Computational aspects of singularity theory.

Patrick. Kirk, Neil (1993) Computational aspects of singularity theory. PhD thesis, University of Liverpool.

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Abstract

In this thesis we develop computational methods suitable for performing the symbolic calculations common to local singularity theory. For classification theory we employ the unipotent determinacy techniques of Bruce, du Plessis, Wall and complete transversal theorems of Bruce, du Plessis. The latter results are, as yet, unpublished and we spend some time reviewing these results, extending them to filtrations of the module m,,,.E (n, p) other than the standard filtration by degree. Weighted filtrations and filtrations induced by the action of a nilpotent Lie algebra are considered. A computer package called Transversal is developed. This is written in the mathematical language Maple and performs calculations such as those mentioned above and those central to unfolding theory. We discuss the package in detail and give examples of calculations performed in this thesis. Several classifications are obtained. The first is an extensive classification of map-germs (R2,0) -p (R4,0) under A-equivalence. We also consider the classification of function-germs (CP, O) -f (C, 0) under R(D)-equivalence: the restriction of R-equivalence to source coordinate changes which preserve a discriminant variety, D. We consider the cases where D is the discriminant of the A2 and A3 singularities, extending the results of Arnol'd. Several other simple singularities are discussed briefly; in particular, we consider the cases where D is the discriminant of the A4, D4, D5, D6, and Ek singularities. The geometry of the singularities (R2,0) -f (R4,0) is investigated by calculating the adjacencies and several geometrical invariants. For the given source and target dimensions, the invariants associated to the double point schemes and L-codimension of the germs are particularly significant. Finally we give an application of computer graphics to singularity theory. A program is written (in C) which calculates and draws the family of profiles of a surface rotating about a fixed axis in R3, the resulting envelope of profiles, and several other geometrical features. The program was used in recent research by Rycroft. We review some of the results and conclude with computer produced images which demonstrate certain transitions of the singularities on the envelope.

Item Type:	Thesis (PhD)
Depositing User:	Symplectic Admin
Date Deposited:	20 Oct 2023 18:30
Last Modified:	20 Oct 2023 18:30
DOI:	10.17638/03175500
Copyright Statement:	Copyright © and Moral Rights for this thesis and any accompanying data (where applicable) are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge.
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3175500