Constraints , Varieties , and Algorithms

Declaration This dissertation is submitted to University College Cork, in accordance with the requirements for the degree of Doctor of Philosophy in the Faculty of Science. The research and thesis presented in this dissertation are entirely my own work and have not been submitted to any other university or higher education institution, or for any other academic award in this university. Where use has been made of other people's work, it has been fully acknowledged and refer-enced. Acknowledgements Working on this thesis was fun but without the help of some people this work would not have been possible. I should like to thank Pat Fitzpatrick for his final and Jim Bowen for his initial supervision. I have greatly benefited from their comments and input. Also, I should like to thank Jim for getting me to Ireland and for allowing me to spend a lot of time on trying out new ideas. Without him I would not have had the opportunity to start this work. Special thanks should also go to my External Supervisors Peter Jeavons and Dave Cohen for providing excellent comments about the thesis and for making the " viva " an enjoyable event. Thanks to Pádraic, Neil, and Ruth for keeping up the morale and for interesting discussions. Finally, I should like to thank Olly for his friendship and hospitality, especially in the two months before finalising this thesis.