## Overview

### Highlights of mathematics at Nottingham

- Expand your study through a wide choice of modules across the variety of disciplines within mathematics
- Benefit from a course accredited by the Institute of Mathematics and its Applications
- Travel while you learn with opportunities to spend a semester studying abroad

The three-year BSc and four year MMath courses have a common programme for the first two years. The first year includes core modules that provide an essential foundation of mathematical skills, as well as more specialised modules in pure mathematics, applied mathematics and probability and statistics.

As you progress you can specialise according to your interests. Both the BSc and MMath allow you to choose from a range of mathematical topics, both vocational and academic. The MMath gives you an insight into problems of current research interest and a deeper mathematical knowledge. On both courses it is possible to take some modules from other schools in the University and to study for a semester abroad at one of our overseas partner institutions.

### BSc Mathematics with Statistics

Students who succeed in a sufficient number of statistics modules may opt for the BSc Mathematics with Statistics degree.

### BSc or MMath?

Our BSc Mathematics provides a broad mathematical education with flexibility in the final year to select from many specialist options. The MMath degree is an integrated masters qualification in which the additional year provides the opportunity for you to study your favourite topics in more depth and develop your research skills in a substantial dissertation. The MMath is excellent preparation for further study, such as a PhD, or a career requiring specialist mathematical skills.

If you choose to study the MMath your student loan will cover tuition fees and living costs for the additional year too (home/EU students only).

You can transfer between the three-year BSc Mathematics and the four-year MMath Mathematics during the first two years of either degree subject to the MMath progression requirements of 55% being achieved in the second year at the first attempt.

## Entry requirements

STEP/MAT/TMUA is not required but may be taken into consideration when offered.

A level General Studies, Critical Thinking and Citizenship Studies are not accepted.

We also accept students who have achieved appropriate grades in the __Engineering and Physical Sciences Foundation Certificate__.

## Course content

### Teaching methods and assessment

You will learn through a variety of methods depending on the module. These may include:

- computer lab
- drop-in classes
- lectures
- problem classes
- tutorials

Assessment varies throughout the course but it is typically a combination of:

- computer assessments or reports
- coursework
- written examination

### Accreditation

This programme is accredited to meet the educational requirements of the Chartered Mathematician designation awarded by the Institute of Mathematics and its Applications.

### Study abroad

You can apply to spend a period of time studying abroad (usually one semester) through either one of our European Erasmus partners or a selected partner from the University-wide exchange programme.

This is an exciting opportunity to gain a global perspective of mathematics, boost your communication skills, and discover a new culture.

## Year one

You will study core mathematics under the three headings of Analytical and Computational Foundations, Calculus, and Linear Mathematics; this includes an introduction to computer programming. You also begin studying the three main subject areas within mathematics, namely pure mathematics, applied mathematics, and probability and statistics.

You will benefit from small-group tutorials with your personal tutor and our Peer-Assisted Study Support (PASS) scheme, designed specifically to help you settle in. PASS Leaders, who are current maths students, will provide you with a friendly face at the start of your first year and then academic support during that year, through regular PASS sessions.

### Compulsory modules

This module introduces students to a broad range of core mathematical concepts and techniques. It has three components.

- Mathematical reasoning (the language of mathematics, the need for rigour, and methods of proof).
- The computer package MATLAB and its applications.
- Elementary analysis.

You will receive an introduction to classical mechanics and modelling in applied mathematics. This will provide you with a foundation in applied mathematics and you will begin to apply your knowledge to real world problems.

You will begin by practising the basic concepts and methods of calculus including limits, functions, and continuity. In the second semester you will move onto more advanced usage of calculus. Topics will be based around the calculus of functions of several variables and include partial derivatives, chain rules, the vector operator grad, Lagrange multipliers and multiple integrals.

This module provides a foundation for all further pure mathematics studies on your course. You will learn that pure mathematics is a language based on the notion of sets, functions and relations, and how to read and write this language. There will also be discussions with examples on the application and use of pure mathematics.

