Mathematics and Physics, _BSc

15 Credit Points

Linear algebra is the study of vector spaces and linear maps between them and it is a central subject within mathematics.

It provides foundations for almost all branches of mathematics and sciences in general. The techniques are used in engineering, physics, computer science, economics and others. For example, special relativity and quantum mechanics are formulated within the framework of linear algebra.

The two courses Linear Algebra I and II aim at providing a solid foundation of the subject.

15 Credit Points

Analysis provides the rigorous, foundational underpinnings of calculus. It is centred around the notion of limits: convergence within the real numbers. Related ideas, such as infinite sums (a.k.a. series) and continuity are also visited in this course.

Care is needed to properly use the delicate formal concept of limits. At the same time, limits are often intuitive, and we aim to reconcile this intuition with correct mathematical reasoning. The emphasis throughout this course is on rigorous mathematical proofs, valid reasoning, and the avoidance of fallacious arguments.

15 Credit Points

For most of us, our perceptions are governed most strongly by our vision. We see because of light, but what is light? It’s been considered a particle, a wave, and in modern physics is somehow both. This course explores the fascinating physics of this phenomenon, at an elementary mathematical level suitable for non-science students. We’ll cover petrological microscopy, of interest to geologists, interference and diffraction, how colour works, see how polarisation can be applied in both scientific fields and every day life, and see how the photon can be used in devices in the increasing prevalent field known as photonics.

15 Credit Points

Understanding oscillatory and wavelike behaviour is of huge importance in comprehending how our natural world works. It seems that everything in nature has its own cycle, rhythm or oscillation. From planets revolving around the sun to waves on the sea, even fundamental particles are treated as waves in modern physics. Accessible to students with some knowledge of calculus, this course will explain the mathematics of this fascinating and important subject. Methods of solving the differential equations that describe waves and oscillatory phenomena will be explored, including numerical techniques.

15 Credit Points

Linear algebra is the study of vector spaces and linear maps between them and it is a central subject within mathematics.

It provides foundations for almost all branches of mathematics and sciences in general. The techniques are used in engineering, physics, computer science, economics and others. For example, special relativity and quantum mechanics are formulated within the framework of linear algebra.

The two courses Linear Algebra I and II aim at providing a solid foundation of the subject.

15 Credit Points

Analysis provides the rigorous, foundational underpinnings of calculus. This course builds on the foundations in Analysis I, and explores the notions of differential calculus, Riemann integrability, sequences of functions, and power series.

The techniques of careful rigorous argument seen in Analysis I will be further developed. Such techniques will be applied to solve problems that would otherwise be inaccessible. As in Analysis I, the emphasis of this course is on valid mathematical proofs and correct reasoning.

15 Credit Points

In the 20th Century, Physics got strange, and this course sets out to explore the foundations of this modern approach. In Special Relativity we will look at the idea that time is not an absolute – that events can happen in different times for different observers – and explore the effects of travelling at close to the speed of light. The quantum mechanics section introduces some of the most exciting and dramatically successful science of all time, and discuss the evolution of this idea from the days of Schrodinger’s cat to quantum tunnelling.

15 Credit Points

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 Credit Points

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 Credit Points

Our world is made of three types of matter, Solids, Liquids and Gases. The first part of this course will explore the physical properties of these forms of matter and investigate important technological phenomena such as the flow of liquids and the causes of catastrophic failure in mechanical components. In the second half of the course, the nature of heat energy in matter will be explored. Thermodynamic behaviour will be understood in terms of Entropy and the operation of engines and their theoretical efficiency limitations will be explained.

15 Credit Points

This course gives an overview of a optical physics, spanning the classical ideas of geometrical optics through to modern photonic devices such as lasers that are quantum mechanical in nature. It touches on a number of wave phenomena such as optical polarisation and how optics is fundamentally linked to the powerful mathematical tool of Fourier theory. Through a range of examples, the course will illustrate how optics is used in the modern world, from microscopes to telecommunications.

15 Credit Points

The course aims to provide the students with the underpinning knowledge that will enable them to think constructively about phenomena that relate to the quantum structure of matter. It is intended that the students will gain a broad appreciation of the hierarchy of interactions that give rise to the energy levels of atoms and the consequent structure of the associated spectroscopic transitions. In comparison to the previous years more emphasis will be put on the general, mathematical structure of quantum theory, tackling topics such as Hilbert spaces and time independent perturbation theory.

15 Credit Points

We are surrounded by electromagnetic phenomena; it is not possible to understand the physical world without them. In this course we will discuss the link between electricity and magnetism, noticing that changing electric magnetic fields generate electric fields and the other way around. This will lead to the introduction of Faraday’s law, hugely relevant to understand how we generate electricity, and to the introduction of Maxwell’s correction to Ampere’s law, which will lead to the astounding result that light is an electromagnetic wave! We will finish the course by exploring how electromagnetic waves propagate and how they are originated.

15 Credit Points

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 Credit Points

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 Credit Points

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.

45 Credit Points

PX4013 provides the opportunity to carry out an independent, open-ended, piece of research work. This can be in an area of physics (astronomy, nuclear physics, superconductors, dynamical systems etc.) or in related subjects where physicists tools can be applied (generation of proteins, biomechanics, infectious diseases etc.). The project can be dissertation based, practical or computational. You will develop: presentation skills; experience of reading and thinking about a specialist topic in depth; critical analysis skills of your own and other people’s scientific work and project management skills. This will help prepare for your future career beyond university.

15 Credit Points

Statistical physics derives the phenomenological laws of thermodynamics from the probabilistic treatment of the underlying microscopic system. Statistical physics, together with quantum mechanics and the theory of relativity, is a cornerstone in our modern understanding of the physical world.

Through this course, you will gain a better understanding of fundamental physical concepts such as entropy and thermodynamic irreversibility, and you will learn how derive some simple thermodynamic properties of gases and solids.

The final part of the course is devoted to an introduction to stochastic systems, which are widely used in many different fields such as physics, biology and economics.

15 Credit Points

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.