Module Outlines
Here are brief outlines of the modules that may be taken by current fourth year MMathPhys mathematics and physics students.
Term 1
|
Dynamical Systems is one of the most active areas of modern mathematics. This course will be a broad introduction to the subject and will attempt to give some of the flavour of this important area.
The course will have two main themes. Firstly, to understand the behaviour of particular classes of transformations. We begin with the study of one dimensional maps: circle homeomorphisms and expanding maps on an interval. These exhibit some of the features of more general maps studied later in the course (e.g., expanding maps, horseshoe maps, toral automorphisms, etc.). A second theme is to understand general features shared by different systems. This leads naturally to their classification, up to conjugacy. An important invariant is entropy, which also serves to quantify the complexity of the system.
Lecturer: John Smillie
|
The module draws together many themes touched upon briefly in other modules such as Analysis III or Applied Analysis, and - for those interested in pursuing analysis further - would be a natural follow-up to the more abstract 'Measure Theory' module in Term 1.
Lecturer: José Luis Rodrigo
|
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.
Lecturer: Hugo van den Berg
|
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 and partner.
ORGANIZER: Geetha Balakrishnan
|
The aim of this module is to complete your training in the use of computers by exploring the use of super-computers to solve super problems. You will learn how to write scalable, portable programs for parallel computer systems and explore how large-scale physics problems are tackled. This module is 100% continuously assessed (there is no examination).
ORGANISER: David Quigley
|
Characterising and, where possible, controlling the structure of materials is one the most important areas of research in science. The microscopic structure of a material has important effects on its mechanical strength, its electrical properties and, at surfaces, the way the material interacts with the outside world (for example as a catalyst, in electrical contacts or as it corrodes). In this module, we will look at the methods for identifying and studying structure in both crystalline and amorphous materials and at surfaces. A large armoury of new techniques have been developed in the last 20 years to study the properties of solids using incident electrons, ions, neutrons and photons (infra-red to X-rays) and to detect a similar set of emergent particles.
LECTURERS: Tom Hase and Chris McConville
|
Einstein's general theory of relativity is the basis for our understanding of black-holes and the Universe on its largest scales. In general relativity the Newtonian concept of a gravitational force is abolished, to be replaced by a new notion, that of the curvature of space-time. This leads in turn to predictions of phenomena such as the bending of light and gravitational time dilation that are well tested, and others, such as gravitational waves, which are only now perhaps coming into the regime of direct detection.
The course will start with a recap of Special Relativity, emphasizing its geometrical significance. The formalism of curved coordinate systems will then be developed. Einstein's equivalence principle will be used to link the two to arrive at the field equations of GR. The remainder of the course will then be spent on the application of general relativity to stellar collapse, neutron stars and black-holes, gravitational waves, including their detection, and finally to cosmology where the origin of the "cosmological constant" -- nowadays called "dark energy" -- will become apparent.
LECTURER: Gareth Alexander
|
The detection of planets orbiting stars other than the sun is technically challenging and it was not achieved until 1995. This module looks at how exoplanets are now being discovered in large numbers and how these discoveries are challenging existing theories of planet formation and evolution. Various methods of planet detection are considered, as well as methods used to determine physical properties such as temperature, density and composition. We explore likely physical explanations for the observed properties and identify questions that remain open in this active research field. Finally, we consider the prospects for detecting life on distant planets.
LECTURER: Peter Wheatley
|
This module presents the theoretical framework that underpins the Standard Model of particle physics, and uses it to make calculations of basic fundamental particle interactions.
LECTURER: Paul Harrison
|
Our knowledge of what is happening in the sun is increasing rapidly, largely as a result of space-based instrumentation. The challenge now is to understand it. The basic process is simple: Heat moves outwards from its source at the centre (nuclear fusion). However, on its way out, this energy drives many processes on many different length scales many of which are not at all well understood. For example, there is still no convincing theory of how the sun's magnetic field is generated and how the atmosphere is heated.
In this module, we will state the basic properties of the sun as deduced from observation and general physical principles, and we will introduce a hydrodynamic model of the sun. This treats the solar matter as a fluid. There are the usual gravitational and pressure gradient forces governing the fluid motion but, because the constituent particles of the fluid are charged, there are also electromagnetic forces. As a result, we need to worry about Maxwell's equations as well as Newton's laws. We will then discuss applications of this theory, called magnetohydrodynamics, to model and understand phenomena like sunspots, coronal loops, prominences, solar flares, coronal mass ejections and space weather.
LECTURER: Valery Nakariakov
Term 2
The project continues from Term 1.
|
Quantum mechanics is one of the most successful and most fundamental scientific theories. It is fundamental to the understanding of a variety of physical phenomena, ranging from atomic spectra and chemical reactions to superfluidity and Bose-Einstein condensation. In this lecture we will start with the fundamental principles of quantum theory: This includes the concept of a wave function in Hilbert space, the stationary and time-dependent Schrödinger equations, the uncertainty principle and the connections with classical mechanics (Ehrenfest theorem).
