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dc.contributor.advisorBaker, Matthew
dc.contributor.authorBackman, Spencer Christopher Foster
dc.date.accessioned2014-05-22T15:36:04Z
dc.date.available2014-05-22T15:36:04Z
dc.date.created2014-05
dc.date.issued2014-04-08
dc.date.submittedMay 2014
dc.identifier.urihttp://hdl.handle.net/1853/51908
dc.description.abstractChip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technology
dc.subjectChip-firing
dc.subjectGraph
dc.subjectTropical curve
dc.subjectRiemann-Roch
dc.subjectOrientation
dc.subject.lcshDivisor theory
dc.subject.lcshCombinatorial analysis
dc.subject.lcshGraph theory
dc.subject.lcshGeometry, Algebraic
dc.subject.lcshNumber theory
dc.titleCombinatorial divisor theory for graphs
dc.typeDissertation
dc.description.degreePh.D.
dc.contributor.departmentMathematics
thesis.degree.levelDoctoral
dc.contributor.committeeMemberThomas, Robin
dc.contributor.committeeMemberYu, Josephine
dc.contributor.committeeMemberPokutta, Sebastian
dc.contributor.committeeMemberSergey Norin
dc.date.updated2014-05-22T15:36:04Z


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