{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/132574"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/132574","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Categorification of the Kirwan map","abstract":"Given a Hamiltonian action of a group on a scheme we can consider the Hamiltonian reduction. This furnishes us with a notion of quotient. From the construction it follows that we have a map from the equivariant cohomology of the scheme to the cohomology of the quotient. It is natural to ask if this map is surjective. This question is known as Kirwan surjectivity. There is much literature surrounding this question but the exact boundary of when Kirwan surjectivity holds is unknown. The document that follows explains a method for approaching the closely related question of Borel-Moore Kirwan surjectivity. The approach relies on deep results from derived algebraic geometry. Along the way it is shown that the equivariant category of D-modules with certain support condition is equivalent to the category of D-modules on the quotient. It is then shown that we can associate particular equivariant D-modules to characters of the group and these equivariant D-modules are compatible in a natural sense with the well studied equivariant O-modules associated to characters. This gives a framework for an approach to Borel-Moore Kirwan surjectivity. A result of McBreen and Webster immediately allows us to realise that Borel-Moore Kirwan surjectivity holds in the case of hypertorics.","abstract_html":"Given a Hamiltonian action of a group on a scheme we can consider the Hamiltonian reduction. This furnishes us with a notion of quotient. From the construction it follows that we have a map from the equivariant cohomology of the scheme to the cohomology of the quotient. It is natural to ask if this map is surjective. This question is known as Kirwan surjectivity. There is much literature surrounding this question but the exact boundary of when Kirwan surjectivity holds is unknown. The document that follows explains a method for approaching the closely related question of Borel-Moore Kirwan surjectivity. The approach relies on deep results from derived algebraic geometry. Along the way it is shown that the equivariant category of D-modules with certain support condition is equivalent to the category of D-modules on the quotient. It is then shown that we can associate particular equivariant D-modules to characters of the group and these equivariant D-modules are compatible in a natural sense with the well studied equivariant O-modules associated to characters. This gives a framework for an approach to Borel-Moore Kirwan surjectivity. A result of McBreen and Webster immediately allows us to realise that Borel-Moore Kirwan surjectivity holds in the case of hypertorics.","abstract_has_math":false,"creators":["O'Neill, Ciaran"],"institution":"University of Illinois Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dodd, Christopher","Katz, Sheldon","Pascaleff, James","Berwick-Evans, Daniel"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-22T22:25:07Z","subjects":["Kirwan surjectivity","Hochshild homology","D-modules,"],"languages":["en"],"rights":["Copyright 2025 Ciaran O'Neill"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/132574","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dodd, Christopher","Katz, Sheldon","Pascaleff, James","Berwick-Evans, Daniel"]},{"key":"dc:creator","label":"Author","values":["O'Neill, Ciaran"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12","2025-12-04"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Kirwan surjectivity","Hochshild homology","D-modules,"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2025 Ciaran O'Neill"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/132574"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Given a Hamiltonian action of a group on a scheme we can consider the Hamiltonian reduction. This furnishes us with a notion of quotient. From the construction it follows that we have a map from the equivariant cohomology of the scheme to the cohomology of the quotient. It is natural to ask if this map is surjective. This question is known as Kirwan surjectivity. There is much literature surrounding this question but the exact boundary of when Kirwan surjectivity holds is unknown. The document that follows explains a method for approaching the closely related question of Borel-Moore Kirwan surjectivity. The approach relies on deep results from derived algebraic geometry. Along the way it is shown that the equivariant category of D-modules with certain support condition is equivalent to the category of D-modules on the quotient. It is then shown that we can associate particular equivariant D-modules to characters of the group and these equivariant D-modules are compatible in a natural sense with the well studied equivariant O-modules associated to characters. This gives a framework for an approach to Borel-Moore Kirwan surjectivity. A result of McBreen and Webster immediately allows us to realise that Borel-Moore Kirwan surjectivity holds in the case of hypertorics.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Ciaran O'Neill, accepted the attached license on 2025-12-04 at 13:16.","The student, Ciaran O'Neill, submitted this Dissertation for approval on 2025-12-04 at 13:24.","This Dissertation was approved for publication on 2025-12-04 at 16:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23061 on 2026-02-19 at 18:29:12"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Categorification of the Kirwan map"]}]}],"canonical_facts":{"dc:contributor":["Dodd, Christopher","Katz, Sheldon","Pascaleff, James","Berwick-Evans, Daniel"],"dc:creator":["O'Neill, Ciaran"],"dc:date":["2025-12","2025-12-04"],"dc:description":["Given a Hamiltonian action of a group on a scheme we can consider the Hamiltonian reduction. This furnishes us with a notion of quotient. From the construction it follows that we have a map from the equivariant cohomology of the scheme to the cohomology of the quotient. It is natural to ask if this map is surjective. This question is known as Kirwan surjectivity. There is much literature surrounding this question but the exact boundary of when Kirwan surjectivity holds is unknown. The document that follows explains a method for approaching the closely related question of Borel-Moore Kirwan surjectivity. The approach relies on deep results from derived algebraic geometry. Along the way it is shown that the equivariant category of D-modules with certain support condition is equivalent to the category of D-modules on the quotient. It is then shown that we can associate particular equivariant D-modules to characters of the group and these equivariant D-modules are compatible in a natural sense with the well studied equivariant O-modules associated to characters. This gives a framework for an approach to Borel-Moore Kirwan surjectivity. A result of McBreen and Webster immediately allows us to realise that Borel-Moore Kirwan surjectivity holds in the case of hypertorics.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms","The student, Ciaran O'Neill, accepted the attached license on 2025-12-04 at 13:16.","The student, Ciaran O'Neill, submitted this Dissertation for approval on 2025-12-04 at 13:24.","This Dissertation was approved for publication on 2025-12-04 at 16:03.","DSpace SAF Submission Ingestion Package generated from Vireo submission #23061 on 2026-02-19 at 18:29:12"],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/132574"],"dc:language":["en"],"dc:rights":["Copyright 2025 Ciaran O'Neill"],"dc:subject":["Kirwan surjectivity","Hochshild homology","D-modules,"],"dc:title":["Categorification of the Kirwan map"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:07Z"}