Category Archives: Pure Mathematics

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

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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

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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,

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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

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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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