{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/44328"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/44328","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The analytic and asymptotic behaviors of vortices","abstract":"\"We study vortex equations with a parameter $s$ on smooth vector bundles $E$ over compact K\\\"\"ahler manifolds $M$. For each $s$, we invoke techniques in \\cite{Br} by turning vortex equations into the elliptic partial differential equations considered in \\cite{kw} and obtain a family of solutions. Our results show that away from a singular set, such a family exhibit well controlled convergent behaviors, leading us to prove conjectures posed by Baptista in \\cite{Ba} concerning dynamic behaviors of vortices. These results are published in \\cite{Li}. We also analyze the analytic singularities on the singular set. The analytic singularities of the PDE's reflect topological inconsistencies as $s \\to \\infty$. On the second part of the thesis, we form a modification of the limiting objects, leading to a phenomenon of energy concentration known as the \"\"bubbling\"\". We briefly survey the established bubbling results in literature.\"","abstract_html":""We study vortex equations with a parameter $s$ on smooth vector bundles $E$ over compact K\\""ahler manifolds $M$. For each $s$, we invoke techniques in \\cite{Br} by turning vortex equations into the elliptic partial differential equations considered in \\cite{kw} and obtain a family of solutions. Our results show that away from a singular set, such a family exhibit well controlled convergent behaviors, leading us to prove conjectures posed by Baptista in \\cite{Ba} concerning dynamic behaviors of vortices. These results are published in \\cite{Li}. We also analyze the analytic singularities on the singular set. The analytic singularities of the PDE's reflect topological inconsistencies as $s \\to \\infty$. On the second part of the thesis, we form a modification of the limiting objects, leading to a phenomenon of energy concentration known as the ""bubbling"". We briefly survey the established bubbling results in literature."","abstract_has_math":true,"creators":["Liu, Chih-Chung"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bradlow, Steven B.","Kerman, Ely","Kirr, Eduard-Wilhelm","La Nave, Gabriele"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013-05-24T22:07:57Z","date_published":"2013-05-24T22:07:57Z","updated_at":"2026-07-22T22:25:34Z","subjects":["Vortex Equations","Mathematical Physics","L^2 Geometry","Differential Geometry."],"languages":["en"],"rights":["Copyright 2013 Chih-Chung Liu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/44328","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bradlow, Steven B.","Kerman, Ely","Kirr, Eduard-Wilhelm","La Nave, Gabriele"]},{"key":"dc:creator","label":"Author","values":["Liu, Chih-Chung"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2013-05-24T22:07:57Z","2013-05"]},{"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":["Vortex Equations","Mathematical Physics","L^2 Geometry","Differential 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 2013 Chih-Chung Liu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/44328"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"We study vortex equations with a parameter $s$ on smooth vector bundles $E$ over compact K\\\"\"ahler manifolds $M$. 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We briefly survey the established bubbling results in literature.\"","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-03-29T15:37:31Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 5 Liu_Chih-Chung.pdf: 568722 bytes, checksum: a534f68eececa0509ed158fe8f81021d (MD5) Liu_Chih-Chung.pdf: 568722 bytes, checksum: a534f68eececa0509ed158fe8f81021d (MD5) Liu_Chih-Chung.pdf: 568722 bytes, checksum: a534f68eececa0509ed158fe8f81021d (MD5) Liu_Chih-Chung.tex: 226665 bytes, checksum: 50159e7a30c808ba03bc2f0b5bd7a0ce (MD5) Liu_Chih-Chung.pdf: 568722 bytes, checksum: a534f68eececa0509ed158fe8f81021d (MD5)","Made available in DSpace on 2013-05-24T22:07:57Z (GMT). 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Our results show that away from a singular set, such a family exhibit well controlled convergent behaviors, leading us to prove conjectures posed by Baptista in \\cite{Ba} concerning dynamic behaviors of vortices. These results are published in \\cite{Li}. We also analyze the analytic singularities on the singular set. The analytic singularities of the PDE's reflect topological inconsistencies as $s \\to \\infty$. On the second part of the thesis, we form a modification of the limiting objects, leading to a phenomenon of energy concentration known as the \"\"bubbling\"\". 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