# Modules

On our two main degrees, core modules will give you a grounding in mathematics, and in your second and third years, you can choose to explore the topics that interest you the most. Whatever you choose, you are taught by staff that lead the field in their chosen disciplines.

## Maths modules

Our core modules in the first and second year, outlined below, will provide you with a strong foundation in mathematics. You will begin exploring option modules in your first year, and then in your second and further years, you can choose from a wide array of optional modules to suit your academic, creative, social, and career interests.

Your tutors are keen to discuss the topics that motivate and excite you, and will carefully guide you to take the direction that’s best for you.

### First-year core Maths modules

Foundations, Differential Equations, Introduction to Abstract Algebra, Analysis I, Analysis II, Linear Algebra, Maths by Computer, Geometry and Motion, Probability A.

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.

For more in-depth information on this module, visit the undergraduate handbook.

Can you predict the trajectory of a tennis ball? In this module, you cover the basic theory of ordinary differential equations (ODEs), the cornerstone of all applied mathematics. ODE theory proves invaluable in branches of pure mathematics, such as geometry and topology. You will be introduced to simple differential and difference equations and methods for their solution. You will cover first-order equations, linear second-order equations and coupled first-order linear systems with constant coefficients, and solutions to differential equations with one-and two-dimensional systems. We will discuss why in three dimensions we see new phenomena and have the first glimpse of chaotic solutions.

For more in-depth information on this module, visit the undergraduate handbook.

This course 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.

For more in-depth information on this module, visit the undergraduate handbook.

You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others, concerning sequences and series in term one, continuity and differentiability in term two and integration in term one of your second year. By the end of the year you will be able to answer many interesting questions: What do we mean by `infinity'? How can you accurately compute the value of π or e or 2–√ ? How can you add up infinitely many numbers, or infinitely many functions? Can all functions be approximated by polynomials?

For more in-depth information on this module, visit the undergraduate handbook.

You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others, concerning sequences and series in term one, continuity and differentiability in term two and integration in term one of your second year. By the end of the year you will be able to answer many interesting questions: What do we mean by `infinity'? How can you accurately compute the value of π or e or 2–√ ? How can you add up infinitely many numbers, or infinitely many functions? Can all functions be approximated by polynomials?

For more in-depth information on this module, visit the undergraduate handbook.

Python is a widely-used programming language that is increasingly essential knowledge for mathematicians and scientists. Python underlies important mathematical software such as Sage. It dominates many modern applications, particularly in Data Science and Machine Learning. This module will rapidly introduce you to some of the most important aspects of Python for mathematical and scientific work. You will learn to use powerful libraries that carry out complex tasks and allow you to concentrate on the "big picture". Topics covered include numerical solutions of ODEs, Monte Carlo integration, machine learning, computer graphics, and image processing.

For more in-depth information on this module, visit the undergraduate handbook.

Geometry and motion are connected as a particle curves through space, and in the relation between curvature and acceleration. In this course you will discover how to integrate vector-valued functions and functions of two and three real variables. You will encounter concepts in particle mechanics, deriving Kepler’s Laws of planetary motion from Newton’s second law of motion and the law of gravitation. You will see how intuitive geometric and physical concepts such as length, area, volume, curvature, mass, circulation and flux can be translated into mathematical formulas, and appreciate the importance of conserved quantities in mechanics.

For more in-depth information on this module, visit the undergraduate handbook.

If you’ve covered mathematical modules MA131 and MA132, this takes you further in your exploration of probability and random outcomes. Starting with examples of discrete and continuous probability spaces, you’ll learn methods of counting (inclusion–exclusion formula and multinomial co-efficients), and examine theoretical topics including independence of events and conditional probabilities. Using Bayes’ theorem and Simpson’s paradox, you’ll reason about a range of problems involving belief updates, and engage with random variables, learning about probability mass, density and cumulative distribution functions, and the important families of distributions. Finally, you’ll study variance and co-variance, including Chebyshev’s and Cauchy-Schwartz inequalities.

For more in-depth information on this module, visit the undergraduate handbook.

#### First- year optional Maths modules

Choose from an extensive list of optional modules offered from a wide range of other departments.

Probability B, Programming for Scientists, Statistical Laboratory.

For more in-depth information on this module, visit the undergraduate handbook.

Classical Mechanics and Special Relativity, Electricity and Magnetism, Introduction to Astronomy, Introduction to Particle Physics, Quantum Phenomena.

For more in-depth information on this module, visit the undergraduate handbook.

