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Group realizations of configurations

 dc.contributor.supervisor Padmanabhan, Ranganathan (Mathematics) en_US dc.contributor.author Lanyon, Jeffrey dc.date.accessioned 2018-03-29T20:41:05Z dc.date.available 2018-03-29T20:41:05Z dc.date.issued 2017 dc.date.submitted 2018-03-19T14:57:49Z en dc.identifier.uri http://hdl.handle.net/1993/32915 dc.description.abstract One of the basic questions in this topic is about the "existence" and the related "realization" of abstract configurations. Motivated by the fact that a non-singular cubic curve in the projective plane over a field admits a geometrically defined group law "⊕" such that three points {P, Q, R} are collinear if and only if the sum P⊕Q⊕R = 0 under the group law, we define the concept of group realization of a given (nk) configuration. Group realizations are in turn used to construct geometric realizations (of (n3)'s and (n4)'s) in the real or the complex projective plane. Using a variety of techniques from algebra and number theory like the resultant of polynomials, Hensel's Lemma on lifting primitive roots to prime powers, companion matrices, and Bunyakovsky's conjecture we prove the existence of infinitely many realizable configurations, including: en_US 1. Realizations of cyclic (n3) configurations using the group structure on cubic curves over the real or the complex plane; 2. Realizations of cyclic (n4) configurations using the group structure on non-circular ellipses where now blocks of 4 points are circles over the real plane. dc.subject Mathematics, Geometry, Configurations, Groups en_US dc.title Group realizations of configurations en_US dc.degree.discipline Mathematics en_US dc.contributor.examiningcommittee Kirkland, Stephen (Mathematics) en_US Zhang, Yang (Mathematics) Kocay, William (Computer Science) dc.degree.level Master of Science (M.Sc.) en_US dc.description.note May 2018 en_US
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