Problems with a lot of Potential: Energy Optimization on Compact Spaces A DISSERTATION SUBMITTED TO THE FACULTY OF THE UNIVERSITY OF MINNESOTA BY Ryan William Matzke IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Dmitriy Bilyk May 2021 © Ryan William Matzke 2021 ALL RIGHTS RESERVED Acknowledgments I do not think that I can ever completely express my gratitude towards my adviser, Professor Dmitriy Bilyk. When I was applying for the NSF Graduate Research Fellowship in my first year of graduate school, with a project in Additive Combinatorics, a subject no one at the University of Minnesota particularly studies, Dmitriy was the only one who was willing to advise the project. When I asked him what he did for research, he offered for me, someone he barely knew, to join his current research, letting me learn about Discrepancy Theory and Energy Optimization by working on new problems, and getting me interested in and excited about the two subjects. He helped me work through the culture shock I had as I switched from undergraduate math courses and research to the much harder work of being a professional mathematician and the impostor syndrome I dealt with as I faced, for the first time in my life, academic challenges I wasn’t sure I could overcome. I am deeply grateful for Dmitriy’s consistent, unyielding support, his thoughtful feedback and advice, and the independence he encouraged, all of which has helped me grow into the mathematician I couldn’t imagine being in my first year. I also appreciate him as a person, bringing life with him wherever he goes, caring about others, and making research such a fun and exciting experience. So, to my mentor, collaborator, role model, and friend, Dmitriy Bilyk, thank you so much. You’ve helped me make my dream job a reality. It has been an honor and a blast being your first student and I hope I continue to make you proud. I don’t think I can fully express my appreciation of Professor Bela´ Bajnok either. He put a great deal of work, thought, and fun into the Abstract Math and research courses he created at Gettysburg College, and by far these courses, as well as Bela´ himself, are most responsible for my decision of going into mathematics, a subject I never considered pursuing when I began college. Through these courses, I learned how much more there was to mathematics that just using formulas or procedures, how it could be so much more fun and challenging, how to discuss and present mathematical ideas, and what it meant to do research. Course by course, I grew immensely as a mathematician and fell more in love i with the subject. Beyond the courses, Bela´ has been a phenomenal and fun collaborator and a pillar of support and mentorship, encouraging my continued growth, giving me guidance on how to do that, and believing in me. When I was stuck deciding between pursuing Physics or Mathematics, Bela´ helped me see that I would benefit most from focusing more on one subject rather than an even split between the two. By letting go of Physics, I was able to do a semester abroad in the Budapest Semesters in Mathematics program, participate in several research projects and programs, and attend more conferences as an undergraduate, all of which helped me get accepted into an excellent doctorate program. So, to my mentor, collaborator, role model, and friend Bela´ Bajnok, thank you so much. You helped me realize what my dream job is and have helped me all along the way. To the many teachers of Neshaminy School District and professors and staff of Get- tysburg College and the University of Minnesota, as well as the various mathematicians worldwide, who have given me several positive experiences as a student and supported me on my journey to this point, thank you. In particular, I want to thank Randi Danziger, who taught a class called Academic Enrichment (A.E.) at my elementary school. A.E. was a course that covered a wide variety of topics not normally covered in standard classes and designed for students that needed something more challenging and academically engaging. It was a class I desperately needed, but for a time it seemed that I couldn’t take it. I have had Attention Deficit Hyperactivity Disorder my entire life, something that made my first few years of school very difficult, and near the end of second grade I was diagnosed and began taking medication that drastically improved my ability to concentrate. However, this seemingly came too late, as I had already failed the test to get into the class for third grade, due to my struggle with focus, and the school’s administration said that it didn’t matter that I had been diagnosed and medicated, I