The Combinatorial PT-DT Correspondence

178 Pages·2021·2.247 MB·English

by  Gautam Webb

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
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(cid:13)c 2021 Gautam Webb
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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This dissertation includes previously unpublished co-authored material.
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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
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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
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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.
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For my parents
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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
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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
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