A highly speculative hypothesis that has been around for some thirty years now is that the fundamental objects in nature are not point particles but strings. For this project you will work on the classical bosonic string and its quantization, proving the 'no-ghost theorem': that such strings are consistent only in 26 spacetime dimensions. Beyond this you can pursue various topics as you choose: scattering amplitudes, fermionic strings and superstrings, compactification on a circle or lattice, or, for a 'hot topic', the noncommutative geometry of string-ends.

Reading:
Green, Schwarz and Witten, 'Superstring Theory', vol.1, CUP 1987
Here are informal introductions to string theory by Mike Green and John Pierre.

Prerequisites:
This is very much a project for a student taking modules in theoretical physics.
054509 Advanced Quantum Mechanics; also 054508 General Relativity, 054510 Introduction to Quantum Field Theory and 059315 Electromagnetism preferred, or equivalent courses from the Physics Dept.
 
 

The fundamental symmetry group of nature is the Poincaré group, the extension of the Lorentz group (of boosts and rotations) to include spatial and time-translations. The natural next step after studying quantum mechanics is to incorporate special relativity: particle states must respect the symmetries of the Poincaré group. It is with an explanation of how this works that Weinberg, one of the architects of quantum field theory, begins his text on the subject. The first part of the project is to work through his introduction to Lie groups in physics, and his derivation of the Poincaré group, going through some problems sheets I've produced.

Central to the next step is the Coleman-Mandula theorem: that, with certain assumptions, the Poincaré group is the largest possible spacetime symmetry group for a physical theory. The symmetry of a physical theory must therefore be a combination of the Poincaré group and a so-called 'internal' symmetry group which does not generate spacetime transformations. There are various ways round the assumptions, however. One is to work in one space dimension, another is to add 'supersymmetry,' which mixes bosons with fermions. The rest of the project is to explore and describe one or more of these possibilities.

Prerequisites:
This project is a natural partner to 054509 `Advanced Quantum Mechanics', and 054510 `Introduction to Quantum Field Theory'. The latter is a co-requisite, whilst 059306, `Quantum Mechanics', is a prerequisite (or equivalent courses from the Physics Dept).

Reference:
S. Weinberg, `The Quantum Theory of Fields', vol.1, CUP 1996.
 
 

If one wishes to tile a plane, an obvious choice is to do so with triangles, squares or hexagons. Only one shape of tile is needed, and the resulting pattern is regular. In 1974 an alternative was devised by Roger Penrose. In his tiling, there are two shapes, usually either kites and darts, or thick and thin rhombuses. With some carefully chosen rules for fitting the tiles together, an arbitrarily large area can again be covered without gaps. The Penrose tiling makes an intriguing pattern to look at (see website link below), because whilst it is non-periodic -- that is, it is somewhat unpredictable and irregular -- it almost has a five-fold rotational symmetry. Come to my office and I can give you a piece of toilet paper featuring one.

The project will require you to do some research on the literature of Penrose tilings, both in the library and on the web. You will need to understand how the tilings are constructed, and how the different constructions are related, and you will need to understand and state various precise mathematical results associated with them. Depending on the level of the project, it may be appropriate to give proofs of some of these.

Prerequisites: none

Reading:
A good introduction can be found in Grünbaum and Shephard, `Tilings and Patterns', Freeman 1987, or at Dr Matrix's website -- there are many other good websites too.
 
 

Anyone who has seen Maxwell's equations will have been struck by the apparent symmetry between electric and magnetic fields - yet magnetic charges (`monopoles') and currents are never seen in nature. Why not? What would be the implications if they did exist?

This project explores some of the answers. In particular, Dirac (1931) showed that the existence of magnetic monopoles leads to a smallest possible unit of electric charge - precisely as seen in nature and carried by the electron. You will need to explain this result, and describe some of the consequences of its generalization, the Schwinger-Zwanziger quantization condition.

Many of these have only been explored in the last twenty years. This project thus enables the student with an interest in mathematical physics to explore results close to the research frontier but which are quite accessible, and do not require a higher-level background in classical and quantum field theory.

Prerequisites:
059315 Electromagnetism, 059306 Quantum Mechanics or equivalent courses in the Physics Dept

References:
Jun Song, 'Theory of Magnetic Monopoles...'
D. Olive, 'Exact Electromagnetic Duality'
Here is a (rather advanced) lecture course.
 
