SHEAVES, COSHEAVES AND APPLICATIONS Justin Michael Curry ADISSERTATION in Mathematics Presented to the Faculties of The University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2014 Robert W. Ghrist, Andrea Mitchell University Professor Professor of Mathematics Professor of Electrical and Systems Engineering Supervisor of Dissertation David Harbater, Christopher H. Browne Distinguished Professor Professor of Mathematics Graduate Group Chairperson Dissertation Committee: Robert Ghrist, Andrea Mitchell University Professor of Mathematics Robert MacPherson, Hermann Weyl Professor of Mathematics Tony Pantev, Professor of Mathematics SHEAVES, COSHEAVES AND APPLICATIONS © 2014 Justin Michael Curry Dedicated to Michael Loyd Curry iii ACKNOWLEDGEMENTS At the current moment in time, a PhD is the highest academic degree awarded in the United States. As such, this thesis reflects over two decades of formal education and schooling across multiple institutions. It also reflects the author’s life experience to date, which is formed in many informal and non-academic ways. Accounting for all of these influences and giving credit where credit is due is an impossible task; however, I would like to take some time to thank the many hands which helped this thesis come to be. Given the public nature of this document, I will not always name names, but I will make clear the contributions of my colleagues and teachers. First and foremost I must thank my parents for bringing me into the world. While my father was in the Navy, my mother had the strongest influence on my education. Instilling a love of reading is, besides giving me life itself, the greatest gift she has given to me. I remember distinctly being told that, given our socio-economic status, receiving a scholarship was the only way I would make it to a university one day, and that reading would take me there. My sense of reverence for reading, among other things, is entirely due to my mother. In contrast, my father engaged me in philosophical dialog at a young age, which is how I gained my first experience with critical thinking. He was never much of a reader; he preferred to sort things out for himself. My entire family — aunts, uncles, cousins and grandparents included — have supported me every step of the way and they know I owe them a great deal. If anyone thinks that obtaining a PhD comes after an endless stream of successes, they are mistaken. I failed many times, and fortunately I was given many second chances. The educators in the Virginia Beach public school system gave me my first second chance by letting me retake a placement exam for the gifted and talented program. Mr. Ausberry, at Thomas Harrison Middle School, requested that I be accelerated a year in mathemat- ics. Mr. Frutuozo made being a scientist seem fashionable, by being a rock star himself. At Harrisonburg High School, I had many excellent instructors, but I felt the strongest direction and guidance from Henry Buhl, Myron Blosser, Andrew Jackson, Patrick Lint- ner and David Loughran. Without these hardworking and underpaid teachers, I don’t think I would have gotten to go to MIT. Attending MIT as an undergrad was one of the most formative experiences of my life. It certainly tested and broke the mental toughness that I thought I had. Sitting as a fresh- man in Denis Auroux’s 18.100B and getting my first taste of point-set topology was like stepping in to another dimension. It was too much, too fast, and for a moment I thought that the gate of mathematics was closed to me. Gerald Sussman helped steer me back to- wards mathematics by preaching the value of the MIT quadrivium: logic/programming, analysis, algebra, geometry, topology, relativity and quantum mechanics. Haynes Miller iv gave me my second second chance by overlooking my shabby mathematical prepara- tion and letting me study for the Part 1B tripos at Churchill College, as part of the Cambridge-MIT exchange program. Cambridge exposed me to one of the greatest math- ematical cultures to ever exist. The integrated nature of the classes and the year-long preparation for the tripos helped me gain independence and synthesize my lessons into a unified whole. It was in the Churchill buttery, where Part II and III students waxed poetic about Riemann surfaces and topoi before I even knew what a ring was, that I decided I had to pursue mathematics for graduate school. Returning to MIT, Haynes exposed me to even more advanced mathematics through summer projects and an IAP project with Aliaa Barakat on integrable systems. Working with Aliaa and, later, Victor Guillemin gave me lots of practice with writing mathematics. All of this has served me well for graduate school. The University of Pennsylvania appealed to my theory-building nature, but it was having to retake the preliminary exams that helped me become