From Computer Science

Mind and Reality, Introduction to Symbolic Logic.

For more in-depth information on this module, visit the undergraduate handbook.

Introduction to Quantitative Economics.

For more in-depth information on this module, visit the undergraduate handbook.

Mathematical Programming I.

For more in-depth information on this module, visit the undergraduate handbook.

### Second-year core Maths modules

During your second year of study, you will have the opportunity to deepen your knowledge of key mathematical areas as well as develop communication skills by writing an essay on a topic of special interest to you, decided in consultation with your tutor.

There are many situations in pure and applied mathematics where the continuity and differentiability of a function f: R n. → R m has to be considered. Yet, partial derivatives, while easy to calculate, are not robust enough to yield a satisfactory differentiation theory. In this module you will establish the basic properties of this derivative, which will generalise those of single-variable calculus. The module will review line and surface integrals, introduce div, grad and curl and establish the divergence theorem.

For more in-depth information on this module, visit the undergraduate handbook.

On this course, you will develop and continue your study of linear algebra. You will develop methods for testing whether two general matrices are similar, and study quadratic forms. Finally, you will investigate matrices over the integers, and investigate what happens when we restrict methods of linear algebra to operations over the integers. This leads, perhaps unexpectedly, to a complete classification of finitely generated abelian groups. You will be familiarised with the Jordan canonical form of matrices and linear maps, bilinear forms, quadratic forms, and choosing canonical bases for these, and the theory and computation of the Smith normal form for matrices over the integers.

For more in-depth information on this module, visit the undergraduate handbook.

In this module, you will learn methods to prove that every continuous function can be integrated, and prove the fundamental theorem of calculus. You will discuss how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties. You will develop a working knowledge of the Riemann integral; understand uniform and pointwise convergence of functions; study complex differentiability (Cauchy-Riemann equations) and complex power series; study contour integrals, Cauchy's integral formulas and applications.

For more in-depth information on this module, visit the undergraduate handbook.

This course focuses on developing your understanding and application of the theories of groups and rings, improving your ability to manipulate them. Some of the results proved in MA242 Algebra I: Advanced Linear Algebra for abelian groups are true for groups in general. These include Lagrange's theorem, which says that the order of a subgroup of a finite group divides the order of the group. 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’ theorem on unique factorisation in polynomial rings.

For more in-depth information on this module, visit the undergraduate handbook.

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.

For more in-depth information on this module, visit the undergraduate handbook.

This module is made up of an essay and presentation. You will be given the opportunity of independent study with guidance from a Personal Tutor. It will provide you with an opportunity to learn some mathematics directly from books and other sources. It will allow you to develop your written and oral exposition skills. You will be able to develop your research skills, including planning, use of library and of the internet.

For more in-depth information on this module, visit the undergraduate handbook.

#### Second-year optional Maths modules

Choose from an more extensive array of optional modules. Explore new concepts and topics, and broaden your perspective.

Combinatorics, Geometry, Introduction to Partial Differential Equations, Combinatorial Optimization, Theory of ODEs, Introduction to Systems Biology, Introduction to Number Theory, Variational Principles, Games Decisions and Behaviour, Introduction to Mathematical Statistics, Stochastic Processes.

For more in-depth information on this module, visit the undergraduate handbook.

Hamiltonian Mechanics, Computational Physics, Quantum Mechanics and its Applications, Electromagnetic Theory and Optics, Physics of Fluids, Stars, Methods of Mathematical Physics.

For more in-depth information on this module, visit the undergraduate handbook.

Algorithms, Logic and Verification, Algorithmic Graph Theory.

For more in-depth information on this module, visit the undergraduate handbook.

Mathematical Economics 1A, Mathematical Economics 1B.

For more in-depth information on this module, visit the undergraduate handbook.

Foundations of Accounting, Foundations of Finance, Starting a Business, The Practice of Operational Research, Mathematical Programming II.

For more in-depth information on this module, visit the undergraduate handbook.

Logic II, History of Modern Philosophy.

For more in-depth information on this module, visit the undergraduate handbook.

Applied Imagination, Challenges of Climate Change, Genetics: Science and Society.

For more in-depth information on this module, visit the undergraduate handbook.

Introduction to Secondary School Teaching.

For more in-depth information on this module, visit the undergraduate handbook.

at Warwick offers academic modules in Arabic, Chinese, French, German, Japanese, Russian and Spanish at a wide range of levels.

For more in-depth information on this module, visit the undergraduate handbook.