This module introduces you to the methods and practices of linear mathematics that you will need in subsequent modules on your course, such as complex numbers, vector algebra and matrix algebra. You will then expand your knowledge to include vector spaces, linear transformations and inner product spaces.

This module provides an introduction to axiomatic systems in mathematics.It will cover the basic concepts of some key mathematical structures from algebra (groups, rings and fields) together with the basic properties of permutations, integers and polynomials.

This module provides an introduction to probability by developing a framework for the logic of uncertainty. Random variables and the topics surrounding them will also be introduced.

This module offers you the chance to learn about a range of statistical ideas and skills, along with concepts and techniques for modelling and practical data analysis. You will learn to write reports based on these topics which will help you in further studies.

The above is a sample of the typical modules that we offer at the date of publication but is not intended to be construed and/or relied upon as a definitive list of the modules that will be available in any given year. This prospectus may be updated over the duration of the course, as modules may change due to developments in the curriculum or in the research interests of staff.

## Year two

Choosing from a range of optional modules, you will typically continue to study two of the three main mathematical subject areas. You will also have the option to choose some modules from outside mathematics if you wish.

### Optional modules

This course will develop in more detail the fundamental concepts in algebra such as groups and rings and will provide an introduction to elementary number theory.

We will consider how general algebraic concepts can be applied in concrete situations in number theory and after a review of primes, integer factorization and module arithmetic, the focus will be on classical problems. This includes Fermats Little Theorem and its application to primality testing, methods of factorization, primitive roots, discrete logarithms, some classical Diophantine equations (linear and polynomial), Fibonacci numbers, and continued fractions.

In this module you will learn about the theory and applications of functions of a complex variable using a method and applications approach. You will develop an understanding of the theory of complex functions and evaluate certain real integrals using your new skills.

This course is an introduction to Fourier series and integral transforms and to methods of solving some standard ordinary and partial differential equations which occur in applied mathematics and mathematical physics.

The course describes the solution of ordinary differential equations using series and introduces Fourier series and Fourier and Laplace transforms, with applications to differential equations and signal analysis. Standard examples of partial differential equations are introduced and solution using separation of variables is discussed.

This course explores the classical and quantum mechanical description of motion. The laws of classical mechanics are investigated both in their original formulation due to Newton and in the mathematically equivalent but more powerful formulations due to Lagrange and Hamilton. Applications are made to problems such as planetary motion, rigid body motion and vibrating systems. Quantum mechanics is developed in terms of a wave function obeying Schroedinger's equation, and the appropriate mathematical notions of Hermitian operators and probability densities are introduced. Applications include problems such as the harmonic oscillator and a particle in a three-dimensional central force field.

This module introduces basic techniques in numerical methods and numerical analysis which can be used to generate approximate solutions to problems that may not be amenable to analysis. Specific topics include:

- Implementing algorithms in Matlab
- Discussion of errors (including rounding errors)
- Iterative methods for nonlinear equations (simple iteration, bisection, Newton, convergence)
- Gaussian elimination, matrix factorisation, and pivoting
- Iterative methods for linear systems, matrix norms, convergence, Jacobi, Gauss-Siedel
- Interpolation (Lagrange polynomials, orthogonal polynomials, splines)
- Numerical differentiation & integration (Difference formulae, Richardson extrapolation, simple and composite quadrature rules)
- Introduction to numerical ODEs (Euler and Runge-Kutta methods, consistency, stability)

In this module you will build on the foundation of knowledge gained from your core year one modules in Analytical and Computational Foundations and Calculus. You will learn to follow a rigorous approach needed to produce concrete proof of your workings.

This course aims to provide students with tools which enable them to develop and analyse linear and nonlinear mathematical models based on ordinary and partial differential equations. Furthermore, it aims to introduce students to the fundamental mathematical concepts required to model the flow of liquids and gases and to apply the resulting theory to model physical situations.

