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- Mathematics MMath (G103)
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## Find out more about our Mathematics MMath at Warwick

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### Page updates

We have revised the information on this page since publication. See the edits we have made and content history.

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G103

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Master of Mathematics (MMath)

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4 years full-time

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26 September 2022

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Warwick Mathematics Institute

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University of Warwick

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Mathematics enhances your ability to think clearly, learn new ideas quickly, manipulate precise and intricate concepts, follow complex reasoning, construct logical arguments and expose dubious ones.

Our four-year Mathematics (MMath) shares the same core as our BSc but enables you to explore in greater depth areas of interest, both through specialised fourth-year modules and via a substantial Research or Maths-in-Action project.

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Our challenging degree will harness your strong mathematical ability and commitment, enabling you to explore your passion for mathematics. You will be taught by world-leading researchers in a supportive environment, where learning spaces – including breakout areas and common spaces – are all geared towards you sharing, collaborating and exploring your academic curiosity.

Pure Mathematics modules combine the work of some of the world’s greatest thinkers, while Applied Mathematics addresses real-world problems in biology, data science, climate science and finance. Many third and fourth-year modules offer glimpses of the latest research.

The four-year MMath shares the same core as the BSc but enables you to explore areas of interest in greater depth, both through specialised fourth year modules and via a substantial Research or Maths-in-Action project.

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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 comprises 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).

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Most of our teaching is through lectures delivered by a member of academic staff. Undergraduates usually take four or five modules in each of Term One and Term Two. Term Three is mostly for revision and examinations. Each module is usually taught in three one-hour lectures per week.

In your first year, you meet your supervisor (a graduate student or final year undergraduate) twice a week to discuss the course material and go over submitted work.

In your second and third years, lecture modules are accompanied by weekly support classes. Your personal tutor provides a further layer of learning and pastoral support.

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- Lectures vary from 10 to 400. Supervisions and tutorials are typically in groups of five.
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- Typical contact hours across lectures, seminars, supervisions etc: 18 hours/week during Term One and Term Two (15 hours of lectures and 3 hours of supervisions, problem classes and tutorials)
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Most modules are assessed by 85% exam and 15% homework, or by 100% exam.

The Second Year Essay, Third Year Essay, and the MMath Project are assessed on the basis of an essay/dissertation and oral presentation.

Weighted 10:20:30:40

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### Study abroad

We encourage students to consider spending Year Three at one of 23 European partner universities in Belgium, France, Germany, Italy, The Netherlands, Spain and Switzerland.

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## Placements and work experience

After Year Two, students can take a year’s placement to experience mathematics in an employment setting. The job must be deemed to provide learning experiences related to the degree course. A satisfactory placement leads to the award of a ‘BSc with Intercalated Year’ (and often to a potential job offer after graduation). The maths department is unfortunately unable to help with finding such placements.

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#### Admissions tests

Please note that the majority of applicants for 2022 entry will be required to take one of the following admissions tests. You can register for these tests via the following links:

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##### A level typical offer

A*A*A including A* in both Mathematics and Further Mathematics, plus grade 2 in any STEP (or a suitable grade in MAT or TMUA)

*Students who achieve a sufficiently high score in MAT or TMUA will have the STEP condition removed. As guidance, in previous years our requirements have been 6.5 in TMUA or 64 in MAT. However, the exact thresholds will only be set after results for these qualifications have been released.*##### A level contextual offer

We welcome applications from candidates who meet the contextual eligibility criteria. The typical contextual offer is A*A*A with an A* in Mathematics and Further Mathematics. See if you're eligible.

##### General GCSE requirements

Unless specified differently above, you will also need a minimum of GCSE grade 4 or C (or an equivalent qualification) in English Language and either Mathematics or a Science subject. Find out more about our entry requirements and the qualifications we accept. We advise that you also check the English Language requirements for your course which may specify a higher GCSE English requirement. Please find the information about this below.

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##### IB typical offer

39 with 6 in three Higher Level subjects to include Mathematics ('Analysis and Approaches' only), plus grade 2 in any STEP (or a suitable grade in MAT or TMUA)

*Students who achieve a sufficiently high score in MAT or TMUA will have the STEP condition removed. As guidance, in previous years our requirements have been 6.5 in TMUA or 64 in MAT. However, the exact thresholds will only be set after results for these qualifications have been released.*##### IB contextual offer

We welcome applications from candidates who meet the contextual eligibility criteria. The typical contextual offer is 38 with Higher Level 6,6,6 including Mathematics ('Analysis and Approaches' only). See if you're eligible.

