Module Outlines
Here are brief outlines of the modules that may be taken by current 3rd year Mathematics and Physics students.
Term 1
|
The modern notion of measure, developed in the late 19th century, is an extension of the notion of area, developed by the Greeks. A measure m is a law which assigns a number m(A) to certain subsets A of a given space and is a natural generalization of the following notions: 1) length of an interval, 2) area of a plane figure, 3) volume of a solid, 4) integral of a non-negative function over some space, e.t.c.
It originated in the real analysis and is used now in many areas of mathematics like, for instance, geometry, probability theory, dynamical systems, functional analysis, etc.
Given a measure m, one can define the integral of suitable real valued functions with respect to m. Riemann integral is applied to continuous functions or functions with ''few'' points of discontinuity. For measurable functions that can be discontinuous ''almost everywhere'' Riemann integral does not make sense. However it is possible to define more flexible and powerful Lebesgue's integral (integral with respect to Lebesgue's measure) which is one of the key notions of modern analysis.
Lecturer: Roman Kotecky
|
The course is split into 6 sections:
1. Phase Plane Methods, 2. Reaction Kinetics, 3. Biological Waves 4. Multi-Species Waves 5. Animal coat patterns 6. Neural models for pattern formation/storage and recognition
Mathematical Biology is a relatively new area of applied mathematics in which mathematical models are used to study biological phenomena in areas such as ecology, epidemiology, biochemistry, development and, medicine. The course begins by discussing application of phase plane methods for ODEs to ecology and medicine, followed by a study of ODEs applied to enzymatic processes such as digestion; some introductory perturbation theory will be taught in this section.
Wave phenomena are important in biology and will be considered in the next section of the course with an introduction to the Reaction Diffusion equation for dilute systems. Wave solutions of reaction-diffusion equations will be discussed and their existence proved. Various applications including wound healing and, spread of disease are discussed here.
The course will also discuss reaction-diffusion equations applied to pattern formation, including an explanation of why animals such as leopards with spotted bodies tend to have striped tails. The last part of the course is on mathematical modelling of pattern formation in the brain.
The course is split into 6 sections:
1. Phase Plane Methods, 2. Reaction Kinetics, 3. Biological Waves 4. Multi-Species Waves 5. Animal coat patterns 6. Neural models for pattern formation/storage and recognition
Lecturer: Hugo van den Berg
|
The course offers an overview of the rise of modern algebra, a sense of what doing history is, and an acquaintance with primary sources (in translation). There is some overlap with MA3A6 Algebraic Number Theory and MA3D5 Galois Theory, to which this course might serve as an historical introduction.
Lecturer: Jeremy Gray
|
This is essentially a module about infinite-dimensional Hilbert spaces, which arise naturally in many areas of applied mathematics. The ideas presented here allow for a rigorous understanding of Fourier series and more generally the theory of Sturm-Liouville boundary value problems. They also form the cornerstone of the modern theory of partial differential equations.
Hilbert spaces retain many of the familiar properties of finite-dimensional Euclidean spaces Rn - in particular the inner product and the derived notions of length and distance - while requiring an infinite number of basis elements. The fact that the spaces are infinite-dimensional introduces new possibilities, and much of the theory is devoted to reasserting control over these under suitable conditions.
The module falls, roughly, into three parts. In the first we will introduce Hilbert spaces via a number of canonical examples, and investigate the geometric parallels with Euclidean spaces (inner product, expansion in terms of basis elements, etc.). We will then consider various different notions of convergence in a Hilbert space, which although equivalent in finite-dimensional spaces differ in this context. Finally we consider properties of linear operators between Hilbert spaces (corresponding to the theory of matrices between finite-dimensional spaces), in particular recovering for a special class of such operators (compact self-adjoint operators) very similar results to those available in the finite-dimensional setting.
Throughout the abstract theory will be motivated and illustrated by more concrete examples.
Lecturer: Richard Sharp
|
The project is designed to provide you with the opportunity to make an in depth investigation of a particular area of physics in collaboration with your project supervisor.
The project is only available to BSc students.
ORGANIZER: Geetha Balakrishnan
|
Employers look for many things in would-be employees. Sometimes they will be looking for specific knowledge, but often they will be more interested in more general skills frequently referred to as transferable skills. One such transferable skill is the ability to communicate effectively, both orally and in writing. The aim of this module is to introduce you to the different approaches required to write for non-specialist audiences.
This module is core for BSc students but is unavailable to MMathPhys students.