This module will give you an introduction to the theory of probability and random variables, with particular attention paid to continuous random variables. Fundamental concepts relating to probability will be discussed in detail, including well-known limit theorems and the multivariate normal distribution. You will then progress onto complex topics such as transition matrices, one-dimensional random walks and absorption probabilities.

This module will equip you with valuable skills needed for graduate employment. You will work on two group projects based on open-ended mathematical topics agreed by your group. You will also work independently to improve your communication skills and learn how to summarise technical mathematical data for a general audience. You will be provided with some commercial and business awareness and explore how to use your mathematical sciences degree for your future career.

The first part of this module provides an introduction to statistical concepts and methods and the second part introduces a wide range of techniques used in a variety of quantitative subjects. The key concepts of inference including estimation and hypothesis testing will be described as well as practical data analysis and assessment of model adequacy.

This course aims to give students a sound grounding in the application of both differential and integral calculus to vectors, and to apply vector calculus methods and separation of variables to the solution of partial differential equations. The module is an important pre-requisite for a wide range of other courses in Applied Mathematics.

The above is a sample of the typical modules that we offer at the date of publication but is not intended to be construed and/or relied upon as a definitive list of the modules that will be available in any given year. This prospectus may be updated over the duration of the course, as modules may change due to developments in the curriculum or in the research interests of staff.

## Year three

You will choose from a wide range of advanced optional modules, one of which may involve project work. You will be able to specialise in one of the three main subject areas, or choose a broad course which may include modules from all three areas. You will also have the option to choose some modules from outside mathematics if you wish.

### Optional modules

In this module you will apply the general theory you learnt in Introduction to Mathematical Physics to more general problems. New topics will be introduced such as the quantum theory of the hydrogen atom and aspects of angular momentum such as spin.

In this module you will build on your theoretical knowledge of statistical inference by a practical implementation of the generalised linear model. You will move on to enhance your understanding of statistical methodology including the analysis of discrete and survival data. You will also be trained in the use of a high-level statistical computer program.

The course introduces and explores methods, concepts and paradigm models for classical and quantum mechanical dynamics exploring how classical concepts enter quantum mechanics, and how they can be used to find approximate semi-classical solutions.

In classical dynamics we discuss full integrability and basic notions of chaos in the framework of Hamiltonian systems, together with advanced methods like canonical transformations, generating functions and Hamiltonian-Jacobi theory. In quantum mechanics we recall Schrödinger's equation and introduce the semi-classical approximation. We derive the Bohr-Sommerfeld quantization conditions based on a WKB-approch to the eigenstates. We will discuss some quantum signatures of classical chaos and relate them to predictions of random-matrix theory. We will also introduce Gaussian states and coherent states and discuss their semi-classical dynamics and how it is related to the corresponding classical dynamics. An elementary introduction to complete descriptions of quantum mechanics in terms of functions on the classical phase space will be given.

This course provides an introduction to coding theory in particular to error-correcting codes and their uses and applications. It also provides an introduction to to cryptography, including classical mono- and polyalphabetic ciphers as well as modern public key cryptography and digital signatures, their uses and applications.

This course provides an opportunity for third-year students taking G100 and G103 to gain first-hand experience of being involved with providing mathematical education.

Students will work at local schools alongside practising mathematics teachers in a classroom environment and will improve their skills at communicating mathematics. Typically, each student will work with a class (or classes) for half a day a week for about sixteen weeks. Students will be given a range of responsibilities from classroom assistant to leading a self-originated mathematical activity or project. The assessment is carried out by a variety of means: on-going reflective log, contribution to reflective seminar, oral presentation and a final written report.

This module involves the application of probability and statistics to a variety of practical, open-ended problems, typical of those that statisticians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities.

The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork. Students will have an opportunity to apply a variety of statistical methods and knowledge learned in previous modules.

.

This course introduces various analytical methods for the solution of ordinary and partial differential equations, focussing on asymptotic techniques and dynamical systems theory. Students taking this course will build on their understanding of differential equations covered in MATH2012.

