I am willing to supervise any of the following projects.
(In brackets at the end are the initials of the probable second examiner.)
Students should see me first and obtain my agreement before embarking on the project.
Vortices are points in a fluid where the fluid is spinning (like an idealized whirlpool).
In an ideal fluid, one can study how the vortices interact without any reference to the
background fluid, and over the past 100 or so years there has been much interest in
studying this system, which is in fact a Hamiltonian system. In this project the student will
discover why it is Hamiltonian, and how to use this fact to investigate the many symmetric
configurations of vortices that can occur
(see the applet on my website for examples).
Prerequisite: Classical Mechanics MATH20512 [AH]
The dynamics of rigid bodies has a long history, and is still important today in, for
example, satellite dynamics. This project would begin by looking at the free rigid
body (Euler's equations) and proceed by studying motions in various non-free settings
(such as gravitational fields).
Prerequisite: Classical Mechanics MATH20512 [MM]
The so-called wallpaper groups are the symmetry groups of 2-dimensional patterns.
The aim of this project would be firstly to understand and explain (prove) why there
are just 17 different wallpaper patterns, and secondly to be able to recognize effectively
which symmetry group corresponds to a given pattern.
Prerequisites: A good familiarity with Algebraic Structures 1 MATH20202 (and MATH20212 preferred) [PJR]
Symmetry is the raison-d'etre of groups and group actions.
There is a well-known algebraic structure associated to a finite group called the
Burnside Ring which encodes the different ways a group can act on sets.
This project would investigate this ring, and some of its applications
(and perhaps venture where previous applications haven't gone).
Prerequisites: A good familiarity with Algebraic Structures 1 and 2: MATH20202 and MATH20212 [MYP]
This project would begin with the basics of group actions: orbits, istropy, etc and proceed probably
with the local geometry of group actions (tubular neighbourhood theorem). There are then many directions
the project could evolve in, and this would depend on the background and interests of the student. Examples might be:
invariants and spherical harmonics, symplectic/Poisson actions, equivariant transversality, structure of compact Lie groups, ...
Pre(co)requisite: Calculus on manifolds MATH31431 (or MATH41431). [HK]
What is geometry? The Erlangen programme, introduced by Klein in 19th century, views geometry in
terms of symmetry groups. In fact the idea is not so strange: for example two triangles are congruent whenever one can be
translated, rotated or reflected to make the other, and these translations, rotations and reflections are precisely the symmetries
of Euclidean geometry. The purpose of this project is to investigate this link between geometry and symmetry.
and apply it in a standard example or two (such as spherical geometry) and in a less standard example such as affine geometry.
Prerequisites: MATH20202 and MATH20212 [HK]
If students have other suggestions along similar lines, I would be happy to discuss them.