Lower Bound Frontiers in Arithmetical Circuit Complexity

Acknowledgement I would like to thank my advisor Kenneth Regan for introducing me to the study of computational complexity. Thank you for your guidance and encouragement. The last six years have been great working together. Thank you also for teaching me how to teach. I would like to thank Alan Selman and Xin He for being on the Ph.D. committee and Martin Lotz for his work as the outside reader. Thank you for your help in improving this work. Prof. Selman I must also thank for his engaging seminars that I had the pleasure of attending, and for all those fun times being his teaching assistant for the theory of computation course. Thanks also to fellow theory students Charles Xavier Dennis, Samik Sengupta and Pavan Aduri for creating an enjoyable atmosphere. At the current time they have all vanished from UB, but they were here during most of my stay. I will be remembering those glorious days of taking U-turns in New Jersey! Finally, I would like to thank my wife, for her support, and, together with my son Rohan, for putting up with all of this. v Even within the limits of the possible, the possibilities are limitless.