Computational aspects of singularity theory.
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unpublished and we spend some time reviewing these results, extending them to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.