codes – 20 credits**

Number theory explores the natural numbers. Central themes include primes, arithmetic modulo n, and Diophantine equations as in Fermat's Last Theorem. It is a wide-ranging current field with many applications, e.g. in cryptography.

Error-correcting codes tackle the problem of reliably transmitting digital data through a noisy channel. Applications include transmitting satellite pictures, designing registration numbers and storing data. The theory uses methods from algebra and combinatorics.

This module introduces both subjects. It emphasises common features, such as algebraic underpinnings, and applications to information theory, both in cryptography (involving secrecy) and in error-correcting codes (involving errors in transmission).

**Pure and Applied Mathematics pathway**

**Optional modules**

**Methods of Applied Mathematics – 20 credits**

Develop techniques to solve ordinary and partial differential equations arising in mathematical physics. For the important case of second-order PDEs, we distinguish between elliptic equations (e.g., Laplace's equation), parabolic equations (e.g., heat equation) and hyperbolic equations (e.g., wave equation), and physically interpret the solutions. When there is not an exact solution in closed form, approximate solutions (so-called perturbation expansions) can be constructed if there is a small or large parameter.

**Groups and Symmetry – 20 credits**

Group theory is the mathematical theory of symmetry. Groups arise naturally in pure and applied mathematics, for example in the study of permutations of sets, rotations and reflections of geometric objects, symmetries of physical systems and the description of molecules, crystals and materials. Groups have beautiful applications to counting problems, answering questions like: "How many ways are there to colour the faces of a cube with m colours, up to rotation of the cube?"

**Graph Theory and Combinatorics – 20 credits**

Graph theory is one of the primary subjects in discrete mathematics. It arises wherever networks as seen in computers or transportation are found, and it has applications to fields diverse as chemistry, computing, linguistics, navigation and more. More generally, combinatorics concerns finding patterns in discrete mathematical structures, often with the goal of counting the occurrences of such patterns. This module provides a foundation in graph theory and combinatorics.

**Entropy and Quantum Mechanics – 20 credits**

The material world is composed of countless microscopic particles. When three or more particles interact, their dynamics is chaotic, and impossible to predict in detail. Further, at the microscopic atomic-scale particles behave like waves, with dynamics that is known only statistically. So, why is it that the materials around us behave in predictable and regular ways? One reason is that random behaviour on the microscopic scale gives rise to collective behaviour that can be predicted with practical certainty, guided by the principle that the total disorder (or entropy) of the universe never decreases. A second reason is that the mathematics of quantum mechanics provides incredibly accurate predictions at the atomic scale. This module studies calculations involving both entropy and quantum mechanics, as applied to the matter that makes up our world.

**Fluid Dynamics – 20 credits**

Fluid dynamics is the science that describes the motion of materials that flow. It constitutes a significant mathematical challenge with important implications in an enormous range of fields in science and engineering, including aerodynamics, astrophysics, climate modelling, and physiology. This module sets out the fundamental concepts of fluid dynamics, for both inviscid and viscous flows. It includes a formal mathematical description of fluid flow and the derivation of the governing equations, using techniques from vector calculus. Solutions of the governing equations are derived for a range of simple flows, giving you a feel for how fluids behave and experience in modelling everyday phenomena.

**Proof and Computation – 20 credits**

The main goal of this module is to prove Gödel's First Incompleteness Theorem (1931) which shows that, if any reasonable formal theory has strong enough axioms, there are statements which it can neither prove nor refute. This module will also provide background to the impact of Gödel's Theorem on the modern world and the way it sets an agenda for further research.

**Numbers and Codes – 20 credits**

Number theory explores the natural numbers. Central themes include primes, arithmetic modulo n, and Diophantine equations as in Fermat's Last Theorem. It is a wide-ranging current field with many applications, e.g. in cryptography.

Error-correcting codes tackle the problem of reliably transmitting digital data through a noisy channel. Applications include transmitting satellite pictures, designing registration numbers and storing data. The theory uses methods from algebra and combinatorics.

This module introduces both subjects. It emphasises common features, such as algebraic underpinnings, and applications to information theory, both in cryptography (involving secrecy) and in error-correcting codes (involving errors in transmission).

**Mathematical Biology – 20 credits**

Mathematics is increasingly important in biological and medical research. This module aims to introduce you to some areas of mathematical biology and medicine, using tools from applied mathematics.

**Nonlinear Dynamical Systems and Chaos – 20 credits**

Many applications, ranging from biology to physics and engineering, are described by nonlinear dynamical systems, in which a change in output is not proportional to a change in input. Nonlinear dynamical systems can exhibit sudden changes in behaviour as parameters are varied and even unpredictable, chaotic dynamics. This module will provide you with the mathematical tools to analyse nonlinear dynamical systems, including identifying bifurcations and chaotic dynamics.

### Year 4 (MNatSci, BSc)

For year 4, you’ll continue on the same pathway you chose in year 3:

**Applied Mathematics****Pure Mathematics****Pure and Applied Mathematics**

**Applied Mathematics pathway**

**Optional modules**

**Evolutionary Modelling – 15 credits**

Darwin’s natural selection theory is a cornerstone of modern science. Recently, mathematical and computational modelling has led to significant advances in our understanding of evolutionary puzzles, such as what determines biodiversity or the origin of cooperative behaviour. On this module, you will be exposed to fundamental ideas of evolutionary modelling, and to the mathematical tools needed to pursue their study. These will be illustrated by numerous examples motivated by exciting developments in mathematical biology.

