Triangle

Course overview

The MSc provides you with in depth knowledge of algebra, number theory, group theory and analysis. Your interest in mathematical theorems will be fuelled by studying optional modules in topics ranging from Elliptic Curves to Linear Analysis.

You'll learn from mathematicians with research interests ranging from algebraic geometry to applied analysis. This is reflected in the wide range of previous dissertation projects. They have included a broad spectrum of subjects, including:

  • Homotopic topology
  • The sieve of Eratosthenes
  • Reduced models for ferromagnetic materials.

Teaching is influenced by the research work of the Algebra and Analysis and the Number Theory and Geometry groups. Some of the topics studied by group members include:

  • Analytic number theory
  • Arithmetic, algebraic and anabelian geometry
  • Categorical representation theory
  • Data and image analysis

The masters course provides flexibility for a range of careers whether your interests lie in pure or applied maths. Many of our graduates go on to study a PhD. Others use it more widely for jobs in industry, teaching, government and finance.

Why choose this course?

Top 10 UK ranking

research power and quality. Many 4* level papers in pure maths featured in REF submission

Research Excellence Framework 2014

Optional modules

allow you to tailor the degree to suit your interests

Acclaimed lecturers

regularly contribute to articles in mathematical journals including Compositio Mathematica, Crelle's Journal

One-to-one supervision

by faculty staff during dissertation project

No programming assumed

yet some staff are programming experts who contribute to Sage and other PC algebra systems

Flexibility to focus

by specialising in one of the strands of algebra, analysis or number theory and geometry

Course content

You will study six modules across the year, split between three modules per semester, plus a dissertation. Students must take 120 credits worth of optional modules and a dissertation worth 60 credits.

Central to the course are pairs of modules each consisting of an autumn and a spring module. In 2019/20, students selected modules from at least one of the following pairs, plus additional optional modules to suit their specific interests.

  • Advanced Group Theory and Combinational Group Theory
  • Higher Number Theory and Algebraic Number Theory

By completing one of these topic combinations, you will gain advanced knowledge and expertise in one of the key areas of group theory or number theory.

Modules

Core modules

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. You will undertake 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.

Choose one combination from following:

Advanced Group Theory

This module starts with 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.

Topics for this module include:

  • review of finite groups
  • introduction to group actions
  • basic examples and main results on actions of finite groups
  • applications to symmetric group and Young's diagrams
  • actions of p-groups, simplicity tests
  • finitely generated abelian groups
  • basic commutative harmonic analysis and discrete Fourier transform
  • introduction to non-commutative harmonic analysis.
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, you will study the fundamental Nielson-Schreier Theorem in some detail. You will also study an introduction to the advanced topics of free products with amalgamation and corresponding extensions.

Or:

Higher Number Theory

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. Equations of this sort are called Diophantine equations, and were first studied by the Greeks. What makes the study of these equations so fascinating is the seemingly chaotic distribution of prime numbers within the integers. We shall establish the basic properties of the Riemann zeta-function in order to find out how evenly these primes are distributed in nature. This module will present several methods to solve Diophantine equations including analytic methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.

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. The module discusses some of the central results in the theory which were obtained by several generations of mathematicians in the nineteenth and the first two decades of the twentieth century.

The theory of finite extensions of the field of rational numbers, the structure of their rings of integers, ideal classes and unique factorisation, Dirichlet's unit theorem, the splitting of prime ideals in field extensions, and applications to various classical problems of number theory will be discussed.

Remaining optional modules to be chosen from:

Advanced Group Theory

This module starts with 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.

Topics for this module include:

  • review of finite groups
  • introduction to group actions
  • basic examples and main results on actions of finite groups
  • applications to symmetric group and Young's diagrams
  • actions of p-groups, simplicity tests
  • finitely generated abelian groups
  • basic commutative harmonic analysis and discrete Fourier transform
  • introduction to non-commutative harmonic analysis.
Advanced Linear Analysis

This module provides 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. normed, Banach and Hilbert spaces, basic examples; Lebesgue integral in R^n; Hilbert space of square integrable functions as main example; bases in Hilbert spaces, orthonormal bases; linear operators and the operator norm; compact operators;integral operators on L^2 as main example; the Spectral Theorem for compact self-adjoint and normal operators.

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. The module discusses some of the central results in the theory which were obtained by several generations of mathematicians in the nineteenth and the first two decades of the twentieth century.

The theory of finite extensions of the field of rational numbers, the structure of their rings of integers, ideal classes and unique factorisation, Dirichlet's unit theorem, the splitting of prime ideals in field extensions, and applications to various classical problems of number theory will be discussed.

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, you will study the fundamental Nielson-Schreier Theorem in some detail. You will also study an introduction to the advanced topics of free products with amalgamation and corresponding extensions.

Elliptic Curves

The module 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
Foundations of Advanced Analysis

This course introduces and studies the abstract and more advanced concepts of metric and topological spaces, leading on to topics such as Functional Analysis. Metric space generalises the concept of distance familiar from Euclidean space and applies in many different contexts. It provides a notion of continuity for functions between quite general spaces. 

The course 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. Coursework and presentation which form part of the assessment will deepen your understanding and explore further related special topics.

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.

Higher Number Theory

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. Equations of this sort are called Diophantine equations, and were first studied by the Greeks. What makes the study of these equations so fascinating is the seemingly chaotic distribution of prime numbers within the integers. We shall establish the basic properties of the Riemann zeta-function in order to find out how evenly these primes are distributed in nature. This module will present several methods to solve Diophantine equations including analytic methods using zeta-functions and Dirichlet series, theta functions and their applications to arithmetic problems, and an introduction to more general modular forms.