The course will start with several topics from the perspective of what can be explicitly calculated with an emphasis on applications to geometry and number theory.

Topics include:

- basic notions of projective geometry
- plane algebraic curves including elliptic curves
- addition of points on elliptic curves
- results on the group of rational points on an elliptic curve
- properties of elliptic curves and their applications.

The course provides an introduction to electromagnetism and the electrodynamics of charged particles. The aims of this course are:

- to develop an appropriate mathematical model of electromagnetic phenomena that is informed by observations
- to understand electromagnetic configurations of practical importance and to relate predictions made to everyday phenomena
- to illustrate the use of solutions of certain canonical partial differential equations for determining electrostatic fields and electromagnetic waves in vacuum and in matter
- to illustrate the interplay between experimental input and the development of a mathematical model, and the use of various mathematical techniques for solving relevant problems.

Number theory concerns the solution of polynomial equations in whole numbers, or fractions. For example, the cubic equation x3 + y3 = z3 with x, y, z non-zero has infinitely many real solutions yet not a single solution in whole numbers.

We shall establish the basic properties of the Riemann zeta-function to find out how evenly these primes are distributed in nature. This course will present several methods to solve Diophantine equations including analytical methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.

A graph (in the sense used in Graph Theory) consists of vertices and edges, each edge joining two vertices. Graph Theory has become increasingly important recently through its connections with computer science and its ability to model many practical situations.

Topics covered in the course include:

- paths and cycles
- the resolution of Eulers Königsberg Bridge Problem
- Hamiltonian cycles
- trees and forests
- labelled trees,
- the Prüfer correspondence
- planar graphs
- Demoucron et al. algorithm
- Kruskal's algorithm
- the Travelling Salesman's problem
- the statement of the four-colour map theorem
- colourings of vertices
- chromatic polynomial
- colourings of edges.

This course builds on the basic ideas of group theory. It covers a number of key results such as the simplicity of the alternating groups, the Sylow theorems (of fundamental importance in abstract group theory), and the classification of finitely generated abelian groups (required in algebraic number theory, combinatorial group theory and elsewhere). Other topics to be covered are group actions, used to prove the Sylow theorems, and series for groups, including the notion of solvable groups that will be used in Galois theory.

This module gives an introduction into some basic ideas of functional analysis with an emphasis on Hilbert spaces and operators on them.

Many concepts from linear algebra in finite dimensional vector spaces (e.g. writing a vector in terms of a basis, eigenvalues of a linear map, diagonalisation etc.) have generalisations in the setting of infinite dimensional spaces making this theory a powerful tool with many applications in pure and applied mathematics

In this module the concepts of discrete time Markov chains are explored and used to provide an introduction to probabilistic and stochastic modelling for investment strategies, and for the pricing of financial derivatives in risky markets. You will gain well-rounded knowledge of contemporary issues which are of importance in research and applications.

This module consists of a self-directed investigation of a project selected from a list of projects or, subject to prior approval of the School, from elsewhere.

Project modules are carried out in the Autumn and Spring semesters.

The project will be supervised by a member of staff and will be based on a substantial mathematical problem, an application of mathematics or investigation of an area of mathematics not previously studied by the student. The course includes training in the use of IT resources, the word-processing of mathematics and report writing.

Metric space generalises the concept of distance familiar from Euclidean space. It provides a notion of continuity for functions between quite general spaces.

The module covers metric spaces, topological spaces, compactness, separation properties like Hausdorffness and normality, Urysohn’s lemma, quotient and product topologies, and connectedness. Finally, Borel sets and measurable spaces are introduced.

This module is concerned with the analysis of multivariate data, in which the response is a vector of random variables rather than a single random variable. A theme running through the module is that of dimension reduction. Key topics to be covered include: principal components analysis, whose purpose is to identify the main modes of variation in a multivariate dataset; modelling and inference for multivariate data, including multivariate regression data, based on the multivariate normal distribution; classification of observation vectors into sub-populations using a training sample; canonical correlation analysis, whose purpose is to identify dependencies between two or more sets of random variables. Further topics to be covered include factor analysis, methods of clustering and multidimensional scaling.

