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Square tilings of surfaces from discrete harmonic 1-chains
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Chien, Edward Der-Yu. Square tilings of surfaces from discrete harmonic 1-chains. Retrieved from https://doi.org/doi:10.7282/T31G0P7N

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TitleSquare tilings of surfaces from discrete harmonic 1-chains
NameChien, Edward Der-Yu (author); Luo, Feng (chair); Han, Zheng-cao (internal member); Rong, Xiaochun (internal member); Kofman, Ilya (outside member); Rutgers University; Graduate School - New Brunswick
Date Created2015
Other Date2015-10 (degree)
SubjectMathematics
Extent1 online resource (viii, 53 p. : ill.)
DescriptionWorks of Dehn [De] and Brooks, Smith, Stone, and Tutte [BSST] have noted a classical correspondence between planar electrical networks and square tilings of a Euclidean rectangle. In this thesis, a generalization of this result is proven for a closed, orientable surface S_g of genus g greater than or equal to 1 with a CW decomposition Gamma and a generic non-zero 1st homology class. From these data, a square-tiled flat cone metric on S_g is produced. The construction utilizes a discrete harmonic representative, and a discrete Poincar e-Hopf theorem is obtained in the process. Similar results for non-generic non-zero homology class are also obtained where the metric and tiling are constructed on a surface of equal or lower genus. Additionally, we obtain similar constructions of square-tiled metrics on surfaces with boundary via modi cation of the harmonicity conditions. Finally, it is simple to obtain rectangle-tiled flat cone metrics with the extra data of a resistance function r on the edges of Gamma, and the necessary changes are described.
NotePh.D.
NoteIncludes bibliographical references
Noteby Edward Der-Yu Chien
Genretheses, ETD doctoral
Persistent URLhttps://doi.org/doi:10.7282/T31G0P7N
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.
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