production

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. Sylow p-subgroups are introduced and the three Sylow theorems are proved. Throughout symmetric groups are consulted as a source of examples.

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.

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.

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.

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.

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.

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.

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.

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.

15 credits

Level 4

First Term

This two-part course covers the fundamentals of nonlinear dynamical systems. Often no analytical solutions exist for such systems, yet they are essential to describe many phenomena in physics, chemistry, engineering, and biology.

Part I lays out the mathematical foundations required for understanding nonlinear dynamical systems. The focus is on the dynamical behaviours exhibited by linear systems, how these describe nonlinear systems locally, and how these can model time varying natural systems.

15 credits

Level 4

First Term

Financial mathematics is a fundamental tool in modeling problems in economics and finance. The tools and techniques are employed by almost all large financial institutions in order to obtain predictive models of the market. It is also a necessary tool in actuarial science. The course enhances problem solving and abstraction skills.

15 credits

Level 4

Second Term

A knot is a closed curve in three dimensions. How can we tell if two knots are the same? How can we tell if they are different? This course answers these questions by developing many different "invariants" of knots. It is a pure mathematics course, drawing on simple techniques from a variety of places, but with an emphasis on examples, computations and visual reasoning.

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.

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.

15 credits

Level 4

Second Term

This two-part course covers the fundamentals of nonlinear dynamical systems. Often no analytical solutions exist for such systems, yet they are essential to describe many phenomena in physics, chemistry, engineering, and biology.

In Part II we focus on bifurcations and chaos. After relativity and quantum mechanics, chaos is often thought of as the third major advance in physics during the 20th century. This paradigm-changing theory has major implications for the predictability of natural systems.

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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