2:1 (or international equivalent) in mathematics, or a closely related subject, with substantial pure mathematics content

## Overview

** Key facts**

- Gain a broad understanding of several exciting areas of pure mathematics that are the subject of current research
- Learn from internationally recognised researchers of algebra, analysis, and number theory and geometry
- Develop your mathematical ability in one of the top 10 schools for Mathematical Sciences in the UK (according to the assessment of 'research power' and 'research quality' by the 2014 Research Excellence Framework (REF)).

- On completion, the MSc offers a solid foundation for a career in research in pure mathematics

To enhance your experience on the course you should have a strong interest in pure mathematics and specifically a good background in at least two to three of the following subject areas: algebra, number theory, group theory or analysis.

This course is designed for both future PhD students and for those who intend to successfully enter the job market upon graduation.

This is reflected in the structure of the course which has two parts: The first one consists of taught modules which provide the skills and background for further work in mathematics and its applications. The second is a substantial dissertation which introduces the student to independent, research-informed, study.

You will benefit from an excellent course infrastructure and the expertise of active researchers working in a variety of pure mathematics subjects. The School has a very successful record of training outstanding pure mathematicians, many of whom hold high-profile posts outside academia and others leading international careers in mathematical research.

## Course structure

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.

Students must take 120 credits from the group listed under optional 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**

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

### Core

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.

### Optional

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.

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 geometry is one of the great 20th century achievements in mathematics. This module describes geometrical structures using the language of algebra and 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 also discussed.

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.

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.

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

Topics will normally include the following:

- Riemann sphere
- Moebius transformations and their properties
- Sequences and series of analytic functions
- Topological properties of analytic functions
- Riemann mapping theorem
- the remarkable theorem of Picard
- the connection between Riemann's result and Picard's
- the elements of the theory of complex dynamics, in which fractals arise, such as the famous Julia and Mandelbrot sets

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

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.

This module develops the fundamental theory of metric and topological spaces. The module will include the study of the main concepts of mathematical analysis in these settings by investigating the properties of numerous examples, and by developing the associated theory, with a strong emphasis on problem-solving and rigorous, axiomatic proof. You will also cover a selection of more advanced topics in analysis.

Topics covered will include the following:

- normed spaces
- Banach spaces and bounded linear operators
- Baire Category Theorem for complete metric spaces
- linear functionals and dual spaces of normed spaces

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 concerns the study of the group of symmetries associated with polynomial equations. We start by laying a theoretical foundation to the construction of splitting fields, and therefore the factorization 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 is 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, eg cyclotomic fields, are studied in detail.

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 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. This course page 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.

### Teaching methods and assessment

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.

## Careers and professional development

### Careers

On completion, the MSc provides a solid foundation for a career in research in pure mathematics.

### Average starting salary and career progression

100% of postgraduates in the School of Mathematical Sciences 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 who were available for employment, 2016/17. Salaries are calculated based on the median of those in full-time paid employment within the UK.

### Careers support and advice

We offer individual careers support for all postgraduate students whatever your course, mode of study or future career plans.

You can access our Careers and Employability Service during your studies and after you graduate. Expert staff will help you research career options and job vacancies, build your CV or résumé, develop your interview skills and meet employers.

More than 1,500 employers advertise graduate jobs and internships through our online vacancy service. We host regular careers fairs, including specialist fairs for different sectors.

## Fees and funding

### Additional costs

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.

### Scholarships and bursaries

**School scholarships for UoN UK alumni**

For 2020/21 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

Masters student loans of up to £10,906 are available for taught and research masters courses. Applicants must ordinarily live in the UK or EU.

### International and EU students

Masters scholarships are available for international and EU 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 you apply for your course with enough time. We provide guidance on funding your degree, living costs and working while you study. You can also access specific funding opportunities, entry requirements and other resources for students from specific countries.

## 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.