Module Outlines
Here are brief outlines of all the modules that may be taken by current 2nd year Mathematics and Physics students.
Term 1
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The first part of the module provides an introduction to vector calculus which is an essential toolkit for differential geometry and for mathematical modelling. After a brief review of line and surface integrals, div, grad and curl are introduced and followed by the two main results, namely, Gauss's Divergence Theorem and Stokes' Theorem. These theorems will be proved only in simple cases; complete proofs are best deferred until one has learned about manifolds and differential forms. The usefulness of these results in applications to flow problems and to the representation of vector fields with special properties by means of potentials will be emphasized. This leads to Laplace's and Poisson's equations which will be discussed briefly. The solution of these equations are discussed more fully in modules on partial differential equations. Cartesian coordinates are in many cases not well suited to a particular problem: for example, polar coordinates yield simpler equations for the flow of water in a cylindrical pipe. We will show how to represent div, grad and curl in general curvilinear coordinates, paying particular attention to spherical and cylindrical geometries.
The second part of the module introduces the rudiments of complex analysis leading up to the calculus of residues. The link with the first part of the module is achieved by considering a complex valued function of one complex variable as a vector field in the plane. This idea is particularly useful in the study of two-dimensional fluid flow. Complex differentiability leads to the Cauchy-Riemann equations which are interpreted as conditions for the vector field to have both zero divergence and zero curl. Cauchy's theorem for complex differentiable functions is then established by means of the main integral theorems of vector calculus. Cauchy's integral formula which expresses the value of a complex differentiable function at a point as a line integral of the function on a contour surrounding the point is the key result from which the stunning properties of complex differentiable functions follow. In some sense, Cauchy's integral formula is the mathematical formulation of what physicists call action at a distance.
Lecturer: Mario Micallef
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This covers two topics: (1) integration, (2) sequences and series of functions.
The idea behind integration is to compute the area under a curve. The fundamental theorem of calculus gives the precise relation between integration and differentiation. However, integration involves taking a limit, and the deeper properties of integration require a precise and careful analysis of this limiting process. This module proves that every continuous function can be integrated, and proves the fundamental theorem of calculus. It also discusses how integration can be applied to define some of the basic functions of analysis and to establish their fundamental properties.
Many functions can be written as limits of sequences of simpler functions (or as sums of series): thus a power series is a limit of polynomials, and a Fourier series is the sum of a trigonometric series with coefficients given by certain integrals. The second part of the module develops methods for deciding when a function defined as the limit of a sequence of other functions is continuous, differentiable, integrable, and for differentiating and integrating this limit. Norms are used at several stages and finally applied to show that a Differential Equation has a solution.
Lecturer: Sergey Nazarenko
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This module is a continuation of First Year Linear Algebra. In that course we studied conditions under which a matrix is similar to a diagonal matrix, but we did not develop methods for testing whether two general matrices are similar. Our first aim is to fill this gap for matrices over C. Not all matrices are similar to a diagonal matrix, but they are all similar to one in Jordan canonical form; that is, to a matrix which is almost diagonal, but may have some entries equal to 1 on the superdiagonal.
We next study quadratic forms. A quadratic form is a homogeneous quadratic expression aij xixj in several variables. Quadratic forms occur in geometry as the equation of a quadratic cone, or as the leading term of the equation of a plane conic or a quadric hypersurface. By a change of coordinates, we can always write q(x) in the diagonal form. For a quadratic form over R, the number of positive or negative diagonal coefficients ai is an invariant of the quadratic form which is very important in applications.
Finally, we study matrices over the integers {Z}, and investigate what happens when we restrict methods of linear algebra, such as elementary row and column operations, to operations over {Z}. This leads, perhaps unexpectedly, to a complete classification of finitely generated abelian groups.
Lecturer: Daan Krammer
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Geometry is the attempt to understand and describe the world around us and all that is in it; it is the central activity in many branches of math and physics, and offers a whole range of views on the nature and meaning of the universe.
