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Sheaves, Cosheaves and Applications Justin Michael Curry University of Pennsylvania, [email protected]

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University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 1-1-2014 Sheaves, Cosheaves and Applications Justin Michael Curry University of Pennsylvania, [email protected] Follow this and additional works at: http://repository.upenn.edu/edissertations Part of the Applied Mathematics Commons, and the Mathematics Commons Recommended Citation Curry, Justin Michael, "Sheaves, Cosheaves and Applications" (2014). Publicly Accessible Penn Dissertations. 1249. http://repository.upenn.edu/edissertations/1249 This paper is posted at ScholarlyCommons. http://repository.upenn.edu/edissertations/1249 For more information, please contact [email protected]. Sheaves, Cosheaves and Applications Abstract This thesis develops the theory of sheaves and cosheaves with an eye towards applications 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 persistent homology is laid via constructible cosheaves, which are equivalent to representations 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 i'th 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 ot a derived equivalence, which in turn recovers Verdier duality. Compactly-supported sheaf cohomology is expressed as the coend with the image of the constant sheaf through this equivalence. The quive alence is further used to establish relations between sheaf cohomology and a herein newly introduced theory of cellular sheaf homology. Inspired to provide fast algorithms for persistence, we prove that the derived category of cellular sheaves over a 1D cell complex is equivalent to a category of graded sheaves. Finally, we introduce the interleaving distance as an extended metric on the category of sheaves. We prove that global sections partition the space of sheaves into connected components. We conclude with an investigation into the geometry of the space of constructible sheaves over the real line, which we relate to the bottleneck distance in persistence. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Robert W. Ghrist Keywords algebraic topology, applied topology, cosheaves, sheaves, stratification theory Subject Categories Applied Mathematics | Mathematics This dissertation is available at ScholarlyCommons: http://repository.upenn.edu/edissertations/1249 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 suc- cesses, 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, re- quested that I be accelerated a year in mathematics. 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 Lintner 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 freshman 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, iv and for a moment I thought that the gate of mathematics was closed to me. Ger- ald Sussman helped steer me back towards mathematics by preaching the value of the MIT quadrivium: logic/programming, analysis, algebra, geometry, topology, relativity and quantum mechanics. Haynes Miller gave me my second second chance by overlooking my shabby mathematical preparation 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 mathematical 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 de- cided 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 bet- ter problem-solver. While drudging through the Berkeley Problems in Mathemat- ics [dSS04] book, my classes gave me something to look forward to. Tony Pan- tev made the first-year algebra sequence geometric for me, by introducing us to the Serre-Swan correspondence, categories, simplicial sets, spectra and sheaves. Jonathan Block balanced the algebraic and the geometric in Penn’s lengthy topol- ogy 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 stu- dent, 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 geographi- cally 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 think- ing 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 lecture was like getting to peer through a telescope into the far v reaches of the mathematical kingdom. The attendees of this seminar were a mot- ley crew of thinkers and Bob was our shepherd.
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