{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/113016"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/113016","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Gopakumar-Vafa invariant and Macdonald formula","abstract":"In this thesis, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicist that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I will use this definition on the total space of canonical bundle of P2 and compute the associated invariants. I will introduce a Gopakumar-Vafa/Pandharipande-Thomas correspondence on the level of perverse sheaves, inspired by the work of Migliorini-Shende-Viviani. I will verify that my calculation actually proves part of the conjecture. I have shown a strong evidence for this conjecture in the case of degree 2.","abstract_html":"In this thesis, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicist that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I will use this definition on the total space of canonical bundle of P2 and compute the associated invariants. I will introduce a Gopakumar-Vafa/Pandharipande-Thomas correspondence on the level of perverse sheaves, inspired by the work of Migliorini-Shende-Viviani. I will verify that my calculation actually proves part of the conjecture. I have shown a strong evidence for this conjecture in the case of degree 2.","abstract_has_math":false,"creators":["Zhao, Lutian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Katz, Sheldon","Bradlow, Steven","Dodd, Christopher","Haboush, William"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-01-12T21:45:36Z","date_published":"2022-01-12T21:45:36Z","updated_at":"2026-07-22T22:24:52Z","subjects":["Moduli spaces, Gopakumar-Vafa invariants, Macdonald Formula, Hilbert Scheme, Decomposition Theorem"],"languages":["en"],"rights":["Copyright 2021, Lutian Zhao"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/113016","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Katz, Sheldon","Bradlow, Steven","Dodd, Christopher","Haboush, William"]},{"key":"dc:creator","label":"Author","values":["Zhao, Lutian"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022-01-12T21:45:36Z","2021-07-14","2021-08"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"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":["Moduli spaces, Gopakumar-Vafa invariants, Macdonald Formula, Hilbert Scheme, Decomposition Theorem"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2021, Lutian Zhao"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/113016"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicist that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I will use this definition on the total space of canonical bundle of P2 and compute the associated invariants. I will introduce a Gopakumar-Vafa/Pandharipande-Thomas correspondence on the level of perverse sheaves, inspired by the work of Migliorini-Shende-Viviani. I will verify that my calculation actually proves part of the conjecture. I have shown a strong evidence for this conjecture in the case of degree 2.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-01-12 without embargo terms","The student, Lutian Zhao, accepted the attached license on 2021-07-12 at 13:02.","The student, Lutian Zhao, submitted this Dissertation for approval on 2021-07-12 at 13:06.","This Dissertation was approved for publication on 2021-07-14 at 17:10.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16850 on 2022-01-12 at 12:44:54","Made available in DSpace on 2022-01-12T21:45:36Z (GMT). No. of bitstreams: 3 ZHAO-DISSERTATION-2021.pdf: 715071 bytes, checksum: 01e6ce8e237d80d8186efef892fcfc66 (MD5) LICENSE.txt: 4208 bytes, checksum: 6cec8bb9aa319980e52b7ee4596d89d5 (MD5) PROQUEST_LICENSE.txt: 4554 bytes, checksum: e93984587eaa467385e527cf2fb8cda8 (MD5) Previous issue date: 2021-07-14"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Gopakumar-Vafa invariant and Macdonald formula"]}]}],"canonical_facts":{"dc:contributor":["Katz, Sheldon","Bradlow, Steven","Dodd, Christopher","Haboush, William"],"dc:creator":["Zhao, Lutian"],"dc:date":["2022-01-12T21:45:36Z","2021-07-14","2021-08"],"dc:description":["In this thesis, I will introduce the Gopakumar-Vafa(GV) invariant and show one calculation on the nonreduced cycle. The GV invariant is an integral invariant predicted by physicist that counts the number of curves inside a given Calabi-Yau threefold. The definition has been conjectured by Maulik-Toda in 2016 in terms of perverse sheaf. I will use this definition on the total space of canonical bundle of P2 and compute the associated invariants. I will introduce a Gopakumar-Vafa/Pandharipande-Thomas correspondence on the level of perverse sheaves, inspired by the work of Migliorini-Shende-Viviani. I will verify that my calculation actually proves part of the conjecture. I have shown a strong evidence for this conjecture in the case of degree 2.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2022-01-12 without embargo terms","The student, Lutian Zhao, accepted the attached license on 2021-07-12 at 13:02.","The student, Lutian Zhao, submitted this Dissertation for approval on 2021-07-12 at 13:06.","This Dissertation was approved for publication on 2021-07-14 at 17:10.","DSpace SAF Submission Ingestion Package generated from Vireo submission #16850 on 2022-01-12 at 12:44:54","Made available in DSpace on 2022-01-12T21:45:36Z (GMT). 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