We will give simple, exactly soluble examples of both time-dependent and time-independent Schrödinger equations. We will then turn to a special topic: we study cases where probability theory can be used to analyze quantum mechanical systems. While in general, the language of quantum mechanics is mainly functional analysis and spectral theory, there are important cases where the use of probabilistic methods gives surprising insights and allows for elegant proofs. The most prominent examples are Feynman-Kac path integrals, which link quantum mechanics to the theory of Brownian motion and diffusion processes. We will give a proof of this formula, and use it to study ground states of quantum systems: we will give upper and lower bounds on the position density of a bound particle, and obtain powerful regularity results for solutions of the stationary Schrödinger equation. Another application of probability theory occurs in the study of Bose-Einstein condensation.
Again the starting point is the Feynman-Kac formula, but this time a model of interacting random permutations emerges. Bose-Einstein condensation then corresponds to the occurrence of infinitely long cycles in the random permutaitons. This is a very recent research topic, and while some aspects are clear, others are less so. We will give an overview over what is known and what is open, and go into details according to time available.
Lecturer: Vassili Gelfreich
|
Neuroscience is a highly active multidisciplinary field of research in which analytical methods play an important role. This course introduces some of the basic concepts in cellular neuroscience and provides an overview of the fundamental approaches used in the modelling of electrical activity in the nervous system. The topics covered span the spatial and temporal scale from rapid signalling at synapses to emergent dynamic states of networks of coupled neurons, specifically: propagation of action potentials in axonal fibres; synaptic transduction including the statistics of vesicle release, short- term synaptic dynamics and spike-time-dependent learning; the subthreshold neuronal response to excitatory and inhibitory synaptic drive, dendritic filtering and cable theory, the role of voltage-gated channels and the firing properties of neurons; and finally, the dynamic states of networks of coupled neurons such as persistent memory and oscillations.
Lecturer: Magnus Richardson.
|
Topics will include several of the following themes:
• Linear and nonlinear waves in fluids and other continuous media, such as plasmas, MHD fluids, Bose-Einstein condensates, superfluid helium, nonlinear optics crystals. Waves in inhomogeneous or/and moving media, scale separation, WKB and ray tracing approach, Born approximation for wave scattering on inhomogeneities and vortices. Hamiltonian and Lagrangian formulations for nonlinear waves. Solitons. Waves in excitable media, eg. spiral waves in cardiac tissue.
• Classical turbulence theory. Richarson cascade and Kolmogorov spectrum. Single and dual cascade systems. Structure functions and intermittency. Scalings in stationary and in evolving turbulence. Near-wall turbulence. Pipe turbulence. Rapid distortion theory.
• Quantum turbulence. Polarised and unpolarised tangles of quantized vortex lines. Biot-Savart-Rios description. Vortex line reconnections. Kelvin waves on vortex lines. Classical-quantum crossover scales. Sound emission by moving vortices.
• Turbulence in Bose-Einstein condensates. Gross-Pitaevskii equation model. Dark solitons and quantized vortices. Inverse cascade and condensation phenomenon. Wave turbulence description. Bogoliubov transformation. Berezinskii_Kousterlitz-Thouless and Kibble-Zurek phase transitions.
• Astrophysical and plasma turbulence. Alfen waves and drift waves. Wave turbulence approach to weak MHD and drift turbulence. Strong turbulence and critical balance.
• Large-scale waves and vortices in atmosphere and oceans. Quasi-geostrophic model. Planetary Rossby waves. Anisotropic cascades. Generation of zonal jets. Transport barriers. Two-layer model. Interaction of barotropic and baroclinic modes.
Lecturer: Sergei Nazarenko
|
Many problems in Mathematics lead to linear problems on infinite-dimensional spaces. In this course we shall mainly study infinite-dimensional normed linear spaces and continuous linear transformations between such spaces. We will study Banach spaces and prove the main theorems of this subject (Hahn-Banach, open mapping, uniform boundedness). The last part of the course will be devoted to bounded and unbounded operators (differential and integral operators in L2 spaces) with some of their applications.
Lecturer: James Robinson
|
Why bother with relativistic quantum mechanics? Well, there are many experimental phenomena which cannot be explained in the non-relativistic domain. There are also theoretical reasons why one would expect new effects at relativistic velocities. Finally, it would be profoundly unsatisfactory, both intellectually and aesthetically, if relativity and quantum mechanics could not be united.
The module will set up the relativistic analogues of the Schrodinger equation and analyse their consequences. We will find that constructing the equations is not trivial - knowing the form of the ordinary Schrödinger equations turns out not to be much help. We will show that the correct equation for the electron, due to Dirac, predicts antiparticles, spin and other surprising phenomena. One is the 'Klein Paradox': When a beam of particles is incident on a high potential barrier, more particles can be 'reflected' than are actually incident on the barrier.