This module will help you develop your knowledge of the basic theory of fields, their extensions and their automorphism groups with applications to classical problems. Particular emphasis is laid on finite fields and number fields as you prove the basic propositions concerning Galois Theory. You will build a theoretical foundation to the construction of splitting fields and then move onto the factorization of polynomials.

In this module a variety of techniques and areas of mathematical optimisation will be covered including Lagrangian methods for optimisation, simplex algorithm linear programming and dynamic programming. You’ll develop techniques for application which can be used outside the mathematical arena.

In this module you’ll have an introduction to Einstein’s theory of general and special relativity. The relativistic laws of mechanics will be described within a unified framework of space and time. You’ll learn how to compare other theories against this work and you’ll be able to explain new phenomena which occur in relativity.

Commutative rings and modules over them are the fundamental objects of what is often referred to as commutative algebra. Already encountered key examples of commutative rings are polynomials in one variable over a field and number rings such as the usual integers or the Gaussian integers.

There are many close parallels between these two types of rings, for example the similarities between the prime factorization of integers and the factorization of polynomials into irreducibles. In this module, these ideas are extended and generalized to cover polynomials in several variables and power series, and algebraic numbers.

You will learn how to use numerical techniques for determining the approximate solution of ordinary and partial differential equations where a solution cannot be found through analytical methods alone. You will also cover topics in numerical linear algebra, discovering how to solve very large systems of equations and find their eigenvalues and eigenvectors using a computer.

In this module you will explore two main concepts of statistical inference; classical (frequentist) and Bayesian. Topics such as sufficiency, estimating equations, likelihood ratio tests and best-unbiased estimators will be discussed in detail. You will gain knowledge of the theory and concepts underpinning contemporary research in statistical inference and methodology.

In this module you will develop your knowledge of discrete-time Markov chains by applying them to a range of stochastic models. You will be introduced to Poisson and birth-and-death processes and then you will move onto more extensive studies of epidemic models and queuing models with introductions to component and system reliability.

This module involves the application of mathematics to a variety of practical, open-ended problems, typical of those that mathematicians encounter in industry and commerce. Specific projects are tackled through workshops and student-led group activities. The real-life nature of the problems requires students to develop skills in model development and refinement, report writing and teamwork.

The above is a sample of the typical modules that we offer at the date of publication but is not intended to be construed and/or relied upon as a definitive list of the modules that will be available in any given year. This prospectus may be updated over the duration of the course, as modules may change due to developments in the curriculum or in the research interests of staff.

## Student support

All students have a personal tutor. Personal tutors are members of staff in the school who will:

- monitor your academic progress and check on your wellbeing
- provide exam marks and help you reflect on feedback
- act as a first point of contact for any guidance on academic or personal matters
- lead weekly, small-group, tutorials supporting core first-year mathematics modules

This ensures you have enough time to build a relationship with your tutor and benefit from their support. Your fellow tutees also provide peer support.

Additionally, the school has a dedicated Teaching Officer who is available to help you adapt to university life and provide advice on more complex issues.

### Peer mentoring

PASS (Peer-Assisted Study Support) is our award-winning scheme, in which current students mentor first-year students and support them in the early stages of university life. As well as helping first-year students to settle in, PASS leaders run a number of informal study-support sessions over the course of the year, in which first-year students can discuss academic work and general university life, and in which PASS leaders can help answer any queries.

PASS helps the school to maintain a good sense of community.

Many of the students who participate in PASS sessions during their first year then apply to become a PASS leader in the following year. PASS leaders enhance their own employability skills, such as teamwork, leadership and communication.

Becoming a PASS leader also gives students the chance to gain recognition through the Nottingham Advantage Award.

