# Mathematics (MMath) (Full-Time, 2021 Entry)

**UCAS Code**

G103

**Qualification**

Master of Mathematics (MMath)

**Duration**

4 years full-time

**Start Date**

27 September 2021

**Department of Study**

Warwick Mathematics Institute

**Location of Study**

University of Warwick

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.

## Course overview

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, computing, 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.

### Course structure

Pure Mathematics modules combine the work of some of the world’s greatest thinkers, while Applied Mathematics addresses real-world problems in biology, computing, climate science and finance. Flexible options enable you to explore in depth your love of mathematics, while studying other subjects you enjoy. Optional modules cover the entire range of mathematical sciences, including algebra, number theory, geometry, topology, pure and applied analysis, differential equations, and applications to physical and life sciences.

**Year One:** 8 core modules (75% of normal load).

**Year Two:** 5 core modules plus essay (55% of normal load).

**Year Three:** no core, but do at least 75% maths.

The remaining modules can be chosen from mathematics or one of many subjects. Students must do at least 75% maths each year.

### How will I learn?

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.

### Class size

Lectures vary from 10 to 400. Supervisions and tutorials are typically in groups of five.

### Contact hours

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)

### How will I be assessed?

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

### Study abroad

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

### Work experience

After Year Two, students can take a year’s placement to experience mathematics in action. 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.

### General entry requirements

#### A level:

- A*A*A including A* in both Mathematics and Further Mathematics, plus grade 1 in any STEP
**OR**A*A*A* including Mathematics and Further Mathematics**OR**A*A*AA including A* in both Mathematics and Further Mathematics

#### IB:

- 39 + STEP (grade 1) with 6 in three Higher Level subjects to include Mathematics ('Analysis and Approaches' only)
**OR**39 with 7, 6, 6 in three Higher Level subjects to include Mathematics ('Analysis and Approaches' only)

#### BTEC:

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

**Additional requirements:**

You can take the Mathematics Admissions Test (MAT) or the Test of Mathematics for University Admissions (TMUA) before applying. Depending on your performance in your test/s, you may then be eligible for a reduced offer.

Find out more about MAT, TMUA and STEP.

You will also need to meet our English Language requirements.

### International Students

We welcome applications from students with other internationally recognised qualifications.

Find out more about international entry requirements.

### Contextual data and differential offers

Warwick may make differential offers to students in a number of circumstances. These include students participating in the Realising Opportunities programme, or who meet two of the contextual data criteria. Differential offers will be one or two grades below Warwick’s standard offer (to a minimum of BBB).

### Warwick International Foundation Programme (IFP)

All students who successfully complete the Warwick IFP and apply to Warwick through UCAS will receive a guaranteed conditional offer for a related undergraduate programme (selected courses only).

Find out more about standard offers and conditions for the IFP.

### Taking a gap year

Applications for deferred entry welcomed.

### Interviews

We do not typically interview applicants. Offers are made based on your UCAS form which includes predicted and actual grades, your personal statement and school reference.

### Year One

###### Linear Algebra

Linear algebra addresses simultaneous linear equations. You will learn about the properties of vector space, linear mapping and its representation by a matrix. 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.

###### Differential Equations

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 a first glimpse of chaotic solutions.

###### Mathematics by Computer

This course aims to show how the computer may be used, throughout all of mathematics, to enhance understanding, make predictions and test hypotheses. The module will be taught using the Matlab software package. Through using this software tool you will be introduced to the rudiments of computer programming. You will learn how to graph functions, study vectors and matrices graphically and numerically, how to iterate and use iteration to study sequences and series, how to solve algebraic and differential equations numerically and how to study statistical properties of sets of numbers.

###### Geometry and Motion

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.

###### 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 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 calculation, and gradually moving towards abstraction and proof.

###### Introduction to Abstract Algebra

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.

**Analysis I and II**

###### Probability

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.

### Year Two

###### Multivariable Calculus

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.

###### Algebra I: Advanced Linear Algebra

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.

###### Analysis III

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.

###### Algebra II: Groups and Rings

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.

###### 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.

###### Second Year Essay

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.

### 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.

#### Examples of optional modules/options for current students

**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)

### Tuition fees

Find out more about fees and funding.

### Additional course costs

There may be costs associated with other items or services such as academic texts, course notes, and trips associated with your course. Students who choose to complete a work placement or study abroad will pay reduced tuition fees for their third year.

**Warwick Undergraduate Global Excellence Scholarship 2021**

We believe there should be no barrier to talent. That's why we are committed to offering a scholarship that makes it easier for gifted, ambitious international learners to pursue their academic interests at one of the UK's most prestigious universities. This new scheme will offer international fee-paying students **250 tuition fee discounts** ranging from full fees to awards of £13,000 to £2,000 for the **full duration of your Undergraduate degree course**.

Find out more about the Warwick Undergraduate Global Excellence Scholarship 2021

### Your career

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

### Helping you find the right career

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

"You have to be prepared to work hard but once you’re okay with that, it’s the most exciting place to study Maths.”

AddayBSc Mathematics with Intercalated Year graduate