Klein's Erlangen program describes geometry as the study of properties invariant under a group of transformations. Affine and projective geometries consider properties such as collinearity of points, and the typical group is the full n x n matrix group. Metric geometries, such as Euclidean geometry and hyperbolic geometry (the non-Euclidean geometry of Gauss, Lobachevsky and Bolyai) include the property of distance between two points, and the typical group is the group of rigid motions (isometries or congruences) of 3-space. The study of the group of motions throws light on the chosen model of the world.
The module includes a diversity of topics, such as the rules of life and self-consistency of the non-Euclidean world, symmetries of bodies both Euclidean and otherwise, tilings of Escher and the regular solids, and the geometric rules of perspective in photography and art.
Lecturer: Saul Schleimer
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The first year module PX101 Quantum Phenomena discussed some of the phenomena that led to quantum theory and showed that most of these are a consequence of wave-particle duality. The first part of this year's module uses the simple ideas introduced in PX101 to explore atomic structure. The module goes on to cover the mathematical tools needed in quantum mechanics and outlines the fundamental postulates that form the basis of the theory. The module will discuss the time-independent and the time-dependent Schrödinger equations for spherically symmetric and harmonic potentials, angular momentum and hydrogenic atoms.
The second half of the module looks at many-particle systems and aspects of the standard model of particle physics. The module will introduce the quantum mechanics of free fermions and discuss how it accounts for the conductivity and heat capacity of metals, the state of electrons in white dwarf stars and that of neutrons in neutron stars. Introducing the lattice and the scattering of electrons off ions then allows us to describe the properties of semiconductors and insulators. The standard model of particle physics is a quantum field theory and beyond simple quantum mechanics. However, using ideas you have already met, we will be able to discuss a number of aspects of the standard model such as antiparticles and particle oscillations.
LECTURERS: Gavin Bell (term 1) and Nicholas d'Ambrumenil (term 2)
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This module introduces the Hamiltonian formulation of classical mechanics. This elegant theory has provided the natural framework for several important developments in theoretical physics including quantum mechanics.
The module will start by covering the general "spirit" of the theory and then go on to introduce the details. The module will use a lot of examples. Many of these will be familiar from earlier studies of mechanics while others, which would be much harder to deal with using traditional techniques, can be dealt with easily using the language and methods of Hamiltonian mechanics.
LECTURER: James Lloyd-Hughes
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The aim of this introductory module is to present an understanding of the behaviour of the solid Earth in terms of simple physical principles. The topics which will be covered to some extent include: the age of the Earth, plate tectonics, seismology, gravity and the shape of the Earth, oceanic and continental heat the Earth's core and magnetic field.
LECTURER: Nicholas Hine
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Power generation is a very topical issue. This module introduces this topic from a physicist's perspective. It will consider the conventional (coal/oil/gas) generation process in detail, before moving on to look at fission reactors. Various forms of renewable power will then be explored and the final part of the module will look at the potential of fusion.
LECTURER: Vasily Kantsler
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You do not need to be told that global warming (climate change) is a serious issue. This module will look at the challenges posed by the measurement and attempted prediction of climate change and the economics and politics associated with any actions (or inaction).
LECTURERS: Michael Pounds and David Mond
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This module will develop your facility with the Python programming language which you obtained in PX150 Physics Programming Workshop.
LECTURER: Michal Kreps
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The theory of partial differential equations (PDE) is important both in pure and applied mathematics. On the one hand they are used to mathematically formulate many phenomena from the natural sciences (electromagnetism, Maxwell's equations) or social sciences (financial markets, Black-Scholes model). On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics (geometry, Poincare-conjecture).
Subject of the module are four significant partial differential equations (PDEs) which feature as basic components in many applications: The transport equation, the wave equation, the heat equation, and the Laplace equation. We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.
Lecturers: Hugo Van den Berg (Term 1) and Florian Theil (Term 2)
Term 2
The PDEs module continues from term 1.
