{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98085"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98085","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Symplectic toric stratified spaces with isolated singularities","abstract":"We provide a classification of two types of toric objects: symplectic toric cones and symplectic toric stratified spaces with isolated singularities. Both types of object are classified via orbital moment map and a second degree cohomology class. As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every homogeneous unimodular local embedding has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with the same orbital moment map by showing that their isomorphism classes are in bijective correspondence with the first Chern classes of principal G-bundles over W. This generalizes Lerman’s classification of compact connected contact toric manifolds. Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map called stratified unimodular local embeddings. Every stratified unimodular local embedding is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding, the isomorphism classes of symplectic toric stratified spaces with isolated singularities with a given orbital moment map are in bijective correspondence with a collection of cohomology classes dependent on the topology of W. This generalizes Burns, Guillemin, and Lerman’s classification of the compact connected symplectic toric stratified spaces with isolated singularities.","abstract_html":"We provide a classification of two types of toric objects: symplectic toric cones and symplectic toric stratified spaces with isolated singularities. Both types of object are classified via orbital moment map and a second degree cohomology class. As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every homogeneous unimodular local embedding has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with the same orbital moment map by showing that their isomorphism classes are in bijective correspondence with the first Chern classes of principal G-bundles over W. This generalizes Lerman’s classification of compact connected contact toric manifolds. Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map called stratified unimodular local embeddings. Every stratified unimodular local embedding is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding, the isomorphism classes of symplectic toric stratified spaces with isolated singularities with a given orbital moment map are in bijective correspondence with a collection of cohomology classes dependent on the topology of W. This generalizes Burns, Guillemin, and Lerman’s classification of the compact connected symplectic toric stratified spaces with isolated singularities.","abstract_has_math":false,"creators":["Wolbert, Seth P"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Lerman, Eugene","Tolman, Susan","Loja Fernandes, Rui","Kerman, Ely"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T16:37:37Z","date_published":"2017-09-29T16:37:37Z","updated_at":"2026-07-22T22:24:34Z","subjects":["Symplectic geometry"],"languages":["en"],"rights":["Copyright 2017 Seth Wolbert"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98085","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lerman, Eugene","Tolman, Susan","Loja Fernandes, Rui","Kerman, Ely"]},{"key":"dc:creator","label":"Author","values":["Wolbert, Seth P"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T16:37:37Z","2017-07-05","2017-08"]},{"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":["Symplectic geometry"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Seth Wolbert"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98085"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We provide a classification of two types of toric objects: symplectic toric cones and symplectic toric stratified spaces with isolated singularities. Both types of object are classified via orbital moment map and a second degree cohomology class. As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every homogeneous unimodular local embedding has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with the same orbital moment map by showing that their isomorphism classes are in bijective correspondence with the first Chern classes of principal G-bundles over W. This generalizes Lerman’s classification of compact connected contact toric manifolds. Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map called stratified unimodular local embeddings. Every stratified unimodular local embedding is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding, the isomorphism classes of symplectic toric stratified spaces with isolated singularities with a given orbital moment map are in bijective correspondence with a collection of cohomology classes dependent on the topology of W. This generalizes Burns, Guillemin, and Lerman’s classification of the compact connected symplectic toric stratified spaces with isolated singularities.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Seth Wolbert, accepted the attached license on 2017-07-03 at 13:26.","The student, Seth Wolbert, submitted this Dissertation for approval on 2017-07-03 at 13:35.","This Dissertation was approved for publication on 2017-07-05 at 13:06.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10883 on 2017-09-29 at 11:25:09","Made available in DSpace on 2017-09-29T16:37:37Z (GMT). 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As symplectic toric stratified spaces with isolated singularities are locally modeled on symplectic toric cones, we first focus on classifying symplectic toric cones. We show that symplectic toric cones have a certain type of map (called homogeneous unimodular local embeddings) as orbital moment maps. Conversely, every homogeneous unimodular local embedding has a symplectic toric cone for which it is an orbital moment map. We classify the symplectic toric cones with the same orbital moment map by showing that their isomorphism classes are in bijective correspondence with the first Chern classes of principal G-bundles over W. This generalizes Lerman’s classification of compact connected contact toric manifolds. Symplectic toric stratified spaces with isolated singularities are spaces with neighborhoods of singularities modeled on symplectic cones. We first show their quotients W are space stratified by manifolds with corners and their moment maps are a particular type of map called stratified unimodular local embeddings. Every stratified unimodular local embedding is the orbital moment map of a symplectic toric stratified space. Finally, we show that, for any stratified unimodular local embedding, the isomorphism classes of symplectic toric stratified spaces with isolated singularities with a given orbital moment map are in bijective correspondence with a collection of cohomology classes dependent on the topology of W. This generalizes Burns, Guillemin, and Lerman’s classification of the compact connected symplectic toric stratified spaces with isolated singularities.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms","The student, Seth Wolbert, accepted the attached license on 2017-07-03 at 13:26.","The student, Seth Wolbert, submitted this Dissertation for approval on 2017-07-03 at 13:35.","This Dissertation was approved for publication on 2017-07-05 at 13:06.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10883 on 2017-09-29 at 11:25:09","Made available in DSpace on 2017-09-29T16:37:37Z (GMT). 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