Modulo Errors

During the summer of 2005 I carried out research into the geometry of the triangle, exploring consequences of a result on statics and barycentric coordinate systems first demonstrated by my supervisor Geoff Smith. This project was financed by a Nuffield Undergraduate Research Bursary, specifically URB/02196/G. More on that scheme can be found at The Nuffield Foundation site.

The main results of this work are contained in my report, Applications of the GPAT to Triangle Geometry, which you can access in [ps] or [pdf] formats according to preference.

The latter part of that work refers to conjectures based on experimentation with the Cinderella geometry package; from this a java applet has been generated which can be accessed here.

Additional applets demonstrating other geometric results are also available on: The Euler Line, The Brocard Points, The Fermat and Nagel points.

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Discussion of (affine) Varieties, ideals and examples of how they relate. Statement and proof of the Nullstellensatz.

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Integer and polynomial GCD calculations: useful properties, Euclidean techniques,Resultants and the Sylvester matrix, non-Euclidean modular GCD by large/many small primes.

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An algorithm for determining a Gröbner basis for a collection of polynomials.

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A Gröbner basis for a system of polynomials preserves the common roots whilst being simpler relative to an ordering.

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Discussion of canonical forms for polynomials- single variable dense and sparse representations; orderings for multivariate polynomials.

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First introduction to group theory via the properties of the general linear group.

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Definition of the Cholesky factorisation, technique for deriving, problem cases and a worked example.

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Solving linear equations via LU factorisation- definition, solution method, derivation of LU factorisation (with worked example) and considerations of complexity.

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Definition and properties of the adjoint. Proof of uniqueness. Notational issues.