ORGANISER: Michael Pounds
|
The basic principles of quantum mechanics will be applied to a range of problems in atomic physics. The concept of spin will be introduced and the importance of the indistinguishability of identical particles in quantum mechanics discussed. Perturbation theory and variational methods will be described and applied to several problems. The hydrogen and helium atoms will be analysed and the ideas that come out from this work will be applied to obtain a good qualitative understanding of the periodic table.
LECTURER: Martin Lees
|
The module builds on the first and second year modules on electromagnetism by using Maxwell's equations to describe the generation and propagation of electromagnetic waves and their interaction with matter and, in particular, plasmas. The module will introduce at the start the magnetic vector potential, A, which is defined so that the magnetic field B=curl A. Although this appears at first sight to be a technical device, it will prove very useful in later modules particularly in quantum mechanics. We will see that it is also the natural variable to consider when exploring how electric and magnetic fields are affected under Lorentz transformations (special relativity).
The radiation (EM-waves) emitted by accelerating charges will be described in more detail using retarded potentials (these are just time-dependent analogs of the usual electrostatic potential and its vector equivalent used to describe magnetic fields) which have the wave-like nature of light built in. The scattering of light by free electrons (Thompson scattering) and by bound electrons (Rayleigh scattering) will also be described. Understanding the bound electron problem led Rayleigh to his celebrated explanation of why the sky is blue and why sunlight appears redder at sunrise and sunset.
LECTURER: Sandra Chapman
|
Quantum and statistical mechanics are the basis for describing the physics of solids, and in this module we will apply these ideas to problems in condensed matter. We will understand the role of the microscopic structure in determining the properties of macroscopic samples and be able to explain magnetic and conductivity phenomena, and how to measure these experimentally.
LECTURER: Rachel Edwards
|
This module should help you develop the C programming skills beyond the level of the introductory second year module PX270 C Programming. The module will consist of some lectures and a series of programming exercises designed to illustrate important aspects of program design. The module will also cover some important numerical techniques used in data processing in physics, including: the coding and application of fast Fourier transforms. Aspects relating to the reliability, accuracy and efficiency of these techniques will be discussed, as well as other issues such as making software user friendly, and data transfer between platforms. The module will be assessed on the basis of the exercises completed during the module and some project work.
LECTURER: Ben McMillan
|
The diffusion, convection, chemical reactions and the interaction with living organisms which take place in or at the boundaries of the atmosphere determine the weather patterns we observe. The module will look at some of these processes. The lectures will also cover the phenomenon of cloud-formation and the role of the earth's rotation in determining flow patterns in the atmosphere.
LECTURER: David Leadley
|
This module looks at fluctuations and how they may be described using statistical mechanics (see Thermal Physics II). Fluctuations play an essential role in nature. We will see that statistical mechanics can be seen as a description of the role played by these fluctuations. It allows us to understand the physics of such seemingly diverse problems as diffusion, phase transitions and Fermi-Dirac and Bose-Einstein statistics. We will also give examples from Polymer physics where the behaviour of the polymer chains is driven by configurational entropy, rather than energy
LECTURER: Marco Polin
|
Plasmas are 'fluids' of charged particles. The motion of these charged particles (usually electrons) is controlled by the electromagnetic fields which are imposed from outside and by the fields which the moving charged particles themselves set up. This module will cover the key equations which describe such plasmas. It will examine some predictions derived on the basis of these equations and compare these with results from laboratory experiments and with observations from in situ measurements of solar system plasmas and remote observations of astrophysical systems. It will also be important to look at instabilities in plasmas and how electromagnetic waves interact with the plasmas.
LECTURER: Valery Nakariakov
|
To module aims to illustrate how important physical principles, from different areas of physics, can be developed to yield a description of complex physical systems like galaxies. The module should explore some of the properties of galaxies, which yield insights into their formation, evolution and ongoing processes.
LECTURER: Elizabeth Stanway
Term 2
The project and a number of modules taught by outside departments continue from Term 1.
|
The course focuses on the properties of differentiable functions on the complex plane. Unlike real analysis, complex differentiable functions have a large number of amazing properties, and are very "rigid'' objects. Some of these properties have been explored already in Vector Analysis. Our goal will be to push the theory further, hopefully revealing a very beautiful classical subject.
We will start with a review of elementary complex analysis topics from vector analysis. This includes complex differentiability, the Cauchy-Riemann equations, Cauchy's theorem, Taylor's and Liouville's theorem, Laurent expansions. Most of the course will be new topics: Winding numbers, the generalized version of Cauchy's theorem, Morera's theorem, the fundamental theorem of algebra, the identity theorem, classification of singularities, the Riemann sphere and Weierstrass-Casorati theorem, meromorphic functions, Rouche's theorem, integration by residues.
