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The Combinatorial PT-DT Correspondence PDF

178 Pages·2021·2.247 MB·English
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THE COMBINATORIAL PT-DT CORRESPONDENCE by GAUTAM WEBB A DISSERTATION Presented to the Department of Mathematics and the Division of Graduate Studies of the University of Oregon in partial fulfillment of the requirements for the degree of Doctor of Philosophy September 2021 DISSERTATION APPROVAL PAGE Student: Gautam Webb Title: The Combinatorial PT-DT Correspondence This dissertation has been accepted and approved in partial fulfillment of the requirements for the Doctor of Philosophy degree in the Department of Mathematics by: Benjamin Young Chairperson Jonathan Brundan Core Member Patricia Hersh Core Member Nicholas Proudfoot Core Member Boyana Norris Institutional Representative and Andrew Karduna Interim Vice Provost for Graduate Studies Original approval signatures are on file with the University of Oregon Division of Graduate Studies. Degree awarded September 2021 ii (cid:13)c 2021 Gautam Webb iii DISSERTATION ABSTRACT Gautam Webb Doctor of Philosophy Department of Mathematics September 2021 Title: The Combinatorial PT-DT Correspondence We resolve an open conjecture from algebraic geometry, which states that two generating functions for plane partition-like objects (the “box-counting” formulae for the Calabi-Yau topological vertices in Donaldson-Thomas theory and Pandharipande-Thomas theory) are equal up to a factor of MacMahon’s generating function for plane partitions. The main tools in our proof are a Desnanot-Jacobi- type condensation identity, and a novel application of the tripartite double-dimer model of Kenyon-Wilson. This dissertation includes previously unpublished co-authored material. iv CURRICULUM VITAE NAME OF AUTHOR: Gautam Webb GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED: University of Oregon, Eugene, OR Colorado College, Colorado Springs, CO DEGREES AWARDED: Doctor of Philosophy, Mathematics, 2021, University of Oregon Master of Science, Mathematics, 2019, University of Oregon Bachelor of Arts, Mathematics, 2015, Colorado College AREAS OF SPECIAL INTEREST: Algebraic Combinatorics PROFESSIONAL EXPERIENCE: Graduate Employee, University of Oregon, 2016-2021 GRANTS, AWARDS AND HONORS: Anderson Graduate Teaching Award, Department of Mathematics, University of Oregon, 2021 National Merit Scholar, 2011-2015 Goldwater Scholar Honorable Mention, 2014 PUBLICATIONS: Michael Penn, Christopher Sadowski, and Gautam Webb, Principal subspaces of twisted modules for certain lattice vertex operator algebras, Internat. J. Math. 30 (2019), no. 10, 1950048, 47 pp. MR 4013314 v Pedro A. Garc´ıa-Sa´nchez, Christopher O’Neill, and Gautam Webb, The computation of factorization invariants for affine semigroups, J. Algebra Appl. 18 (2019), no. 1, 1950019, 21 pp. MR 3910672 Christopher O’Neill, Vadim Ponomarenko, Reuben Tate, and Gautam Webb, On the set of catenary degrees of finitely generated cancellative commutative monoids, Internat. J. Algebra Comput. 26 (2016), no. 3, 565-576. MR 3506349 vi ACKNOWLEDGEMENTS To my advisor, Ben Young, thank you for your guidance, enthusiasm, and encouragement. I’m grateful to my coauthors, Helen Jenne and Ben Young, without whom this dissertation would not exist. To the faculty who showed me how beautiful mathematics can be, especially Robert Lipshitz, Victor Ostrik, and Jon Brundan, thank you for believing in me and for continually inspiring me. To my family, thank you for your constant love and support. To the friends who climbed with me, went on adventures with me, and made me laugh, thank you for brightening my days. And finally, thank you to Elevation Bouldering Gym for being my island of happiness whenever the sea of mathematics-induced sadness grew too deep. vii For my parents viii TABLE OF CONTENTS Chapter Page I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 III. DT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1. DT Box Configurations . . . . . . . . . . . . . . . . . . . . . . 10 3.2. DT Theory and the Dimer Model . . . . . . . . . . . . . . . . 11 3.3. The Condensation Recurrence in DT Theory . . . . . . . . . . 13 IV. PT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.1. Labelled PT Box Configurations . . . . . . . . . . . . . . . . . 18 4.2. Labelled AB Configurations . . . . . . . . . . . . . . . . . . . 21 4.2.1. The Base AB Configuration . . . . . . . . . . . . . . . . . 23 4.2.2. The Labelling Algorithm for AB Configurations . . . . . . 25 4.2.3. Projection to the Base AB Configuration . . . . . . . . . . 29 4.3. PT Theory and the Labelled Double-Dimer Model . . . . . . . 49 4.4. Proofs of the Equivalence of the Labelling Algorithms . . . . . 56 4.5. The Condensation Recurrence in PT Theory . . . . . . . . . . 93 4.6. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ix Chapter Page V. WEIGHTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.1. Modifying the Partition µ . . . . . . . . . . . . . . . . . . . . 100 5.1.1. The Diagonal of µ . . . . . . . . . . . . . . . . . . . . . . . 100 5.1.2. The Partitions µr and µc . . . . . . . . . . . . . . . . . . . 102 5.1.3. The Partition µrc . . . . . . . . . . . . . . . . . . . . . . . 113 5.2. DT Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.1. Weight of Minimal Configuration . . . . . . . . . . . . . . 116 5.2.2. Algebraic Simplification . . . . . . . . . . . . . . . . . . . . 119 5.3. PT Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.3.1. Edge-Weight of Base Double-Dimer Configuration . . . . . 138 5.3.2. Algebraic Simplification . . . . . . . . . . . . . . . . . . . . 145 REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 x

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