couldn’t join A.E. since I didn’t pass the test. Having had me as a student for A.E. in first and second grade, Randi knew that I needed the class and that I had been diagnosed and medicated. When she found out what we had been told by the administration, she told my parents how I could retake the test with the ii school psychologist, and I passed with flying colors. It’s a good thing too, because while I was bursting with curiosity as a child, A.E. classes helped me develop a true love for learning and critical thinking, qualities that have stuck with me into my adult life. I want to thank Darren Glass, who was my undergraduate adviser and my first math professor at Gettysburg College. Darren and his Honors Calculus 2 course, which focused on only sequences, series, and differential equations, instead of all the tricks for integration, con- vinced me to try the college’s introductory proof-writing course (Abstract Math 1), which was where I truly started to fall in love with mathematics. Darren was the first person to show me how mathematics could involve a higher level of critical thinking and how it was more than just finding answers to problems. He guided me through undergrad, helping me find and partake in many opportunities that furthered my development as a mathematician and supported me throughout and beyond, and I am very grateful to him. I am also very thankful of Professors Pete Johnson and Overtoun Jenda, who gave me my first research experience outside of Gettysburg and outside of Additive Combinatorics through their REU at Auburn University, as well as the many opportunities they gave me to present my work or participate in more research collaborations through the Masamu program with SAMSA. They have been incredibly supportive throughout my mathematical career, and have given me several wonderful experiences with mathematics which have certainly played a role in my growth as a mathematician. I am grateful as well to my collaborators of the work in this dissertation beyond my adviser: Feng Dai, Damir Ferizovic,´ Alexey Glazyrin, Josiah Park, and Oleksandr Vlasiuk. I always have more fun working on research with others, and I appreciate so very much how great of an experience it has been to work with you all so far and how much you’ve helped me learn and grow over the past six years. I would like to thank my Preliminary and Final Oral Exam committee members, Max Engelstein, Svitlana Mayboroda, Andrew Odlyzko, and Vic Reiner for all of their feedback and everything they helped me learn through my two exams. I would also like to thank Greg Blekherman, Henry Cohn, David iii de Laat, Axel Flinth, Michael Lacey, Tongseok Lim, Galyna Livshyts, Svitlana Mayboroda, Robert McCann, Bruno Poggi, Danylo Radchenko, Bruce Reznick, Alexander Reznikov, Daniela Schiefeneder, Alexander Shapiro, Stefan Steinerberger, and Yao Yao for the helpful discussions my collaborators or I have had with them during the course of our work. I asked many friends, family members, and colleagues to proofread this dissertation, and for their effort and incredibly helpful feedback, I would like to thank Angeliki Matzke, Dmitriy Bilyk, Megan Aslanian, Esther Banaian, Katie Behling, Olivia Cannon, Mary Clarke, Jennifer Dorey, Alexey Glazyrin, Taylor Jacovich, Michelle Janossy, Steve Kenyon, Scott Magers, Jessica Matzke, Julie Matzke, Lauren McCabe, Brian McEvoy, Jackie McMillen, Lisa Naples, Amanda Plimpton, Maddie Price, Cecily Santiago, Josie Spencer, Stefan Steinerberger, Eric Stucky, and Oleksandr Vlasiuk. I am also grateful to Adrienne Sands for helping me with the template for this dissertation. Finally, I am incredibly grateful to all of my friends and family for making my life so much brighter and loving and supporting me throughout everything, on both good days and rough days, and helping me make it though all of my struggles, especially my impostor syndrome. To my siblings and parents, thank you so much for helping me become who I am today, and believing in me, whether or not you think you know what I do. Thank you for being there for me when I didn’t think I could become a doctor, and making sure I knew you all were proud of me no matter what, but you still thought I could make it through the program. Thank you for helping me realize what I needed to change in order to tackle the biggest challenge of my life and helping me believe that I could do it again. Thank you for the love and support you show me everyday. To my parents, thank you for always trying to do what was best for me. Thank you for getting me help when I needed it, whether it was speech therapy or getting me tested for ADHD.