If these grades are not met, English preparatory courses may be available

If these grades are not met, English preparatory courses may be available

## Overview

### Key facts

- This course provides students with a broader and deeper understanding of several core areas of pure mathematics that are of strong current interest
- On completion, it offers a solid foundation for a career in research in pure mathematics
- The course is informed by the research interests of the members of the research groups Algebra, Analysis, and Number Theory and Geometry
- One of the largest and strongest mathematics departments in the UK, with over 70 full-time academic staff
- The Research Excellence Framework (REF) 2014 results place the School in the top 10 nationally within Mathematical Sciences for 'research power' and 'research quality'; with 32% of its research recognised as world-leading and a further 56% as internationally excellent
- The research environment was classified as 75% world-leading in vitality and sustainability, with the remaining 25% internationally excellent, reflecting the outstanding setting the School provides for its academic staff as well as its postdoctoral and postgraduate researchers
- The school scored 87% for Student Satisfaction in the National Student Survey, 2018

### Academic English preparation and support

If you require additional support to take your language skills to the required level, you may be able to attend a presessional course at the Centre for English Language Education, which is accredited by the British Council for the teaching of English in the UK.

Students who successfully complete the presessional course to the required level can progress to postgraduate study without retaking IELTS or equivalent. You could be eligible for a joint offer, which means you will only need to apply for your visa once.

## Full course details

### Course Structure

The MSc Pure Mathematics is offered on a full-time basis over one year and is designed for students with a degree in Mathematics with a substantial component in pure mathematics.

Students should have a strong interest in pure mathematics and specifically they should have a good background in at least two to three of the following subject areas: algebra, number theory, group theory or analysis.

Modules are mainly delivered through lectures and example and/or problem classes for smaller groups. You will typically be assessed by a combination of examination, coursework and a class test.

The course comprises 180 credits, split across 120 credits of taught modules and a 60-credit research project.

### Taught modules

This consists of taught modules of which students must take up to 120 credits worth.

Central to the course are three pairs of modules each consisting of an autumn and a spring module. Students must choose at least one such pair of modules. Each pair provides a substantial foundation in analysis, algebra (in particular group theory) and number theory, respectively.

#### Group one

Students must take a 120 credits from the group listed under 'Modules' below. This must include both modules from at least one of the following pairs:

- Foundations of Advanced Analysis and Further Topics in Analysis
- Advanced Group Theory and Combinational Group Theory
- Higher Number Theory and Algebraic Number Theory

### Dissertation

During the summer period, you will conduct an independent research project under the supervision of academic staff, which is worth 60 credits.

### Prerequisite Information

Specific prerequisites and recommended books, where appropriate, are listed below for all the taught modules on the course.

#### Semester one

##### Foundations of Advanced Analysis

You should have a good background in real analysis.

- Sutherland;
*Metric and Topological Spaces*

##### Advanced Group Theory

Basic knowledge in algebra and group theory.

- J B Fraleigh; A First Course in Abstract Algebra
- W Ledermann & A J Weir;
*Introduction to group theory*(2nd edition, Longman Mathematical Series)

##### Algebraic Geometry

Good math background in algebra and commutative ring theory.

- W. Fulton;
*Algebraic Curves* - M Atiyah, R Macdonald;
*Introduction to Commutative Algebra*

##### Complex Analysis

A good first mathematical course in complex analysis and solid background in real analysis as covered in J W Brown; *Complex variables and applications*.

- L V Ahlfors;
*Complex analysis: an introduction to the theory of analytic functions of one complex variable*

##### Higher Number Theory

A solid maths back ground in basic number theory (factorisation, Diophantine equations, classical theorems, multiplicative arithmetic functions, perfect numbers and Mersenne numbers).

- K Rosen;
*Elementary number theory and its applications* - K Ireland, M Rosen;
*A Classical Introduction to Modern Number Theory* - F Gouvêa;
*p-adic numbers: an introduction*

#### Semester two

##### Further Topics in Analysis

A solid background in metric and topological spaces covering completeness etc. as covered in Foundations of Advanced Analysis and a solid knowledge of linear algebra.

- Bollobás, Béla;
*Linear analysis: an introductory course*

##### Algebraic Number Theory

A solid background in advanced number theory, algebra, rings and modules and Galois theory.

- S Lang;
*Algebraic Number Theory*

##### Combinatorial Group Theory

A solid background in group theory.

- D L Johnson;
*Presentations of Groups*

## Modules

#### Advanced Group Theory

This module looks at basic examples of finite groups and their actions on finite sets.

The material is focused on illustrating how the abstract algebraic concept of a group gives rise to numerous applications in a wide range of subjects, such as combinatorics, number theory, geometry, probability and mathematical physics.

#### Advanced Linear Analysis

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.

#### Algebraic Geometry

Algebraic geometry is one of the great twentieth century achievements in mathematics.

This module describes geometrical structures using the language of algebra.

It presents and discusses affine and projective algebraic varieties over algebraically closed fields, and the associated algebraic structures: co-ordinate rings and function fields. Various notions are illustrated on algebraic curves, including elliptic curves.

The module further introduces the concept of the Zariski topology and the spectrum of rings. The structure sheaf on the spectrum of rings and its properties are motivated and explained. Birational geometry notions are discussed.

#### Algebraic Number Theory

This module presents the fundamental features of algebraic number theory, the theory in which numbers are viewed from an algebraic point of view.

So numbers are often treated as elements of rings, fields and modules, and properties of numbers are reformulated in terms of the relevant algebraic structures.

This approach leads to understanding of certain arithmetical properties of numbers (in particular, integers) from a new point of view.

#### Combinatorial Group Theory

This module is largely concerned with infinite groups, especially free groups, although their role in describing and understanding finite groups is emphasized.

