What you will study
Currently you can choose from seven topics for your dissertation.
Algebraic graph theory
Algebraic graph theory is a branch of mathematics that studies graphs and other models of discrete structures by a combined power of spectral methods of linear algebra (with basics treated in M208); group theory (covered in part in Further pure mathematics (M303)); and algebra over finite fields (as encountered in Further pure mathematics and Coding theory (M836)). You will need to get acquainted with appropriate mathematical tools by reading selected chapters from the book Algebraic Graph Theory by C. Godsil and G. Royle (Graduate Texts in Mathematics, Springer, 2001). About halfway through the module you will have the opportunity to choose a particular topic of your interest within algebraic graph theory which you will then develop into a dissertation.
Advances in approximation theory
This topic extends the material in Approximation theory (M832) to the study of splines and piecewise polynomials, and their possible application to the approximate solution of differential equations. You will need to master the appropriate mathematics which includes topics such as orthogonal polynomials, Green’s functions and the linearisation of non-linear differential equations. You will also need to carry out computational studies using the algebraic computing software, Maple, as introduced in Advanced mathematical methods (M833).
Aperiodic tilings and symbolic dynamics
Aperiodic tilings are of interest not only for their aesthetic appeal, but also due to their applications in mathematical crystallography, where they serve as structure models of quasicrystalline materials. In this M840 topic you will explore some concepts of symbolic dynamics, in particular substitution rules on finite alphabets, and the dynamical systems they generate. Their geometric counterparts give rise to inflation tilings. The topic involves reading original literature in the field and offers the option of constructing and exploring tilings.
Dynamical functional equations and applications
Dynamical functional equations arise in the study of critical phenomena in the sciences and in complex social systems such as financial markets. They have been used to model geophysical phenomena (such as volcanic eruptions and earthquakes), financial crashes, stress in materials leading to rupture, and critical behaviour in physical systems, particularly in solid state physics. In this M840 topic you will study the basic theory of linear dynamical functional equations and then study in detail one or two applications, reading the original literature and, if desired, conducting your own explorations theoretically and/or numerically.
History of modern geometry
This topic covers the history of geometry in the nineteenth century. It follows the history of projective geometry and the discovery of non-Euclidean geometry from the 1820s and 1830s. It concentrates on algebraic developments in projective geometry and the work on abstract axiomatic geometry. Differential geometric aspects of non-Euclidean geometry are discussed, as is their influence on Einstein. The module ends with a discussion of geometry and physics, formal geometry and geometry and truth. The module is about geometry, specifically the history of geometry, but it is not a geometry module. ;What will be discussed is the production and reception of ideas, and how this was affected by the social context. The ideas are those of mathematics and the practices those of mathematicians. All the necessary mathematics will be presented but the ideas are to be understood as a historian would treat them and a good standard of English is required
Interfacial flows and microfluidics
Many natural and technological processes involve the understanding and modelling of systems in which a viscous liquid is in contact with other phases (e.g. gas and/or solid). Examples of applications include the coating of a substrate by a liquid, transport processes in falling liquid films, fluid flow in porous media, and many problems in the fields of nano- and micro-fluidics, such as inkjet printing or lab-on-a-chip devices. In this topic you will learn the mathematical modelling of interfacial phenomena. Some problems of current interest will be considered, such as for example, the motion of thin liquid films, droplets evaporating on solid surfaces, or fluid flow in confined systems. Basic knowledge of fluid mechanics (e.g. Mathematical Methods and Fluid Mechanics (MST326)) is desirable but not necessary.
Variational methods applied to eigenvalue problems
This topic extends the theory developed in Calculus of variations and advanced calculus (M820) to deal with some types of linear partial differential equations. After establishing functionals for various types of partial differential equations, the Rayleigh-Ritz method will be extended to obtain upper bounds on the smallest frequency of the oscillations of various shaped membranes. The most important part of your dissertation will involve deriving asymptotic estimates of the number, N(λ), of eigenvalues smaller that a given (large number), λ, for a various types of boundary value problems with arbitrary shaped boundaries. A number of theorems that establish increasingly better estimates of N(λ) will be investigated.
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, working on open-ended problems and communicating mathematical ideas clearly.