{"id":{"repo_id":"ubc","oai_identifier":"oai:circle.library.ubc.ca:2429/731"},"canonical_url":"https://search.dev.ndltd.org/etd/ubc/oai:circle.library.ubc.ca:2429/731","repository":{"repo_id":"ubc","name":"University of British Columbia","base_url":"http://circle.library.ubc.ca/oai/request"},"display":{"title":"Counting coloured boxes","abstract":"This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁ x₂ - x₃ x₄ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, we derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂ x Z₂ , we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.","abstract_html":"This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁ x₂ - x₃ x₄ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, we derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂ x Z₂ , we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition.","abstract_has_math":false,"creators":["Young, Benjamin"],"institution":"University of British Columbia","degree_name":"Doctor of Philosophy - PhD","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-24T05:07:11Z","subjects":[],"languages":["eng"],"rights":["Attribution-NonCommercial-NoDerivatives 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2429/731","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Young, Benjamin"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2008"]},{"key":"dc:publisher","label":"Institution","values":["University of British Columbia"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy - PhD"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of British Columbia"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/","Attribution-NonCommercial-NoDerivatives 4.0 International"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2429/731","http://circle.library.ubc.ca/bitstream/2429/731/1/ubc_2008_spring_young_benjamin.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁ x₂ - x₃ x₄ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, we derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂ x Z₂ , we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition."]},{"key":"dc:format","label":"Dc Format","values":["1356991","application/pdf"]},{"key":"dc:title","label":"Title","values":["Counting coloured boxes"]}]}],"canonical_facts":{"dc:creator":["Young, Benjamin"],"dc:date":["2008"],"dc:description":["This thesis consists of the manuscripts of two research papers. In the first paper, we verify a recent conjecture of Kenyon/Szendroi by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the partition function for the Donaldson-Thomas theory of a non-commutative resolution of the conifold singularity {x₁ x₂ - x₃ x₄ = 0}⊂ C⁴. The proof does not require algebraic geometry; it uses a modified version of the domino (or dimer) shuffling algorithm of Elkies, Kuperberg, Larsen and Propp. In the second paper, we derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite abelian subgroup G of SO(3). These generating functions turn out to be orbifold Donaldson-Thomas partition functions for the orbifold [C³/G]. We need only the vertex operator methods of Okounkov-Reshetikhin-Vafa for the easy case G = Zn; to handle the considerably more difficult case G = Z₂ x Z₂ , we will also use a refinement of the author's recent q-enumeration of pyramid partitions. In the appendix, written by Jim Bryan, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold [C³/G]. We find a relationship between the Donaldson-Thomas partition functions of the orbifold and its G-Hilbert scheme resolution. We formulate a crepant resolution conjecture for the Donaldson-Thomas theory of local orbifolds satisfying the Hard Lefschetz condition."],"dc:format":["1356991","application/pdf"],"dc:identifier":["http://hdl.handle.net/2429/731","http://circle.library.ubc.ca/bitstream/2429/731/1/ubc_2008_spring_young_benjamin.pdf"],"dc:language":["eng"],"dc:publisher":["University of British Columbia"],"dc:rights":["http://creativecommons.org/licenses/by-nc-nd/4.0/","Attribution-NonCommercial-NoDerivatives 4.0 International"],"dc:title":["Counting coloured boxes"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy - PhD"],"thesis:institution_name":["University of British Columbia"]},"updated_at":"2026-07-24T05:07:11Z"}