# Category Archives: Pure Mathematics

## The Inverted Pendulum

Why is it that a solid pendulum can be made to stand on its end when it’s other end is made to oscillate? This project will investigate the underlying mathematics behind this very strange phenomena. You could also try modelling this in Maple or even building your own control system to do it. Supervisor: Dr

## Mondromy of Abelian Integrals

This project is related to the change in an complex integral as its defining parameters change. You will need to learn some theory about Riemann surfaces and integrals on them, but I would like this project to have more of a research nature with plenty of experiments (and pictures) in Maple to find out what

## Hilbert’s 10th Problem

How complicated can it be to find all the integer solutions to a set of polynomial equations? In fact, as complicated as you like! Given any output from a computer program (a list of all prime numbers, for example), you can find a set of equations whose only solutions occur when one of the variables

## Algebraic Curves

We know that x2+y2=1 is the equation of a circle, but what happens when the degree of the equation is greater than two? In this project we investigate some of the geometry of algebraic curves – that is, solutions of polynomial equations P(x,y)=0. In particular, when P is a cubic polynomial, the resulting curve is

## Symmetry Groups and Space

This project will look at the interplay between group theory and symmetry in different spaces. It is deliberately left open for you to explore the areas that you like best – but you will be expected to understand the mathematics behind the pretty pictures too! Supervisor: Dr Colin Christopher

## Polynomial invariants of graphs and matroids

There are some simple and amazing polynomial invariants of graphs, such as the chromatic polynomial, which are special cases of the Tutte-Grothendieck polynomial invariants of matroids. References: Harary, F 1969 Graph Theory Addison-Wesley. Peter Cameron Polynomial aspects of codes, matroids and permutation groups Welsh, DJA 1976 Matroid Theory Academic Press, London. Bollobás, B 1998 Modern

## Braid Groups

There are many intriguing invariants of knots, some only discovered recently. Some of these come from studying the related braids, which lead to extremely interesting groups, relatively simple to define but with open questions. References: Wikipedia Braid group Epple, M 1998 Orbits of Asteroids, a Braid, and the First Link Invariant Mathematical Intelligencer, volume 20,

## Aperiodic tessellations

It was discovered only fairly recently that there is a pair of tiles with which the entire plane can only be tessellated in a non-periodic way. These tiles are very simple, but have remarkably surprising and deep mathematical properties, as well as giving beautiful tessellations. References: Austin, D Penrose tiles Penrose, R 1989 Tilings and

## Cartographic Projections

Cartographers use a wide variety of projections from the sphere to the plane, which have a very rich structure studied in differential geometry. Particularly interesting examples are conformal maps, in which angles are preserved but not lengths. References: Feeman, TG 2002 Portraits of the Earth: A Mathematician Looks at Maps American Mathematical Society. McCleary, J

## Riemann surfaces

Instead of thinking of complex numbers as lying on a plane, it is much better use stereographic projection to put them on a sphere, called the Riemann sphere. The Möbius group acts on this sphere, giving a fascinating set of transformations, crucially important in many branches of mathematics. A natural development is to consider multi-valued

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