# 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

## 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,

## 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

## 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

## 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

## 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

## 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

## 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

## 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

## 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

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