The above is a sample of the typical modules 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. Modules (including methods of assessment) may change or be updated, or modules may be cancelled, over the duration of the course due to a number of reasons such as curriculum developments or staffing changes. Please refer to the module catalogue for information on available modules. This content was last updated on Tuesday 30 March 2021.

Learning and assessment

How you will learn

  • Lectures
  • Problem classes
  • Tutorials

How you will be assessed

  • Examinations
  • Coursework
  • Dissertation

You will be awarded the Master of Science Degree provided you have successfully completed the taught stage by achieving a weighted average mark of at least 50%, with no more than 40 credits below 50% and no more than 20 credits below 40%.

You must achieve a mark of at least 50% in the dissertation.

Candidates for the masters degree who fail to reach the required standard for the award may be awarded a Postgraduate Diploma or a Postgraduate Certificate under certain circumstances.

Contact time and study hours

The number of formal contact hours and class sizes vary depending on the optional modules you are studying. As a guide, in the Autumn and Spring semesters you will typically spend around 12 hours per week between Monday and Friday in classes.

You will work on your research project between June and September, usually based at the University.

Teaching is provided by academic staff within the School of Mathematical Sciences. The majority of modules are typically delivered by Professors, Associate and Assistant Professors. Additional support in small group and practical classes may include PhD students and post-doctoral researchers.

Programming skills are not required for the course, however, several staff members are experts in computational aspects, including Sage to which some regularly contribute.

The majority of your lecturers and tutors will be based within the mathematics building. This means if you need to get in touch with them during office hours, they can be contacted easily as they are close by.

Entry requirements

All candidates are considered on an individual basis and we accept a broad range of qualifications. The entrance requirements below apply to 2022 entry.

Undergraduate degree2:1 in mathematics, or a closely related subject with substantial pure mathematics content.

Applying

Our step-by-step guide covers everything you need to know about applying.

How to apply

Fees

All listed fees are per year of study.

Qualification MSc
Home / UK £10,500 per year
International £20,000 per year

Additional information for international students

If you are a student from the EU, EEA or Switzerland, you will pay international tuition fees in most cases. If you are resident in the UK and have 'settled' or 'pre-settled' status under the EU Settlement Scheme, you will be entitled to 'home' fee status.

Irish students will be charged tuition fees at the same rate as UK students. UK nationals living in the EU, EEA and Switzerland will also continue to be eligible for ‘home’ fee status at UK universities until 31 December 2027.

For further guidance, check our information for applicants from the EU.

These fees are for full-time study. If you are studying part-time, you will be charged a proportion of this fee each year (subject to inflation).

Additional costs

As a student on this course, we do not anticipate any extra significant costs, alongside your tuition fees and living expenses.

Printing

Due to our commitment to sustainability, we don’t print lecture notes but these are available digitally. You will be given £5 worth of printer credits a year. You are welcome to buy more credits if you need them. It costs 4p to print one black and white page.

Books

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.

Computers

Personal laptops are not compulsory as we have computer labs that are open 24 hours a day but you may want to consider one if you wish to work at home.

Funding

School scholarships for UoN international alumni

For 2021/22 entry, 10% alumni scholarships may be offered to former University of Nottingham international graduates who have studied at the UK campus. 

There are many ways to fund your postgraduate course, from scholarships to government loans.

We also offer a range of international masters scholarships for high-achieving international scholars who can put their Nottingham degree to great use in their careers.

Check our guide to find out more about funding your postgraduate degree.

Postgraduate funding

Careers

We offer individual careers support for all postgraduate students.

Expert staff can help you research career options and job vacancies, build your CV or résumé, develop your interview skills and meet employers.

Each year 1,100 employers advertise graduate jobs and internships through our online vacancy service. We host regular careers fairs, including specialist fairs for different sectors.

International students who complete an eligible degree programme in the UK on a student visa can apply to stay and work in the UK after their course under the Graduate immigration route. Eligible courses at the University of Nottingham include bachelors, masters and research degrees, and PGCE courses.

Graduate destinations

Graduates go on to pursue a variety of careers. Some enter roles that have a direct relationship with pure mathematics including cryptography and statistical programming. Others choose to pursue a PhD in mathematics directly related to pure mathematics or even in applied mathematics.

Graduate destinations include:

Career progression

97.5% of postgraduates from the School of Mathematical Sciences secured graduate level employment or further study within 15 months of graduation. The average annual salary for these graduates was £28,131.*

* HESA Graduate Outcomes 2020. The Graduate Outcomes % is derived using The Guardian University Guide methodology. The average annual salary is based on graduates working full-time within the UK.

Two masters graduates proudly holding their certificates
" My area of research includes motives and A^1-homotopy theory, quadratic forms and cohomological operations. Motives permit to translate flexible topological methods into the rigid realm of Algebraic Geometry. The resulting "motivic world" can be considered as a much richer version of the "topological world". The techniques created permitted to solve many long standing problems in Algebraic Geometry, Algebra and Number Theory as well as to get a new insight on classical topological questions, such as the computation of the homotopy groups of spheres. This new and rapidly developing area influences many branches of mathematics and finds new and interesting applications "
Alexander Vishik, Associate Professor and Reader

Related courses

This content was last updated on Tuesday 30 March 2021. Every effort has been made to ensure that this information is accurate, but changes are likely to occur given the interval between the date of publishing and course start date. It is therefore very important to check this website for any updates before you apply.