{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/90599"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/90599","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Transversals to horocycle flow on the moduli space of translation surfaces","abstract":"Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in H(2).","abstract_html":"Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in H(2).","abstract_has_math":false,"creators":["Work, Grace M"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Athreya, Jayadev","Kapovich, Ilya","Dunfield, Nathan","Tyson, Jeremy"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016-07-07T19:54:29Z","date_published":"2016-07-07T19:54:29Z","updated_at":"2026-07-22T22:26:34Z","subjects":["Translation surfaces","gap distribution","saddle connection","Veech surface","horocycle flow"],"languages":["en"],"rights":["Copyright 2016 Grace Work"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/90599","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Athreya, Jayadev","Kapovich, Ilya","Dunfield, Nathan","Tyson, Jeremy"]},{"key":"dc:creator","label":"Author","values":["Work, Grace M"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016-07-07T19:54:29Z","2016-04-21","2016-05"]},{"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":["Translation surfaces","gap distribution","saddle connection","Veech surface","horocycle flow"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2016 Grace Work"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/90599"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in H(2).","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-07-07 without embargo terms","The student, Grace Work, accepted the attached license on 2016-04-20 at 18:06.","The student, Grace Work, submitted this Dissertation for approval on 2016-04-20 at 18:14.","This Dissertation was approved for publication on 2016-04-21 at 08:44.","DSpace SAF Submission Ingestion Package generated from Vireo submission #9389 on 2016-07-07 at 13:32:00","Made available in DSpace on 2016-07-07T19:54:29Z (GMT). No. of bitstreams: 3 WORK-DISSERTATION-2016.pdf: 2058235 bytes, checksum: 2a4396b00fad0f0035f341ba02353550 (MD5) LICENSE.txt: 4207 bytes, checksum: a86aeb035af874142bfa0ba1dc545871 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: f3add2a3f9da6e9383e058ea6796e12c (MD5) Previous issue date: 2016-04-21"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Transversals to horocycle flow on the moduli space of translation surfaces"]}]}],"canonical_facts":{"dc:contributor":["Athreya, Jayadev","Kapovich, Ilya","Dunfield, Nathan","Tyson, Jeremy"],"dc:creator":["Work, Grace M"],"dc:date":["2016-07-07T19:54:29Z","2016-04-21","2016-05"],"dc:description":["Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, the first case where the Veech group has multiple cusps, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic surface in H(2).","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2016-07-07 without embargo terms","The student, Grace Work, accepted the attached license on 2016-04-20 at 18:06.","The student, Grace Work, submitted this Dissertation for approval on 2016-04-20 at 18:14.","This Dissertation was approved for publication on 2016-04-21 at 08:44.","DSpace SAF Submission Ingestion Package generated from Vireo submission #9389 on 2016-07-07 at 13:32:00","Made available in DSpace on 2016-07-07T19:54:29Z (GMT). No. of bitstreams: 3 WORK-DISSERTATION-2016.pdf: 2058235 bytes, checksum: 2a4396b00fad0f0035f341ba02353550 (MD5) LICENSE.txt: 4207 bytes, checksum: a86aeb035af874142bfa0ba1dc545871 (MD5) PROQUEST_LICENSE.txt: 4553 bytes, checksum: f3add2a3f9da6e9383e058ea6796e12c (MD5) Previous issue date: 2016-04-21"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/90599"],"dc:language":["en"],"dc:rights":["Copyright 2016 Grace Work"],"dc:subject":["Translation surfaces","gap distribution","saddle connection","Veech surface","horocycle flow"],"dc:title":["Transversals to horocycle flow on the moduli space of translation surfaces"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:34Z"}