##### General GCSE requirements

Unless specified differently above, you will also need a minimum of GCSE grade 4 or C (or an equivalent qualification) in English Language and either Mathematics or a Science subject. Find out more about our entry requirements and the qualifications we accept. We advise that you also check the English Language requirements for your course which may specify a higher GCSE English requirement. Please find the information about this below.

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We welcome applications from students taking a BTEC alongside A level Mathematics and Further Mathematics.

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### Year One

###### Foundations

It is in its proofs that the strength and richness of mathematics is to be found. University mathematics introduces progressively more abstract ideas and structures and demands more in the way of proof until much 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 calculation, and gradually moving towards abstraction and proof.

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

###### Analysis I/II

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.

**Methods of Mathematical Modelling I and II**In this module you will learn the modelling cycle and learn to analyse simple models, using scaling, non-dimensionalisation and linear stability analysis to understand the main dynamics. 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 their solution and numerical approximation.

In the second term 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.

**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 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.

###### 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 including a video presentation.

###### 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’ll learn methods of counting (inclusion-exclusion formula and multinomial coefficients), and examine theoretical topics including independence of events and conditional probabilities. Using Bayes’ theorem 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-Schwarz inequalities. The module ends with a discussion of the celebrated Central Limit Theorem.

### Year Two

**Methods of Mathematical Modelling 3**This module studies several topics commonly used in mathematical modelling: (i) optimisation problems require an understanding of critical points in multi-dimensions, and methods you will see the techniques of linear programming, least squares and regression, convexity, steepest descent algorithms, optimisation with constraints (applications include neural nets); (ii) The Fast Fourier Transform is used in signal processing and audio and video compression; (iii) Hilbert space theory is a framework for discussing orthogonal functions and their use in approximation problems.

**Algebra 3**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’ theorem on unique factorisation in polynomial rings, and see applications in Number Theory, Geometry and Combinatorics.

###### 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.

###### 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.

**Scientific communication**This module includes an essay and presentation. You will be given the opportunity of independent study with guidance and feedback from your Personal Tutor. It will provide you with an opportunity to investigate some mathematics not covered in other modules, using a range of sources, and then develop your written and oral exposition skills.

**Multilinear Algebra**On this course, 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.

**Multivariable Analysis**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, including the inverse and implicit function theorems. The module will review line and surface integrals, introduce div, grad and curl and establish the divergence theorem, with some applications to partial differential equations.

### 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****Or**###### Maths-in-Action (MiA-Projects)

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. This project is 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.

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#### Optional modules

**Mathematics:**Knot Theory; Fractal Geometry; Population Dynamics - Ecology and Epidemiology; Number Theory**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)

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- Recent graduates have pursued job roles such as: • Actuaries, economists and statisticians • Software developers • Chartered and certified accountants • Finance and investment analysts • Teachers • Telecommunication designers • Data scientists and engineers • Academics UK firms that have employed recent Warwick graduates from the Mathematics and Statistics Departments include: • Adder Technology • Amazon • BlackRock International • Merrill Lynch • Brainlabs • Civil Service • Conduent • Darktrace • Deloitte • Department of Health • eBay • Ford Motor Company • Fore Consulting • Goldman Sachs • Government Actuaries • Investec • Jane Street Capital • KPMG • Lloyds • MBDA • Metaswitch • Met Office • Ministry of Justice • RenaissanceRe (Syndicate 1458) • Oxford Clinical Trials Unit • Softwire • Solid Solutions • Sword Apak • Ten10 • Towers Watson • Xafinity
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- Our department has a dedicated professionally qualified Senior Careers Consultant to support you. They offer impartial advice and guidance, together with workshops and events throughout the year. Previous examples of workshops and events include: • Finding experience to boost your CV in Year One and Two • Careers in Data Science and Artificial Intelligence • Warwick careers fairs throughout the year • Interview skills for Statistics students • Maths and Stats Careers Fair [Find out more about careers support at Warwick. https://warwick.ac.uk/services/careers/careers_skills]