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Bergvelt, Maarten Kedem, Rinat Di Francesco, Philippe Nevins, Thomas Sun, Hao 2018-09-04T20:47:19Z 2018-09-04T20:47:19Z 2020-09-05T09:15:09Z 2018-04-18 2018-05 This thesis is motivated by the W-operators introduced by Mironov et al. We prove that the W-operators are generalizations of the cut-and-join operator studied by Goulden and Jackson. We give a new description of the structure of W-operators, using the combinations of symmetric groups. As an application, we prove new formulas about generating functions of connected Hurwitz numbers and give topological recursion formulas for d-Hurwitz numbers. Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2020-05-01 The student, Hao Sun, accepted the attached license on 2018-04-17 at 14:47. The student, Hao Sun, submitted this Dissertation for approval on 2018-04-17 at 14:59. This Dissertation was approved for publication on 2018-04-18 at 09:36. DSpace SAF Submission Ingestion Package generated from Vireo submission #12313 on 2018-08-31 at 17:29:30 Made available in DSpace on 2018-09-04T20:47:19Z (GMT). No. of bitstreams: 2 SUN-DISSERTATION-2018.pdf: 737302 bytes, checksum: 362aa4e388eceb7e3747336f8c4c5de8 (MD5) LICENSE.txt: 4204 bytes, checksum: f526617d9ccb5112912ea423c97d715e (MD5) Previous issue date: 2018-04-18 Embargo set by: Seth Robbins for item 107421 Lift date: 2020-09-04T20:47:38Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Embargo set by: Seth Robbins for item 107421 Lift date: 2020-09-04T20:50:11Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Limited Restriction Lifted for Item 107421 on 2020-09-05T09:15:09Z. application/pdf http://hdl.handle.net/2142/101336 en Copyright 2018 Hao Sun W-operator, Hurwitz number, Cut-and-join operator W-operators and generating functions of hurwitz numbers text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.