Acknowledgments

First of all, I would like to express my gratitude to my PhD advisor, Prof. Stephen Shipman for inspiring me, guiding me and caring for me throughout all these years. I would like to thank all my colleagues and friends at the Department of Math-ematics for creating an exciting and fun environment in which to study. I am especially grateful to all the present and former fellow graduate students Susanna Dann, Cristina Tugurlan, Maiia Bakhova, Silvia Jimenez, and Megumi Onoda, with whom I share much of my spare time. I would also like to thank Dr. Robert Perlis for the financial support through the Louisiana Education Quality Support Fund (LEQSF), which allowed me t