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Roughly speaking, a metric space is any set provided with a sensible notion of 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. 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. For example, a continuous real-valued function on a compact metric space must be bounded. And such a function on a connected metric space cannot take both positive and negative values without also taking the value zero. Continuity is readily described in terms of open subsets, which leads us naturally to study the above concepts also in the more general context of a topological space, where, instead of a distance, it is declared which subsets are open.
Lecturer: Ian Melbourne
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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
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This is an introductory abstract algebra module. As the title suggests, the two main objects of study are groups and rings. You already know that a group is a set with one binary operation. Examples include groups of permutations and groups of non-singular matrices. Rings are sets with two binary operations, addition and multiplication. The most notable example is the set of integers with addition and multiplication, but you will also be familiar already with rings of polynomials. We will develop the theories of groups and rings.
Some of the results proved in MA242 Algebra I: Advanced Linear Algebra for abelian groups are true for groups in general. These include Lagrange's Theorem, which says that the order of a subgroup of a finite group divides the order of the group. We defined quotient groups G/H for abelian groups in Algebra I, but for general groups these can only be defined for certain special types of subgroups H of G, known as normal subgroups. We can then prove the isomorphism theorems for groups in general. An analogous situation occurs in rings. For certain substructures I of rings R, known as ideals, we can define the quotient ring R/I, and again we get corresponding isomorphism theorems.
Other results to be discussed include the Orbit-Stabiliser Theorem for groups acting as permutations of finite sets, the Chinese Remainder Theorem, and Gauss' theorem on unique factorisation in polynomial rings.
Lecturer: Inna Capdeboscq
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The module will introduce a few model systems to motivate the questions and techniques, which will reappear throughout the module. We will apply the new techniques as we learn them. Examples we will look at include: the Lotka-Volterra equation, which is a simple model of a system of predators and prey, the Duffing equation, which is the usual model of damped oscillators with an additional non-linear term, the Lorenz equations, which were introduced to illustrate convection instabilities in the atmosphere, the Hodgkin-Huxley model of signal propagation in neurons and the Fitzhugh-Nagumo model of neuron excitations. We will also look at general Hamiltonian systems (these are classical mechanical systems when there is no dissipation) and gradient flows.
Lecturer: Nicholas Simm
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Systems biology is the study of systems of biological components, which may be molecules, cells, organisms or entire species. Its principal concern is how the components and, in particular, the links between them determine the behaviour of the system as a function of time and various control parameters like temperature. Our mathematical models are usually systems of ordinary as well as partial differential equations that have the biology and the neuroscience built in. We then solve these equations (often using numerics and approximations) and compare with experimental observation.
In this module we will illustrate the spirit of this evolving discipline. We will look at: elementary models of neurons, modelling the cell cycle and biological clocks, biological oscillations, bioinformatics (how information is stored, processed and transmitted) and some PDEs used to model the spatial and temporal variation observed in biological systems (we will look in particular at reaction-diffusion equations and the general area of pattern formation). We will, of course, have to cover some material from biology. However, you will not be expected to carry out laboratory work!
Lecturer: David Rand and others from SysBio
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This module develops the ideas of first year electricity and magnetism into Maxwell's theory of electromagnetism. Maxwell's equations pulled the various laws of electricity and magnetism (Faraday's law, Ampere's law, Lenz's law, Gauss's law) into one unified and elegant theory. Establishing a complete theory of electromagnetism has proved to be one the greatest achievements of physics. It was the principal motivation for Einstein to develop special relativity, it has served as the model for subsequent theories of the forces of nature and it has been the basis for all of electronics (radios, telephones, computers, the lot...).
The module will also show that Maxwell's equations in free space have time-dependent solutions, which turn out to be the familiar electromagnetic waves (light, radio waves, X-rays etc), and will study their behaviour at material boundaries (Fresnel Equations).
Finally the module will focus on the basic physics of optical instruments and light sources.