**Advanced Mathematical Methods – 15 credits**

Many real-world problems can be modelled by ordinary or partial differential equations or formulated as a complicated integral. In this module, advanced techniques are developed to solve such problems and interpret their solutions, motivated by examples from mathematical physics, continuum mechanics, and mathematical biology. These techniques include so-called asymptotic methods, which yield approximate solutions if the problem contains a small or large parameter.

**Astrophysical and Geophysical Fluids – 15 credits**

This module concerns mathematical modelling of various phenomena observed in astrophysical and geophysical flows, meaning those in planetary and stellar atmospheres and interiors. The focus is on understanding key dynamical processes in such flows, including those due to rotation and density stratification and, in many astrophysical flows, the electrical conductivity of the fluid (which can thus support a magnetic field). These effects lead to various interesting waves and instabilities, with physical and observational significance.

**Environmental and Industrial Flows – 15 credits**

Many flows found in nature such as avalanches and glaciers or in industrial applications such as 3D printing and coatings involve complex fluids, whose properties can be very different from those of simple Newtonian fluids like air and water. This course gives an introduction into the often-surprising behaviour of these fluids and how they can be modelled mathematically using differential equations.

**Classical and Quantum Hamiltonian Systems – 15 credits**

The Hamiltonian formulation of dynamics is the most mathematically beautiful form of mechanics and a stepping stone to quantum mechanics. Hamiltonian systems are conservative dynamical systems with a very interesting algebraic and geometric structure: the Poisson bracket. Hamilton's equations are invariant under a very wide class of transformations and this leads to a number of powerful solution techniques.

**Pure Mathematics pathway**

**Optional modules**

**Models and Sets – 15 credits**

Set Theory is generally accepted as a foundation for mathematics, in an informal sense, but is also a formal axiomatic system. Model Theory is the study of formal axiomatic systems and depends on Set Theory for many of its basic definitions and results. Model Theory and Set Theory constitute two of the basic strands of mathematical logic. In this module, we explain the basic notions of these interrelated subjects.

**Algebras and Representations – 15 credits**

An algebra is a vector space with a compatible binary operation, or multiplication. A natural example is the set of all square matrices of a fixed size with complex entries, under matrix multiplication. Semisimple algebras form an important class of algebras and one of the highlights of the course is the structure theorem classifying the semisimple algebras in terms of matrix algebras. The module will also study representations of general algebras via matrices.

**Topology – 15 credits**

A topological space is a set with the minimal added structure that makes it possible to define continuity of functions. In this module, we will define topological spaces and explain what it means for them to be connected, path-connected and compact. In the second half of this module, we study algebraic topology and show some applications to Euclidean space: e.g any continuous map from a disk to itself must have a fixed point.

**Pure and Applied Mathematics pathway**

**Optional modules**

**Evolutionary Modelling – 15 credits**

Darwin’s natural selection theory is a cornerstone of modern science. Recently, mathematical and computational modelling has led to significant advances in our understanding of evolutionary puzzles, such as what determines biodiversity or the origin of cooperative behaviour. On this module, you will be exposed to fundamental ideas of evolutionary modelling, and to the mathematical tools needed to pursue their study. These will be illustrated by numerous examples motivated by exciting developments in mathematical biology.

**Advanced Mathematical Methods – 15 credits**

Many real-world problems can be modelled by ordinary or partial differential equations or formulated as a complicated integral. In this module, advanced techniques are developed to solve such problems and interpret their solutions, motivated by examples from mathematical physics, continuum mechanics, and mathematical biology. These techniques include so-called asymptotic methods, which yield approximate solutions if the problem contains a small or large parameter.

**Astrophysical and Geophysical Fluids – 15 credits**

This module concerns mathematical modelling of various phenomena observed in astrophysical and geophysical flows, meaning those in planetary and stellar atmospheres and interiors. The focus is on understanding key dynamical processes in such flows, including those due to rotation and density stratification and, in many astrophysical flows, the electrical conductivity of the fluid (which can thus support a magnetic field). These effects lead to various interesting waves and instabilities, with physical and observational significance.

**Environmental and Industrial Flows – 15 credits**

Many flows found in nature such as avalanches and glaciers or in industrial applications such as 3D printing and coatings involve complex fluids, whose properties can be very different from those of simple Newtonian fluids like air and water. This course gives an introduction into the often-surprising behaviour of these fluids and how they can be modelled mathematically using differential equations.

**Classical and Quantum Hamiltonian Systems – 15 credits**

The Hamiltonian formulation of dynamics is the most mathematically beautiful form of mechanics and a stepping stone to quantum mechanics. Hamiltonian systems are conservative dynamical systems with a very interesting algebraic and geometric structure: the Poisson bracket. Hamilton's equations are invariant under a very wide class of transformations and this leads to a number of powerful solution techniques.

**Models and Sets – 15 credits**

Set Theory is generally accepted as a foundation for mathematics, in an informal sense, but is also a formal axiomatic system. Model Theory is the study of formal axiomatic systems and depends on Set Theory for many of its basic definitions and results. Model Theory and Set Theory constitute two of the basic strands of mathematical logic. In this module, we explain the basic notions of these interrelated subjects.

**Algebras and Representations – 15 credits**

An algebra is a vector space with a compatible binary operation, or multiplication. A natural example is the set of all square matrices of a fixed size with complex entries, under matrix multiplication. Semisimple algebras form an important class of algebras and one of the highlights of the course is the structure theorem classifying the semisimple algebras in terms of matrix algebras. The module will also study representations of general algebras via matrices.

**Topology – 15 credits**

A topological space is a set with the minimal added structure that makes it possible to define continuity of functions. In this module, we will define topological spaces and explain what it means for them to be connected, path-connected and compact. In the second half of this module, we study algebraic topology and show some applications to Euclidean space: e.g any continuous map from a disk to itself must have a fixed point.