{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/49531"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/49531","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Rigidity of length functions over strata of flat metrics","abstract":"In this thesis we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Leininger-Rafi. Specifically, for any stratum with more unmarked zeroes than the genus, the Sigma-length-spectrum of a set of simple closed curves Sigma determines the flat metric in the stratum if and only if Sigma is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the Sigma-length-spectrum of a finite set of closed curves Sigma.","abstract_html":"In this thesis we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Leininger-Rafi. Specifically, for any stratum with more unmarked zeroes than the genus, the Sigma-length-spectrum of a set of simple closed curves Sigma determines the flat metric in the stratum if and only if Sigma is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the Sigma-length-spectrum of a finite set of closed curves Sigma.","abstract_has_math":false,"creators":["Fu, Ser-Wei"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Leininger, Christopher J.","Kapovitch, Ilia","Athreya, Jayadev S.","Dowdall, Spencer"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-05-30T16:48:39Z","date_published":"2014-05-30T16:48:39Z","updated_at":"2026-07-22T22:25:38Z","subjects":["Surface","Quadratic differential","Measured foliation","Train track"],"languages":["en"],"rights":["Copyright 2014 Ser-Wei Fu"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/49531","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leininger, Christopher J.","Kapovitch, Ilia","Athreya, Jayadev S.","Dowdall, Spencer"]},{"key":"dc:creator","label":"Author","values":["Fu, Ser-Wei"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-05-30T16:48:39Z","2014-05"]},{"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":["Surface","Quadratic differential","Measured foliation","Train track"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Ser-Wei Fu"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/49531"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Leininger-Rafi. Specifically, for any stratum with more unmarked zeroes than the genus, the Sigma-length-spectrum of a set of simple closed curves Sigma determines the flat metric in the stratum if and only if Sigma is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the Sigma-length-spectrum of a finite set of closed curves Sigma.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-04-23T13:07:33Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 Fu_Ser-Wei.tex: 145067 bytes, checksum: 7d8220ea0ca31f78dc1d21bf217f1972 (MD5) Fu_Sew-Wei.pdf: 471225 bytes, checksum: f1d3bbd89804447a0624c4e66ec4ce17 (MD5) Fu_Ser-Wei.pdf: 471225 bytes, checksum: f1d3bbd89804447a0624c4e66ec4ce17 (MD5)","Made available in DSpace on 2014-05-30T16:48:39Z (GMT). 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Specifically, for any stratum with more unmarked zeroes than the genus, the Sigma-length-spectrum of a set of simple closed curves Sigma determines the flat metric in the stratum if and only if Sigma is dense in the projective measured foliation space. We also prove that flat metrics in any stratum are locally determined by the Sigma-length-spectrum of a finite set of closed curves Sigma.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-04-23T13:07:33Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 3 Fu_Ser-Wei.tex: 145067 bytes, checksum: 7d8220ea0ca31f78dc1d21bf217f1972 (MD5) Fu_Sew-Wei.pdf: 471225 bytes, checksum: f1d3bbd89804447a0624c4e66ec4ce17 (MD5) Fu_Ser-Wei.pdf: 471225 bytes, checksum: f1d3bbd89804447a0624c4e66ec4ce17 (MD5)","Made available in DSpace on 2014-05-30T16:48:39Z (GMT). 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