LECTURER: Yorck Ramachers
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The field of fluids is one of the richest and most easily appreciated in physics. Tidal waves, cloud formation and the weather generally are some of the more spectacular phenomena encountered in fluids. We will establish the basic equations of motion for a fluid - the Navier-Stokes equations - and show that in many cases they can yield simple and intuitively appealing explanations of fluid flows. We will concentrate on incompressible fluids.
LECTURER: Julie Staunton
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People have been studying stars for as long as anything else in science. Yet the subject is advancing faster now than almost every other branch of physics. With the arrival of space-based instruments, the prospects are that the field will continue to advance and that some of the most exciting discoveries reported in physics during your lifetime will be in astrophysics.
The module deals with the physics of the observation of stars and with the understanding of their behaviour and properties that the observations lead to. The module will cover the main classifications of stars by size, age and distance from the earth and the relationships between them. We will also look at how the observations of stars' behaviour allows us to study the evolutionary history of galaxies and of the universe as a whole.
LECTURER: Don Pollacco
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Particle physics experiments are designed to study sub-atomic particles and to test their behaviour against the predictions of the Standard Model (SM). After revising the basics of the SM and Feynman diagrams, we will look at the various sources for elementary particles and the methods for detecting them. These include sources we can build (accelerators) and natural sources such as radioactive nuclei and cosmic rays. The quantities we aim to measure, or infer from measurements, are the particles' velocities, their charge, their lifetimes and their decay modes. A major part of any experiment is the extraction of these quantities from large data sets. As the data relating to the events, that we want to analyse, can be scarce and obscured by other data and noise, the module will also explain the statistical methods used in the study of such data sets.
LECTURER: Bill Murray
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The module starts with the theory of Fourier transforms and the Dirac delta function. Fourier transforms are used to represent functions on the whole real line using linear combinations of sines and cosines. Fourier transforms are a powerful tool in physics and applied mathematics. A Fourier transform will turn a linear differential equation with constant coefficients into a nice algebraic equation which is in general much easier to solve.
The module will explain why diffraction patterns in the far-field limit are the Fourier transforms of the "diffracting" object. It will then go on to look at diffraction generally. The case of a repeated pattern of motifs illustrates beautifully one of the most important theorems in the business - the convolution theorem. The diffraction pattern is simply the product of the Fourier transform of the repeated delta functions and the Fourier transform for a single copy of the motif.
The module also introduces Lagrange multipliers, co-ordinate transformations and cartesian tensors illustrating them briefly with examples of their use in various areas of physics.
Lecturer: Rudolf Roemer
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Any macroscopic object we meet contains a large number of particles, each of which moves according to the laws of mechanics (which can be classical or quantum). Yet we can usually ignore the details of this microscopic motion and use just a few average quantities such as temperature and pressure to describe and predict the behaviour of the object. Why we can do this, when we can do this and how to do it are the subject of this module.
The most important idea in the field is due to Boltzmann, who identified the connection between entropy and disorder. (It had been known that entropy had to exist if the empirical laws of thermodynamics were to be consistent with observation, but there was no microscopic definition for it.) The module will show how the whole structure of equilibrium thermodynamics follows from Boltzmann's definition of the entropy and will show how, in principle, any observable equilibrium quantity can be computed. We will see that this microscopic theory (now called statistical mechanics) provides the basis for predicting and explaining all thermodynamic properties of matter.
LECTURER: Paul Goddard
Term 3
The Thermal Physics module continues from term 2.
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This module consists of a study of the mathematical techniques of variational methods, with applications to problems in physics and geometry. Critical point theory for functionals in finite dimensions is developed and extended to variational problems. The basic problem in the calculus of variations for continuous systems is to minimise the integral (with respect to x) of f(x,y,yx) on a suitable set of differentiable functions: y:[a,b] to R. The Euler-Lagrange theory for this problem is developed and applied to dynamical systems (Hamiltonian mechanics and the least action principle), shortest time (path of light rays and Fermat's principle), shortest length and smallest area problems in geometry. The theory is extended to constrained variational problems using Lagrange multipliers.
Lecturer: Marie-Therese Wolfram
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