Undergraduate Mathematical Sciences 2015-2016

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.

The course begins with a recap of the fundamental ideas from Introduction to Analysis to support and reinforce an understanding of the rigorous language of mathematical analysis.  

Topics include the behaviour of sequences and series of real numbers, power series including Taylor’s theorem and uniform convergence of sequences of functions.

In many “real life” problems one is required to find optimal solutions, namely a solution which, generally speaking, either minimises “cost” or maximises “gain”. To do so, one models the problem mathematically, and then applies the appropriate mathematical techniques to find the optimal solution. In this course students will learn how to formulate optimisation problems mathematically  and study the relevant techniques from analysis and algebra which are useful in solving them. Applications to “real life” problems and the use of computer software to solve them will also be discussed.

The course introduces the following basic concepts of the classical mechanics:

 Newton’s laws of motion;

 the motion of projectiles with and without air resistance;

 the theory of simple vibrations and, in particular, simple harmonic motion;

 the concepts of momentum, angular momentum, energy and corresponding conservation laws;

an inertial frame and related Galilean transformations;

 systems of particles with and without collisions;

Recommended to mathematicians and physicists.

This course builds on the courses Introduction to Analysis (MA2005) and Further Real Analysis (MX3021), and 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.

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.

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.

The aim of the course is to introduce the student to some basic techniques of classical applied mathematics. In particular the main areas of study will be vector calculus (up to the Divergence theorem), partial differential equations and Fourier series.

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.

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.

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.

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.

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.

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.

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.

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.

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.