"I became a PASS leader in my second year, tutoring first year students every fortnight. I really enjoyed attending these sessions during my first-year so I wanted to support others by running them in my second year, they really help encourage student interaction"

**Grace Bolton, ****BSc Mathematics and Economics**

## Careers

From accountant to engineer, analyst to investment banker, studying mathematics opens the door to a wide range of careers. Some graduates make specific use of mathematics while others use the more general skills they have gained, such as analysis and problem solving, high-level numeracy and a capacity to learn independently.

Our graduates are in high demand from prospective employers and have been well received into a broad range of careers in commerce, industry, the professions and government. They pursue a variety of careers which include:

- commerce
- engineering
- financial services
- government
- industry
- information technology
- science

### Postgraduate research

Rather than directly entering the employment market upon graduating, you might decide to continue your studies at higher-degree level. Postgraduate areas of study include:

- business studies
- computer science
- education
- engineering
- finance
- mathematics
- statistics

Each year some of our best students choose to stay at Nottingham and join our lively group of postgraduate research students in the School of Mathematical Sciences.

Each research group within the school offers a large number of diverse and interesting projects, across the specialisations of pure mathematics, applied mathematics and probability and statistics.

### Average starting salary and career progression

In 2017, 94.4% of undergraduates in the School of Mathematical Sciences who were available for employment had secured work or further study within six months of graduation. The average starting salary was £24,500 with the highest being £50,000.*

**Known destinations of full-time home undergraduates 2016/17. Salaries are calculated based on the median of those in full-time paid employment within the UK.*

The School of Mathematical Sciences ranked third in the UK for boosting graduate salaries, with graduates earning an average of £5,935 more than expected five years after graduation *(The Economist British University rankings, 2017).*

### Careers support and advice

Studying for a degree at the University of Nottingham will provide you with the type of skills and experiences that will prove invaluable in any career, whichever direction you decide to take.

Throughout your time with us, our Careers and Employability Service can work with you to improve your employability skills even further; assisting with job or course applications, searching for appropriate work experience placements and hosting events to bring you closer to a wide range of prospective employers.

Have a look at our careers page for an overview of all the employability support and opportunities that we provide to current students.

The University of Nottingham is consistently named as one of the most targeted universities by Britain’s leading graduate employers (Ranked in the top ten in The Graduate Market in 2013-2019, High Fliers Research).

## Fees and funding

### Additional costs

As a student on this course, you should factor some additional costs into your budget, alongside your tuition fees and living expenses. You should be able to access most of the books you’ll need through our libraries, though you may wish to purchase your own copies.

### Scholarships and bursaries

The International Orientation Scholarship is awarded to the best international (full-time, non EU) applicants to the school's courses. The scholarship is awarded in subsequent years to students who perform well academically (at the level of a 2:1 Hons degree or better at the first attempt).

The scholarship value is £2,000 for students on the BSc Mathematics degree course. Please note that the scholarship will be paid once for each year of study, so if you repeat a year for any reason, the scholarship will not be paid for that repeated year.

The scholarship will be paid in December each year, provided you have registered with the university and the school, are on a relevant course on the 1 December census and have paid the first installment of your fee.

### Home students*

Over one third of our UK students receive our means-tested core bursary, worth up to £2,000 a year. Full details can be found on our financial support pages.

* A 'home' student is one who meets certain UK residence criteria. These are the same criteria as apply to eligibility for home funding from Student Finance.

### International/EU students

Our International Baccalaureate Diploma Excellence Scholarship is available for select students paying overseas fees who achieve 38 points or above in the International Baccalaureate Diploma. We also offer a range of High Achiever Prizes for students from selected countries, schools and colleges to help with the cost of tuition fees. Find out more about scholarships, fees and finance for international students.

## Related courses

**Disclaimer**This online prospectus has been drafted in advance of the academic year to which it applies. Every effort has been made to ensure that the information is accurate at the time of publishing, but changes (for example to course content) are likely to occur given the interval between publishing and commencement of the course. It is therefore very important to check this website for any updates before you apply for the course where there has been an interval between you reading this website and applying.