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### MX3020: GROUP THEORY

15 credits

Level 3

First Term

Group theory concerns the study of symmetry. The course begins with the group axioms, which provide an abstract setting for the study of symmetry. We proceed to study subgroups, normal subgroups, and group actions in various guises. Group homomorphisms are introduced and the related isomorphism theorems are proved. Composition series are introduced and the Jordan-Holder theorem is proved. Sylow p-subgroups are introduced and the three Sylow theorems are proved. Throughout symmetric groups are consulted as a source of examples.

### MX3035: ANALYSIS III

15 credits

Level 3

First Term

Analysis provides the rigourous, foundational underpinnings of calculus. The focus of this course is multivariable analysis, building on the single-variable theory from MA2009 Analysis I and MA2509 Analysis II. Concepts and results around multivariable differentiation are comprehensively established, laying the ground for multivariable integration in MX3535 Analysis IV.

As in Analysis I and II, abstract reasoning and proof-authoring are key skills emphasised in this course.

### MX3036: METRIC AND TOPOLOGICAL SPACES

15 credits

Level 3

First Term

The aim of the course is to introduce the basic concepts of metric spaces and their associated topology, and to apply the ideas to Euclidean space and other examples.

An excellent introduction to "serious mathematics" based on the usual geometry of the n dimensional spaces.

### MX3531: RINGS & FIELDS

15 credits

Level 3

Second Term

Many examples of rings will be familiar before entering this course. Examples include the integers modulo n, the complex numbers and n-by-n matrices with real entries. The course develops from the fundamental definition of ring to study particular classes of rings and how they relate to each other. We also encounter generalisations of familiar concepts, such as what is means for a polynomial to be prime.

### MX3535: ANALYSIS IV

15 credits

Level 3

Second Term

Analysis provides the rigourous, foundational underpinnings of calculus. This course builds on MX3035 Analysis III, continuing the development of multivariable calculus, with a focus on multivariable integration. Hilbert spaces (infinite dimensional Euclidean spaces) are also introduced.

Students will see the benefit of having acquired the formal reasoning skills developed in Analysis I, II, and III, as it enables them to work with increasingly abstract concepts and deep results. Techniques of rigourous argumentation continue to be a prominent part of the course.

### MX3536: DIFFERENTIAL EQUATIONS

15 credits

Level 3

Second Term

Differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In this course we will study the concept of a differential equation systematically from a purely mathematical viewpoint. Such abstraction is fundamental to the understanding of this concept.

### MX4023: PROJECT

15 credits

Level 4

First Term

The 4th year project is a good opportunity to do some research in an area of mathematics which is not covered in any other course. A choice of project topics will be made available to students before the start of the semester. Students will be expected to have regular meetings with their project supervisor. A written report should be submitted at the end of the course, with a presentation taking place shortly afterwards. Students should be able to demonstrate in the project that they have a good understanding of the topic they covered, often through working out examples.

### MX4082: GALOIS THEORY

15 credits

Level 4

First Term

Galois theory is based around a simple but ingenious idea: that we can study field extensions by instead studying the structure of certain groups associated to them. This idea can be employed to solve some problems which confounded mathematicians for centuries, including the impossibility of trisecting an angle with ruler and compass alone, and the insolubility of the general quintic equation.

### MX4083: MEASURE THEORY

15 credits

Level 4

First Term

Measure theory provides a systematic framework to the intuitive concepts of the length of a curve, the area of a surface or the volume of a solid body. It is foundational to modern analysis and other branches of mathematics and physics.

### MX4085: NONLINEAR DYNAMICS & CHAOS THEORY I

15 credits

Level 4

First Term

This course covers the fundamental mathematical concepts required for the description of dynamical systems, i.e., systems that change in time. It discusses nonlinear systems, for which typically no analytical solutions can be found; these systems are pivotal for the description of natural systems in physics, engineering, biology etc. Emphasis will be on the study of phase spaces.

Next to the theory of relativity and quantum mechanics, chaos and dynamical systems theory is been considered as one of three major advances in the natural sciences. This course offers the mathematics behind this paradigm changing theory.

### MX4086: OPTIMISATION THEORY

15 credits

Level 4

First Term

Linear optimisation is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It is widely used in business and economics, and is also utilised for some engineering problems. Industries that use linear programming models include transportation, energy, telecommunications and manufacturing. It has proved useful in modeling diverse types routing, scheduling, assignment and design.

### MX4545: NUMBER THEORY

15 credits

Level 4

Second Term

This course concerns the integers, and more generally the ring of algebraic integers in an algebraic number field.  The course begins with statements concerning the rational integers, for example we discuss the Legendre symbol and quadratic reciprocity. We also prove a result concerning the distribution of prime numbers. In the latter part of the course we study the ring of algebraic integers in an algebraic number field. One crucial result is the unique factorisation of a nonzero ideal as a product of primes, generalising classical prime factorisation in the integers.

### MX4546: ALGEBRAIC TOPOLOGY

15 credits

Level 4

Second Term

Algebraic topology is a tool for solving topological or geometric problems with the use of algebra. Typically, a difficult geometric or topological problem is translated into a problem in commutative algebra or group theory. Solutions to the algebraic problem then provide us with a partial solution to the original topological one.

### MX4549: GEOMETRY

15 credits

Level 4

Second Term

One of the aims of the course is to understand the mathematical concept of curvature. We will do this by first studying the geometry of polygonal surfaces, and then by looking at smooth surfaces in Euclidean space.

Polygonal surfaces provide a set of very easy examples with which we can explore the new ideas and quantities. They also allow us to develop the intuition needed in the later part of the course.

### MX4553: MODELLING THEORY

15 credits

Level 4

Second Term

This course was designed to show you what you can do with everything you learnt in your degree. We will use mathematical techniques to describe a fast variety of “real-world” systems: spreading of infectious diseases, onset of war, opinion formation, social systems, reliability of a space craft, patterns on the fur of animals (morphogenesis), formation of galaxies, traffic jams and others. This course will boost your employability and it will be exciting to see how everything you learnt comes together.

### MX4555: NONLINEAR DYNAMICS & CHAOS THEORY II

15 credits

Level 4

Second Term

This second part of the course covers more advanced mathematical concepts required for the description of dynamical systems. It continues the study of nonlinear systems, for which typically no analytical solutions can be found; these systems are pivotal for the description of natural systems.

Emphasis will be on the study of higher dimensional and chaotic systems. This second part of the course introduces stability criteria for more complex systems and outlines several key results that govern the behaviour of nonlinear dynamical system, such as requirements for chaotic behaviour and recurrence properties.

### MX4557: COMPLEX ANALYSIS

15 credits

Level 4

Second Term

This course asks what happens when concepts such as convergence of sequences and series, continuity and differentiability, are applied in the complex plane? The results are much more beautiful, and often, surprisingly, simpler, than over the real numbers. This course also covers contour integration of complex functions, which has important applications in Physics and Engineering.

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