Algorithmic problems in the braid group.
Item
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Title
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Algorithmic problems in the braid group.
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Identifier
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AAI3083660
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identifier
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3083660
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Creator
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Feder, Elie.
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Contributor
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Adviser: Michael Anshel
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Date
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2003
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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The study of braid groups and their applications is a field which has attracted the interest of mathematicians and computer scientists alike. Their basic structure has been studied as far back as Gauss who considered the notion of a braid when studying the orbit of the first observed asteroid, Ceres. Besides for the value in studying the braid group in a theoretical framework, the braid group has been found to have diverse applications. While its applications to knot theory has been known for many years, its applicability to the field of cryptography has only been realized recently [AAG]. Since this time, a study of algorithmic problems in the braid group and their complexity have acquired a great practical significance, in addition to its intrinsic theoretical beauty.;We begin with a review of the notion of a braid group. We then discuss some known solutions to decision problems in braid groups. We then move on to proving new results in braid group algorithmics. We offer a quick solution to the generalized word problem in braid groups, in the special case of cyclic subgroups. We illustrate this solution and its complexity using a multitape Turing machine. We then turn to a discussion of decision problems in cyclic amalgamations of groups. Again using a multitape Turing machine, we solve the word problem for the cyclic amalgamation of two braid groups. We analyze its complexity as well.;We then turn to a more general study of the conjugacy problem in cyclic amalgamations. We revise and prove some theorems of Lipschutz [L1] and show their application to cyclic amalgamations of braid groups. We generalize this application to prove a new theorem regarding the conjugacy problem in cyclic amalgamations.;We then discuss some application of braid groups, culminating in a section devoted to the discussion of braid group cryptography. We conclude with a discussion of some open questions that we would like to pursue in future research.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
