{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/100940"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/100940","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Symplectic foliations, currents, and local Lie groupoids","abstract":"This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham’s theory of currents, and Sullivan’s theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal’cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.","abstract_html":"This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham’s theory of currents, and Sullivan’s theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal’cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.","abstract_has_math":false,"creators":["Michiels, Daan"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Loja Fernandes, Rui","Tolman, Susan","Kerman, Ely","Pascaleff, James"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-04T20:26:54Z","date_published":"2018-09-04T20:26:54Z","updated_at":"2026-07-22T22:24:38Z","subjects":["foliation","symplectic foliation","Poisson structure","current","calibration","structure cycle","local Lie groupoid","associativity","globalizability","associative completion","associators","integrability","monodromy"],"languages":["en"],"rights":["Copyright 2018 Daan Michiels"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/100940","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Loja Fernandes, Rui","Tolman, Susan","Kerman, Ely","Pascaleff, James"]},{"key":"dc:creator","label":"Author","values":["Michiels, Daan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-04T20:26:54Z","2018-04-09","2018-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["foliation","symplectic foliation","Poisson structure","current","calibration","structure cycle","local Lie groupoid","associativity","globalizability","associative completion","associators","integrability","monodromy"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Daan Michiels"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/100940"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham’s theory of currents, and Sullivan’s theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal’cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-08-31 without embargo terms","The student, Daan Michiels, accepted the attached license on 2018-04-08 at 22:17.","The student, Daan Michiels, submitted this Dissertation for approval on 2018-04-08 at 22:29.","This Dissertation was approved for publication on 2018-04-09 at 10:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12148 on 2018-08-31 at 17:11:06","Made available in DSpace on 2018-09-04T20:26:54Z (GMT). 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We then show how de Rham’s theory of currents, and Sullivan’s theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. Next, three main results are obtained. The first is the generalization of a theorem by Mal’cev. The original theorem characterizes the local Lie groups that are part of a (global) Lie group. We give the corresponding result for local Lie groupoids. The second result is the generalization of a theorem by Olver which classifies local Lie groups in terms of Lie groups. Our generalization classifies, in terms of Lie groupoids, those local Lie groupoids that have integrable algebroids. The third and final result demonstrates a relationship between the associativity of a local Lie groupoid, and the integrability of its algebroid. In a certain sense, the monodromy groups of a Lie algebroid manifest themselves combinatorially in a local integration, as a lack of associativity.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-08-31 without embargo terms","The student, Daan Michiels, accepted the attached license on 2018-04-08 at 22:17.","The student, Daan Michiels, submitted this Dissertation for approval on 2018-04-08 at 22:29.","This Dissertation was approved for publication on 2018-04-09 at 10:17.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12148 on 2018-08-31 at 17:11:06","Made available in DSpace on 2018-09-04T20:26:54Z (GMT). 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