Following the basic definitions of free groups and group presentations, the fundamental Nielson-Schreier Theorem is covered in some detail.

Methods for manipulating group presentations, and using them to read off properties of a given group, will be covered: for example, obtaining presentations for subgroups of finite index in a given group.

The interplay between presentations and standard group-theoretic construction (for example, direct products and group extensions) will also be covered.

There will also be an introduction to the advanced topics of free products with amalgamation and corresponding extensions.

Many of these methods may be automated and have been implemented on computers.

#### Complex Analysis

Complex analysis is one of the central areas of pure mathematics, with many remarkable, often surprising, theorems.

Whereas the introductory course emphasizes applications and computational techniques, differentiable functions of a complex variable display striking properties, and the course will investigate these.

#### Elliptic Curves

#### Foundations of Advanced Analysis

This module introduces and studies the abstract and more advanced concepts of metric and topological spaces, leading on to topics such as Functional Analysis.

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.

The coursework and presentation part of the assessment will deepen understanding and explore further related special topics.

#### Further Topics in Analysis

This module will solidify and deepen the student's knowledge and understanding of the concepts and theory of mathematical analysis.

It builds on the fundamental theory of metric and topological spaces and develops this to some more advanced settings.

At the same time, the student will learn further methods of rigorous proof in abstract analysis and will be trained in the use of concepts and techniques relevant to research in mathematicial analysis.

#### Further Topics in Rings and Modules

Commutative rings and course 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.

#### Galois Theory

Galois Theory concerns the study of the group of symmetries associated with polynomial equations (in modern language, the group of automorphisms of field extensions). It shows the impossibility of solving the general quintic equation by radicals, and it can be used to prove that once cannot square the circle, duplicate a cube, nor trisect an angle using only a ruler and a compass! We start by laying a theoretical foundation to the construction of splitting fields, and thence the factorisation of polynomials.

The Fundamental Theorem of Galois Theory lies at the heart of the course, and shows a close connection between field theory and group theory; we show how the question of solvability of a polynomial can be completely settled using group theory.

Another topic will concern finite field, which are of central importance in many applications including coding and cryptography.

Number fields form another large class of fields which, on the one hand, are sufficiently simple so as to allow an initial study by elementary methods, and, on the other hand, are sufficiently rich in structure to represent some major aspects of Galois theory.

Some number fields will be used for illustration purposes throughout, and some of them, e.g. cyclotomic fields, are studied in detail.

The principle objective of studying Galois Theory are not only its applications, but also the appreciation of the beauty of a single brilliant idea.

#### Higher Number Theory

This module develops some of the main aspects of the theory of numbers with emphasis on Dirichlet series and their applications to the study of prime numbers. It builds on elementary number theory and uses techniques of complex analysis.

This will provide the springboard for the study of more advanced topics which will be useful for further study of algebra, number theory and other areas of Mathematics.

Students should also learn to present and develop a mathematical argument on a self-directed basis.

#### Pure Mathematics Dissertation

In this module a substantial investigation will be carried out on a topic in Pure Mathematics. The study will be largely self-directed, although a supervisor will provide oversight and input where necessary.

The topic will be chosen from a list provided by the School. The topic will be in algebra, analysis or number theory.

The dissertation will consist of a review of some advanced theory and results in one of these areas, or an investigation into a research topic in algebra, analysis or number theory using research-based methodologies appropriate for the chosen topic.

Some projects may contain an element of computing using existing software packages.

The above is a sample of the typical modules that we offer 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. Due to the passage of time between commencement of the course and subsequent years of the course, modules may change due to developments in the curriculum and information is provided for indicative purposes only.

## Fees and funding

### UK/EU students

#### Tuition fees

Information on current course tuition fees can be found on the finance pages.

As a student on this course, we do not anticipate any extra significant costs, 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 which you would need to factor into your budget.

**Graduate School**

The Graduate School website provides more information on internal and external sources of postgraduate funding.

### International students

#### Tuition fees

Information on current course tuition fees can be found on the finance pages.

#### School scholarships for UoN UK alumni

For 2019-19 entry, 10% Alumni scholarships may be offered to former University of Nottingham graduates who have studied at the UK campus.

### Government loans for masters courses

The Government offers postgraduate student loans for students studying a taught or research masters course. Applicants must ordinarily live in England or the EU. Student loans are also available for students from Wales, Northern Ireland and Scotland.

### International and EU students

Masters scholarships are available for international students from a wide variety of countries and areas of study. You must already have an offer to study at Nottingham to apply. Please note closing dates to ensure your course application is submitted in good time.

Information and advice on funding your degree, living costs and working while you study is available on our website, as well as country-specific resources.

## Careers and professional development

Graduates of our mathematics MScs have gone into:

- Industry
- Business
- Commerce
- Statistics (environment, forensic, government, medical)
- Medical research
- The pharmaceutical industry
- Biometrics
- PhD study

### Average starting salary and career progression

In 2017, 100% of postgraduates in the school who were available for employment had secured work or further study within six months of graduation. The average starting salary was £30,800 with the highest being £60,000.*

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

**Career prospects and employability**

University of Nottingham is consistently named as one of the most targeted universities by Britain’s leading graduate employers – ranked in the top 10 in* The Graduate Market 2013-2019, *High Fliers Research.

Those who take up a postgraduate research opportunity with us will not only receive support in terms of close contact with supervisors and specific training related to your area of research, you will also benefit from dedicated careers advice from our Careers and Employability Service.

Our Careers and Employability Service offers a range of services including advice sessions, employer events, recruitment fairs and skills workshops – and once you have graduated, you will have access to the service for life.

**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.

## Contact

**Dr Nikolaos Diamantis**

School of Mathematical Sciences

The University of Nottingham

University Park

Nottingham

NG7 2RD