 

This project is for go-getting high-achievers who plan to work in the City, earn shedloads of money and then possibly drop dead of a heart attack at 40. Such dynamic self-starters do not need to be told what this project is about; they will soon find it out for themselves.

Reading:
Chicago Board Options Exchange Learning Center
P. Wilmott, 'Derivatives', Wiley
J. Hull, 'Options, Futures and Other Derivative Securities', Prentice-Hall

Prerequisites:
059209 and 059210 Mathematical Finance, 059318 Stochastic Calculus
 
 

As part of his Gaia hypothesis -- that the Earth is a robust self-regulating system capable of maintaining life despite changing external conditions -- James Lovelock came up with his 'Daisyworld' model. Imagine a planet whose ecosystem consists of only two species, black and white daisies. White daisies reflect more of the sun's rays, and so cope well with warmer temperatures; black daisies absorb more, and so prefer cooler temperatures. If the temperature increases, so will the number of white daisies -- which then reflect more of the sun's rays and the temperature falls. This leads to more black daisies, which absorb more heat and the temperature rises. The whole forms a system which maintains an optimal temperature for the daisies, even if the intensity of the sunlight varies greatly.

This is a particularly simple system, but it can easily be made to include many more variables and species -- the key idea is to lock together ecosystem and climate in a single strongly-coupled dynamical system. For the project you can explain the results of the original Lovelock and Watson paper, and then explore some of the literature and many websites (including various Daisyworld simulators), choosing material you find interesting.

Prerequisites: 059301 Dynamical Systems, and preferably 059320 Numerical Solutions of DEs

Reading:
Lovelock and Watson, 'Biological homeostasis of the global environment:
the parable of Daisyworld', Tellus 35B(1983)284-289
Some sample websites are    1   2    3
Peter Saunders' excellent Daisyworld tutorial
 
 

The Carbon-60 molecule known as the Buckyball consists of 60 carbon atoms at the vertices of a truncated icosahedron, a polyhedron formed from 20 hexagons and 12 pentagons -- just like a soccer ball. Unlike the well-known forms of carbon (graphite and diamond), buckyball carbon is yellow. The reason for this is connected to the ways in which the molecule can vibrate, which can be analysed using the 'representations' of its symmetries. For this project you will need to teach yourself about the representation theory of finite groups, and work out how it applies to some simple symmetric molecules, working up to the representations of the buckyball's symmetry group.

Prerequisites: an interest in groups and algebra.

Reading:
S. Sternberg, 'Group Theory and Physics', CUP 1994; S 2.86 STE
Chung and Sternberg, 'Mathematics and the Buckyball' and many other websites.
 
 

Occasionally, mariners encounter 'freak' or 'rogue' waves -- lone waves so much bigger, steeper and generally nastier than all the others that they do not fall into the expected statistical distribution based on superpositions of typical  waves. A partial explanation is given by the focusing of waves by currents, but a more subtle possible explanation is that, when waves are steep, the usual wave equation, correct for flat(tish) waves, no longer applies, and is replaced by a nonlinear equation with 'soliton' solutions which describe the freak waves. The project is to describe the mechanism of this, review what is currently known, and possibly, by doing numerical simulations or otherwise, make some progress.

Reading
This is an excellent introductory website, while this lecture explains the central mathematical point.
Here's a recent conference on the subject, although the links are only to abstracts, not full papers.

Prerequisites
059408 Topics in Mathematical Physics, 054526 Solitons
 
 

The making of codes became highly mathematical during the twentieth century, and cryptanalysis -- the cracking of codes -- followed suit. For this project you will explain the making and breaking of some of the most important. The most obvious example is the German 'Enigma' code, whose breaking by Polish and English mathematicians helped the Allies win the Second World War. Another is the 'public-key cryptography' of the RSA system, which uses simple ideas from number theory. This hasn't been broken yet, but there is some history of attempts to do so, which you could explore -- for example, for students who have done some quantum mechanics, you could explain how a quantum computer might be able to factorize large numbers and thereby break RSA.

Reading
There are many websites devoted to these topics.
A good general introduction can be found in Simon Singh, 'The Code Book'

Prerequisites
059017 Intro to Number Theory may be useful for RSA.