Approved as to style and content by:

There are many people who deserve acknowledgement in this work. Greatest recog-nition belongs to my advisor, Professor Andrea Nahmod, for giving me the motivation and drive to come this far. She has offered me so much help, confidence, advice, and support, and I am eternally grateful to have her as a mentor. This thesis would not be possible without Viktor Grigoryan and I am greatly indebted for all of his time and hard work. In addition to mathematical assistance, he has shown me a great deal of kindness, patience, and understanding, which have been invaluable. There are several professors whose teaching and guidance have helped me over the years. First, I credit Professor Jeffrey McGowan for giving me the confidence to pursue graduate studies. Second, I would like to thank Professors Sridhar Mahadevan, Luc Rey-Bellet, and Rodolfo Torres for their time serving on my dissertation committee and their helpful comments and suggestions. I am also very grateful to Professor Rey-Bellet for his kindness as my professor, especially during my first year of graduate school. I would also like to thank Professors Tom Braden, Markos Katsoulakis, Franz Pedit, and Tom Weston for their help and encouraging words