a better problem-solver. While drudging through the Berkeley Problems in Mathematics [dSS04] book, my classes gave me something to look forward to. Tony Pantev made the first-year algebra sequence geometric for me, by introducing us to the Serre-Swan correspondence, categories, sim- plicial sets, spectra and sheaves. Jonathan Block balanced the algebraic and the geometric in Penn’s lengthy topology sequence and introduced us to “Brave New Algebra.” The graduate student body at Penn helped contextualize my mathematical lessons, while my roommate, Elaine So, gave me lessons in how to be a better human. My advisor, Robert Ghrist, believed in me when I did not believe in myself. He taught me to have good taste in mathematics and introduced me to Morse theory, Euler calculus, integral geometry and much more. When I first became his student, the idea that no mathematical object is too abstract to be incarnate resonated deeply with me then, as it does today. Rob outlined a beautiful vision for applied mathematics and worked very hard to realize his ambitious plan. By bringing Yasu Hiraoka, Sanjeevi Krishnan, David Lipsky, Michael Robinson and Radmila Sazdanovic together, Rob augmented my graduate training in profound ways. Given this investment, Rob was extremely generous to let me wander geographically and intellectually. Because of him and Penn’s Exchange Scholar program, I was able to live in Princeton for the last few years of my graduate career. At Princeton, I approached Bob MacPherson in person, who luckily was thinking about applied sheaf theory because of my advisor and Amit Patel, and he agreed to organize a seminar at the Institute for Advanced Study. Listening and watching Bob lec- ture was like getting to peer through a telescope into the far reaches of the mathematical kingdom. The attendees of this seminar were a motley crew of thinkers and Bob was our shepherd. Bob never said more than was necessary, never wanted his own perspec- tive or understanding to crowd out a newly forming one, and did his best to cultivate each individual’s diverse set of mental connections, life experiences and accompanying insights. v Many people helped me directly and indirectly while finishing my thesis. Mark Goresky taught me the subtleties of stratification theory, set a high standard for mathe- matical precision and was enthusiastic and supportive of all my efforts. David Treumann and Jon Woolf both clarified details concerning this work via email. Greg Henselman, Sefi Ladkani, Michael Lesnick, and Jim McClure all provided editorial comments on early drafts of this thesis. Vin de Silva, Matthew Kahle, Dmitriy Morozov, Vidit Nanda, Primoz Skraba and Mikael Vejdemo-Johansson all provided moral support. Ryan and Cate Hodgen kept me sane during my frequent trips to Virginia, where I helped my Dad through the painful process of fighting, and losing to, bladder cancer. My fiancée, Sasha Rahlin, encouraged me to pursue a math major when we first started dating as sopho- mores, made my junior year abroad doubly wonderful, navigated the stressful two-body aspect of picking a graduate school as a senior, helped me through all of the ups and downs of graduate school along with losing my father, and continues to dazzle me with her focus, drive, beauty and brains. You and Simone are the best. vi ABSTRACT SHEAVES, COSHEAVES AND APPLICATIONS Justin Michael Curry Robert W. Ghrist This thesis develops the theory of sheaves and cosheaves with an eye towards applica- tions in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular sheaves and cosheaves, which are finite families of vector spaces and maps parametrized by a cell complex. We develop cellular (co)sheaves as a new tool for topological data analysis, network coding and sensor networks. We utilize the barcode descriptor from persistent homology to interpret cellular cosheaf homology in terms of Borel-Moore homology of the barcode. We associate barcodes to network coding sheaves and prove a duality theorem there. A new approach to multi-modal sensing is introduced, where sheaves and cosheaves model detection and evasion sets. A foundation for multi-dimensional level-set persis- tent homology is laid via constructible cosheaves, which are equivalent to representa- tions of MacPherson’s entrance path category. By proving a van Kampen theorem, we give a direct proof of this equivalence. A cosheaf version of the ith derived pushforward of the constant sheaf along a definable map is constructed directly as a representation of this category. We go on to clarify the relationship of cellular sheaves to cosheaves by providing a formula that takes a cellular sheaf and produces a complex of cellular cosheaves. This formula lifts to a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the im- age of the constant sheaf through this equivalence.