{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/117873"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/117873","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Power operations and central maps in representation theory","abstract":"The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie.","abstract_html":"The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie.","abstract_has_math":false,"creators":["Lonergan, Gus (Gus C.)"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics.","school":null,"contributors":[],"advisors":["Roman Bezrukavnikov."],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018","date_published":"2018","updated_at":"2026-07-22T22:21:15Z","subjects":["Mathematics."],"languages":["eng"],"rights":["MIT theses are protected by copyright. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/117873"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 153-155)."]},{"key":"dc:description.abstract","label":"Abstract","values":["The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Power operations and central maps in representation theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Roman Bezrukavnikov."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Department of Mathematics."],"dc:creator":["Lonergan, Gus (Gus C.)"],"dc:date.accessioned":["2018-09-17T15:48:01Z"],"dc:date.available":["2018-09-17T15:48:01Z"],"dc:date.issued":["2018"],"dc:description":["Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 153-155)."],"dc:description.abstract":["The theme of this thesis is the novel application of techniques of algebraic topology (specifically, Steenrod's operations and Smith's localization theory) to representation theory (especially in the context of the geometric Satake equivalence). In Chapter 2, we use Steenrod's construction to prove that the quantum Coulomb branch is a Frobenius-constant quantization. We also demonstrate the corresponding result for the K-theoretic version of the quantum Coulomb branch. In Chapter 3, we develop the theory of parity sheaves with coefficients in the Tate spectrum, and use it to give a geometric construction of the Frobenius-contraction functor. In Chapter 4, we discuss some related results, including a geometric construction of the Frobenius twist functor, and also discuss future directions of research. The content of Chapter 3 is joint work with S. Leslie."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/117873"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Power operations and central maps in representation theory"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:15Z"}