Lecturer: Xue-Mei Li
|
The lectures will provide the student with a solid background in the mathematical description of fluid dynamics. You will be introduced to the method of deducing the equations of motion from conservation laws (mass, momentum, energy), the value of dimensional analysis in finding scale-invariant solutions and universal turbulence spectra, role of the gravity and rotation in atmospheric and oceanic dynamics, and deriving approximate equations of motion (e.g. boundary layer equations).
The module will cover the following topics:
Kinematics of Fluid Motion. Specification of the flow by field variables; vorticity; stream function; strain tensor; stress tensor. Euler's equation, Navier-Stokes equation.
Conservation Laws. Conservation of mass, momentum and energy and equations of motion deduced from these laws; Bernoulli's equation.
Vorticity. Kelvin's circulation theorem, 3D vorticity equation, vortex lines, vortex tubes and vortex stretching. Cauchy-Lagrange. Special properties of the two-dimensional Euler and Navier-Stokes equations including the interaction of point vortices and vortex sheet.
Dimensional analysis. Reynolds number, Rayleigh number, Ekman number, Rossby number, etc.
Laminar flow. Flow in a pipe; shear flows; flow due to an oscillating plate; Stokes flows of very viscous fluids.
Boundary layers. Prandtl's boundary layer theory.
Instability and waves. Raleigh-Taylor and Kelvin-Helmholtz instabilities; stability of parallel flows. Inertia-gravity and internal waves. Waves on a deep water. Sound.
Geophysical Fluid Dynamics. Time-permitting there will be selections from the following: Stratified flow and hydrostatic equilibrium. Rotating reference frames; Taylor-Proudman theorem and Ekman boundary layer in rotating fluids. Shallow water equations. Geostrophic equations. Rossby waves.
Aerodynamics and Turbulence. Those with an interest in pursuing these topics are advised to take in Term 2: ES441, Advanced Fluid Dynamics.
Lecturer: James Sprittles
|
The important and pervasive role played by pdes in both pure and applied mathematics is described in MA250 Introduction to Partial Differential Equations. In this module I will introduce methods for solving (or at least establishing the existence of a solution!) various types of pdes. Unlike odes, the domain on which a pde is to be solved plays an important role. In the second year course MA250, most pdes were solved on domains with symmetry (eg round disk or square) by using special methods (like separation of variables) which are not applicable on general domains. You will see in this module the essential role that much of the analysis you have been taught in the first two years plays in the general theory of pdes. You will also see how advanced topics in analysis such as Linear Analysis which grew out of an abstract formulation of pdes. Topics in this module include:
- Fundamental solution of Laplacian, Green's function.
- Harmonic functions and their properties; sequential compactness of a bounded family of harmonic functions.
- The Gaussian heat kernel, inhomogeneous diffusion equations.
- Comparison and maximum principles.
- Poincare's inequality and energy methods.
- Existence of solution to nonconstant coefficient linear parabolic equations and/or semilinear equations.
Lecturer: Manh Hong Duong
|
Roughly speaking, a metric space is any set in which one can sensibly measure the ''distance'' between points. The ways in which distance is measured and the sets involved may be very diverse. For example, the set could be the sphere, and we could measure distance either along great circles or along straight lines through the globe; or the set could be New York and we could measure distance ''as the crow flies'' or by counting blocks. The set might be all possible sequences of human DNA, in which case the number of places where the sequences differ would be a measure of the distance between people's genes; or the set might be the set of real valued continuous functions on the unit interval, in which case we could take as a measure of the distance between two functions either the maximum of their difference, or alternatively its ``root mean square''.
This module examines how the important concepts introduced in first year analysis, such as convergence of sequences, continuity of functions, completeness, etc, can be extended to general metric spaces. Applying these ideas we will be able to prove some powerful and important results, used in many parts of mathematics.
Lecturer: Ian Melbourne
|
There are many situations in pure and applied mathematics where one has to consider the continuity and differentiability of a function f: Rn → Rm, such as the determinant of an n × n matrix as a function of its entries, or the wind velocity as a function of space and time. It turns out that partial derivatives, while easy to calculate, are not robust enough to yield a satisfactory differentiation theory.