### Third-year Maths modules

There are no core modules in the third year. You have the freedom to choose from an extensive list of approximately 30 mathematics modules in areas of algebra, analysis, number theory, geometry, as well as a whole host of topical applications of mathematics. Further module information in the Undergraduate Handbook.

You will also have an opportunity to pursue optional modules in another subject of your choice.

#### Third-year optional modules from Maths and Statistics

Galois Theory, Rings and Modules, Groups and Representations, Commutative Algebra, Algebraic Number Theory, Set Theory, Combinatorics II.

For more in-depth information on this module, visit the undergraduate handbook.

Complex Analysis, Functional Analysis I, Functional Analysis II, Manifolds, Measure Theory, Markov Processes and Percolation Theory.

For more in-depth information on this module, visit the undergraduate handbook.

Fractal Geometry, Geometry of Curves and Surfaces, Introduction to Topology, Algebraic Topology.

For more in-depth information on this module, visit the undergraduate handbook.

Topics in Mathematical Biology, Bifurcations Catastrophes and Symmetry, Fluid Dynamics, Numerical Analysis and PDEs, Control Theory, Variational Principles.

For more in-depth information on this module, visit the undergraduate handbook.

Bayesian Statistics and Decision Theory, Applied Stochastic Processes, Monte Carlo Methods, Mathematical Finance, Designed Experiments, Probability Theory, Multivariate Statistics, Topics in Statistics, Medical Statistics, Topics in Data Science, Bayesian Forecasting and Intervention.

For more in-depth information on this module, visit the undergraduate handbook.

Problem Solving, Essay.

For more in-depth information on this module, visit the undergraduate handbook.

#### Third-year optional modules from other subjects

Statistical Physics, Weather and the Environment, Physics in Medicine, Quantum Physics of Atoms, Electrodynamics, Scientific Programming, Plasma Electrodynamics, Galaxies, Optoelectronics and Laser Physics, Cosmology, Nuclear Physics.

For more in-depth information on this module, visit the undergraduate handbook.

Complexity of Algorithms, Computer Graphics, Compiler Design, Principles of Programming Languages, Approximation and Randomised Algorithms, Algorithmic Game Theory.

For more in-depth information on this module, visit the undergraduate handbook.

Systems Modelling and Control.

For more in-depth information on this module, visit the undergraduate handbook.

Business Studies I, Operational Research for Strategic Planning, Business Studies II, Simulation, Mathematical Programming III, The Practice of Operational Research.

For more in-depth information on this module, visit the undergraduate handbook.

Warwick offers academic modules in Arabic, Chinese, French, German, Japanese, Russian, and Spanish at a wide range of levels.

For more in-depth information on this module, visit the undergraduate handbook.

### Fourth- year core Maths module

The distinguishing feature of the fourth year is a substantial core project, either working toward open mathematical research or else exploring the deep underpinnings of mathematics in society, science, technology, or industry.

Research Project

For more in-depth information on this module, visit the undergraduate handbook.

Maths in Action Project.

For more in-depth information on this module, visit the undergraduate handbook.

#### Fourth-year Optional Modules from Mathematics and Statistics

You will also have the opportunity to choose from an array of approximately 20 modules at an advanced level, aimed to prepare you for further graduate study or research opportunities.

Lie Groups, Graph Theory, Analytic Number Theory, Elliptic Curves.

For more in-depth information on this module, visit the undergraduate handbook.

For more in-depth information on this module, visit the undergraduate handbook.

Algebraic Geometry, Differential Geometry, Geometric Group Theory, Cohomology and Poincare Duality, Algebraic Curves.

For more in-depth information on this module, visit the undergraduate handbook.

Dynamical Systems, Applied Dynamical Systems, Population Dynamics, Atmospheric Dynamics, Topics in Complexity Science, Mathematical Acoustics, Structures of Complex Systems.

For more in-depth information on this module, visit the undergraduate handbook.

Brownian Motion, Bayesian Forecasting and Intervention, Applied Stochastic Processes, Monte Carlo Methods, Multivariate Statistics.

For more in-depth information on this module, visit the undergraduate handbook.

Relativistic Quantum Mechanics, High Performance Computing in Physics, Gauge Theories in Particle Physics, General Relativity, Quantum Mechanics Basic Principles and Probabilistic Methods, Statistical Mechanics.

For more in-depth information on this module, visit the undergraduate handbook.

Please note: We update our modules every year based on availability and demand, and we update our course content too. The content on this page gives you a really strong indication of what your course will offer, but given the interval between the publication of courses and enrolment, some of the information may change. Read our terms and conditions to find out more.