No summary of my thesis would be complete without acknowledging the many people that have helped me along the way. My thesis is dedicated to my father, mother, and two brothers, who all helped make my childhood a wonderful and superbly educational one.
As for the other acknowledgments, below I have copied the acknowledgments page, since I feel people may be more likely to read it here than in the 94-page thesis document itself. (I can also provide links to some of their awesome web pages!)
I could not have completed this work without the patient and insightful guidance of my advisor, Mark Haiman. For the last five years, one Yali’s Cafe session at a time, he has taught me the secrets of the world of symmetric functions, Macdonald polynomials, diagonal harmonics, Hilbert schemes, and countless other topics. I am very grateful for his eagerness to talk about mathematics at any time, for his frank assessment of the significance of my results, and for giving me the freedom to choose my own directions in my research while also providing guidance along the way.
There are many other people who I am indebted to as well. I thank Sami Assaf and Angela Hicks for their generous insights on parking functions, quasisymmetric functions, and Macdonald positivity, and more generally for our many collaborative discussions. Sami and Angela have become like mathematical older sisters to me, and I am grateful for their advice and inspiration.
I thank Steven Sam for co-organizing a seminar on Macdonald polynomials with me in my second year of graduate school, without which I would have had a harder time learning the material. I also thank Monica Vazirani for being involved with that seminar, and more generally for her help throughout my graduate studies.
I thank Anne Schilling for suggesting potential new connections of this work to representation theory and crystal base theory, and I look forward to working with her on these connections in the future.
I also thank (in no particular order) Adriano Garsia, Jonah Blasiak, Jim Haglund, Andy Wilson, Sara Billey, and Austin Roberts for insightful discussions pertaining directly to this work. Thanks also to Federico Castillo, Steven Karp, Olya Mandelshtam, Anastasia Chavez, Lauren Williams, Thomas Lam, Jake Levinson, David Harden, and Allen Knutson for in-depth mathematical discussions on related topics throughout my studies at Berkeley. Thanks to Mark Haiman, Lauren Williams, and Christos Papadimitriou for serving on my dissertation committee.
Finally, I thank my husband, Bryan Gillespie, for being by my side for the last wonderful four years, and providing stability through the ups and downs of mathematical research.