The central object of study in this module is the Fréchet derivative: the derivative of f at a point x ∈ Rn is interpreted as a linear transformation df(x): Rn → Rm, or m × n matrix. This module establishes the basic properties of this derivative, which generalise those of single-variable calculus: the usual algebraic rules for differentiation hold true, as do appropriate versions of the chain rule, mean value theorem, Taylor's theorem, and the use of the derivative to find local minima and maxima of a real-valued function. Highlights of the module include the statement and proof of the inverse and implicit function theorems, which have many applications in both geometry and the study of solutions of nonlinear equations, and the Lagrange multiplier theorem for the minimization/maximization of constrained functions.
We will also study norms on infinite-dimensional vector spaces and some applications.
Lecturer: Oleg Pikhurko
|
In recent years considerable progress has been made in the application of physics and physical measurement techniques to medicine. This module concentrates on five major areas of medical physics: X-rays, radionuclide imaging, ultrasound, neuroelectromagnetism and radiotherapy. The aim of the module is to demonstrate the application of basic physical principles to these important areas of medical physics.
LECTURER: Michael Pounds
|
This module introduces the most important physical processes and detection methods required for understanding the broad range emission spectra of astrophysical objects from the radio regime to X-rays and gamma rays. It will provide a basis for any further studies in observational astrophysics.
LECTURER: Boris Gaensicke
|
This module shows how the properties of the stable nucleus can be understood in terms of elementary models using basic physics from earlier modules, but with the introduction of the strong nuclear force. It is shown that the main features of the decay of unstable nuclei can also be understood on the basis of these ideas, but that a further interaction, the weak interaction, has to be postulated.
LECTURER: Michal Kreps
|
Experiment and simulation are central to all sciences except (pure) mathematics, as they generate the data which science attempts to explain and predict. In this module you will gain some experience of the practical work involved. You will work through one experiment in the MPhys laboratory and one computer simulation of a physical system. You will then write a report and present your results to your colleagues. The module is largely about developing skills including group working, report writing, and presentation skills. There will also be a paperwork exercise on the analysis of data.
The module is core for MMathPhys students but is unavailable to BSc students.
ORGANISER: Oleg Petrenko
|
Questions about the origin of the universe, where it is going and how it may get there are the domain of cosmology. One of the questions we will address is whether the universe will continue to expand or ultimately contract. Relevant experimental data relate to the cosmic microwave radiation, the distribution of galaxies and the distribution of mass in the universe. We will discuss the implications of these in some detail. The module will outline Einstein's General Theory of Relativity and the setting up of Einstein's field equations. This is the 'full' theory of relativity. It starts from the apparently simple Principle of Equivalence, which states (roughly speaking) that the laws of physics are the same in all frames of reference including those accelerating in a gravitational field (special relativity is special as it is restricted to frames moving at constant speed with respect to one another).
The cosmological problem will be approached through the Cosmological Principle. This will lead us to the Robertson-Walker metric, Hubble's law and the Cosmological Red Shift. The application of the Einstein equations is then shown to lead irrevocably to the Big Bang Model, with singular behaviour at the origin of the universe. The evolution of the Primeval fireball and the synthesis of Helium is described and the module concludes with a discussion of gravitational collapse, event horizons and black holes.
LECTURER: Andrew Levan
|
The Standard Model (SM) describes elementary particles (the quarks, leptons, and bosons) using gauge theories. Although a full quantitative description of the SM requires the machinery of quantum field theory and is not easily accessible, it is quite possible to develop a good qualitative understanding of what is meant by a gauge theory and how this contrains the predictions of the model. A lot follows from symmetry. We will look at Noether's theorem (for any continuous symmetry property there is a conserved quantity, eg conservation of charge and invariance under gauge transformations are the same thing), flavour symmetry, parity (P) and others. We will show how these aspects of the model are tested against experiment. We will also look at the reasons for quark confinement and the concept of a momentum-transfer dependent coupling, the Higgs mechanism, quark mixing and questions about unification.
LECTURER: Sinead Farrington
|
'Kinetic Theory' is the theory of how things change and is therefore essentially about non-equilibrium phenomena. The description of such phenomena is statistical and is based on Boltzmann's equation (the same Boltzmann who sorted out the equilibrium statistical mechanics you met in Thermal Physics II) and on the related Fokker-Planck equation. These study the evolution in time of a distribution function, which gives the density of particles in the system's phase space. (Phase space is the space of states of the system, which means specifying particles' position and momenta.) We will establish relations between conductivity, diffusion constants and viscosity in gases. We will also look at molecular simulation and applications to financial modelling (many of the concepts are also the basis for models of the 'motion' of stock and option prices in financial markets).
An additional motivation of this module is to illustrate how some of the mathematics you learnt in second year applied mathematics modules is used in theoretical physics.
LECTURER: David Quigley
[ Top of Document ]