<mods:mods xmlns:mods="http://www.loc.gov/mods/v3" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-7.xsd"><mods:titleInfo><mods:title>A Computational Commutative Algebra Approach to Tilings</mods:title></mods:titleInfo><mods:abstract>In 1990, Thurston proved that for any two-dimensional simply connected region, the space of tilings is flip-connected. Later, Yamzon reproved this result using a commutative algebra approach developed by Gross. In the first portion of this thesis, we further explore the algebraic structures from Gross and Yamzon's work. Specifically, we prove a primary decomposition of the flip ideal for small (3 x n) regions, and give a conjecture for the primary decomposition in larger regions.&#13;
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In the second portion of the thesis, we give preliminary computational results on the flip-connected components of three-dimensional tilings.</mods:abstract><mods:name type="personal"><mods:namePart>Chin, Tracy</mods:namePart><mods:role><mods:roleTerm authority="marcrelator" authorityURI="http://id.loc.gov/vocabulary/relators" valueURI="http://id.loc.gov/vocabulary/relators/cre">creator</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Klivans, Caroline</mods:namePart><mods:role><mods:roleTerm authority="marcrelator" authorityURI="http://id.loc.gov/vocabulary/relators" valueURI="http://id.loc.gov/vocabulary/relators/ths">thesis advisor</mods:roleTerm></mods:role></mods:name><mods:name type="personal"><mods:namePart>Valiant, Paul</mods:namePart><mods:role><mods:roleTerm>reader</mods:roleTerm></mods:role></mods:name><mods:name type="corporate"><mods:namePart>Brown University. Applied Mathematics</mods:namePart><mods:role><mods:roleTerm type="text">sponsor</mods:roleTerm></mods:role></mods:name><mods:originInfo><mods:copyrightDate>2019</mods:copyrightDate></mods:originInfo><mods:typeOfResource>text</mods:typeOfResource><mods:physicalDescription><mods:digitalOrigin>born digital</mods:digitalOrigin></mods:physicalDescription><mods:language><mods:languageTerm authority="iso639-2b" type="text" authorityURI="http://id.loc.gov/vocabulary/iso639-2.html" valueURI="http://id.loc.gov/vocabulary/iso639-2/eng">English</mods:languageTerm></mods:language><mods:note type="thesis">Senior thesis (ScB)--Brown University, 2019</mods:note><mods:note displayLabel="Concentration">Applied Math-Computer Science</mods:note><mods:genre authority="aat">theses</mods:genre><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/01150951"><mods:topic>Tiling (Mathematics)</mods:topic></mods:subject><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/01745876"><mods:topic>Tiling spaces</mods:topic></mods:subject><mods:subject authority="fast" authorityURI="http://id.worldcat.org/fast" valueURI="http://id.worldcat.org/fast/00871202"><mods:topic>Commutative algebra</mods:topic></mods:subject><mods:identifier type="doi">10.26300/wqmt-9a94</mods:identifier><mods:accessCondition type="rights statement" xlink:href="http://rightsstatements.org/vocab/InC/1.0/">In Copyright</mods:accessCondition><mods:accessCondition type="restriction on access">Collection is open for research.</mods:accessCondition></mods:mods>