2026-09-22T14:08:18Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/113016 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Katz, Sheldon Bradlow, Steven Dodd, Christopher Haboush, William Zhao, Lutian 2022-01-12T21:45:36Z 2022-01-12T21:45:36Z 2021-07-14 2021-08 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 application/pdf http://hdl.handle.net/2142/113016 en Copyright 2021, Lutian Zhao Moduli spaces, Gopakumar-Vafa invariants, Macdonald Formula, Hilbert Scheme, Decomposition Theorem Gopakumar-Vafa invariant and Macdonald formula text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.