LECTURER: Tim Gershon
|
There is currently considerable research activity in the area of high energy cosmic rays. This module will be concerned with the experimental observation of high energy cosmic rays and how they can be understood in terms of the acceleration of particles to very high energies in plasmas.
LECTURER: Richard Dendy
|
This module concentrates on electronic properties of materials. The module splits roughly into three parts: (i) methods used to produce and characterise materials, (ii) semiconductors and (iii) magnetism and superconductivity. We will explain in particular how electrical properties, such as conductivity and optical absorption, can often be tailored in some materials by varying the composition as a function of position.
The physics of semiconductors is dominated by the existence of a gap in allowed energies for electrons. The Fermi energy lies in or close to this gap, which is the result of the interaction of the electrons with the background lattice of positively charged ions. The module will look at how the dispersion (energy vs k) of the quantum states close to this gap, the statistics of their occupation and the scattering of electrons between these states by lattice vibrations, determine most of the properties we see in optical and transport measurements.
Magnetism and superconductivity are strongly connected as both are the result of interactions between electrons. We will look at how to identify magnetism in materials and the interplay between magnetism and superconductivity.
LECTURER: Marin Alexe and Don Paul
|
Recent observations are beginning to reach back into the Cosmic Dawn - the era when the first stars and galaxies formed. The physical conditions at the time of their formation set the properties of these first objects, and their evolution in turn sets the properties of the stars, galaxies and planets that follow. This module will investigate the formation of structure in the early Universe, starting from the Cosmic Microwave Background and moving through the first generations of stars, and onto the large scale structures that we observe.
The module will discuss the theory behind the formation of the first stars and galaxies from primordial density perturbations, the build-up of mass through hierarchical structure formation and the importance of feedback in shaping galaxies. It will also highlight the state of the art observations that are now being conducted to directly observe distant structures forming when the Universe was less than 5% of its current age, outline the insight that arises from them, and discuss how new observations with missions set for launch in the next few years might answer the remaining, central questions in the field.
LECTURER: Andrew Levan
|
As you may know, only the effects of phase differences in a wavefunction have measurable consequences, while the phase of a wavefunction itself is not measurable. Multiplying a wavefunction by a phase factor makes no difference to any measurable quantity. In fact, provided that we introduce what is called a gauge field to 'keep track' of the phases involved, no physical quantity changes even if the wavefunction is multiplied by a time- and space-dependent phase factor. This property is called a local gauge symmetry. What does matter is when phase differences in a wavefunction for a particle moving between different positions depend on the path the particle takes between the two positions. This is what happens in the presence of a magnetic field and (in the time-dependent case) in the presence of an electromagnetic (em) field. The em field turns out to be the gauge field for this problem.
In gauge theories for particle physics, the corresponding thing to multiplication by a phase factor in the quantum mechanics of the electron, is multiplication by a unitary matrix which mixes different components of a vector-valued wavefunction. As we will see, this simple generalization of the theory of an electron in an electromagnetic field is the basis for all elementary particle physics. We will start with the theory of the electron in the electromagnetic field making the gauge symmetry explicit. We will then discuss the gauge symmetries appropriate for the various theories and approximate theories used to describe other elementary particles and their interactions with their corresponding gauge fields.
LECTURER: Tim Gershon
|
Neutrinos are very interesting particles. Originally they fitted into the standard model quite neatly. There are three flavours associated with the electron, muon and tau and all were supposed to have zero mass. However, observations of flavour oscillations (muon neutrinos turning into electron neutrinos for example) meant that flavour eigenstates and mass eigenstates couldn't be the same and that the idea of massless neutrinos was a non-starter. Although this meant that yet more parameters (some angles and some masses) had to be introduced into the standard model, it also provided a possible explanation of the matter/antimatter asymmetry in the universe. In this module we will look at the observation of neutrinos (they are very hard to detect as they interact only very weakly with other matter), the discovery of the flavour oscillations and how their properties (as currently known) can be accommodated within the framework of the standard model.
LECTURER: Steve Boyd
|
This module discusses the physics of thermonuclear fusion, which is a candidate solution for the energy demands of our society. Nuclear fusion is promising due to the unlimited amount of fuel, the fact that it is CO2 neutral, the limited amount of long lived radioactive waste, and the inherent safety of the approach. As a 'minor' drawback, one could mention that a working concept for this approach still needs to be demonstrated. For reasons we will discuss, the construction of a working fusion reactor is hindered by several, in themselves rather interesting, physics phenomena.
The module discusses the two main approaches: inertial confinement and magnetic confinement, with the emphasis on the latter since it is further developed. The module will deal with both the physics phenomena as well as with the boundary conditions that must be satisfied for a working reactor. At the end of the module you should have an understanding of the main physics effects, the current concepts used and the reasons behind the choices made in the current experimental designs.
LECTURER: Ben MacMillan
[ Top of Document ]