{"id":{"repo_id":"toronto-retro","oai_identifier":"oai:utoronto.scholaris.ca:1807/140996"},"canonical_url":"https://search.dev.ndltd.org/etd/toronto-retro/oai:utoronto.scholaris.ca:1807/140996","repository":{"repo_id":"toronto-retro","name":"University of Toronto","base_url":"https://utoronto.scholaris.ca/server/oai/request"},"display":{"title":"Forcing in Analysis and Combinatorics","abstract":"In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by: (1) Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy\\'nski and Judah. (2) Constructing a model of Set Theory in which Jensen's $\\diamondsuit_{\\aleph_1}$ fails, there is a counterexample to Naimark's Problem, and there is a separably represented $\\textrm{C}^\\ast$-algebra with exactly two inequivalent irreducible representations. Such a $\\textrm{C}^\\ast$-algebra cannot satisfy the conclusion of Glimm's Dichotomy Theorem. (3) Studying families of infinite block sequences of elements of the space $\\textrm{FIN}_k$, Ramsey properties of such families, and Ramsey properties localized on selective or semiselective coideals and ultrafilters.","abstract_html":"In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by: (1) Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy\\&#x27;nski and Judah. (2) Constructing a model of Set Theory in which Jensen&#x27;s <span class=\"etd-inline-math\">\\diamondsuit<sub>\\aleph<sub>1</sub></sub></span> fails, there is a counterexample to Naimark&#x27;s Problem, and there is a separably represented <span class=\"etd-inline-math\">\\textrm{C}<sup>\\</sup>ast</span>-algebra with exactly two inequivalent irreducible representations. Such a <span class=\"etd-inline-math\">\\textrm{C}<sup>\\</sup>ast</span>-algebra cannot satisfy the conclusion of Glimm&#x27;s Dichotomy Theorem. (3) Studying families of infinite block sequences of elements of the space <span class=\"etd-inline-math\">\\textrm{FIN}<sub>k</sub></span>, Ramsey properties of such families, and Ramsey properties localized on selective or semiselective coideals and ultrafilters.","abstract_has_math":true,"creators":["Calderon Wilches, Daniel"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Todorcevic, Stevo","Farah, Ilijas"],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-11","date_published":"2024-11","updated_at":"2026-07-27T21:28:13Z","subjects":["Combinatorics","Forcing","Functional Analysis","Operator Algebras","Ramsey Theory","Real Analysis"],"languages":[],"rights":["Attribution 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1807/140996","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Todorcevic, Stevo","Farah, Ilijas"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Calderon Wilches, Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2024-11"]},{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2024-11-13T18:12:20Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-11-13T18:12:20Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-11"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Combinatorics","Forcing","Functional Analysis","Operator Algebras","Ramsey Theory","Real Analysis"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["Attribution 4.0 International"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://creativecommons.org/licenses/by/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1807/140996"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by: (1) Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy\\'nski and Judah. (2) Constructing a model of Set Theory in which Jensen's $\\diamondsuit_{\\aleph_1}$ fails, there is a counterexample to Naimark's Problem, and there is a separably represented $\\textrm{C}^\\ast$-algebra with exactly two inequivalent irreducible representations. Such a $\\textrm{C}^\\ast$-algebra cannot satisfy the conclusion of Glimm's Dichotomy Theorem. (3) Studying families of infinite block sequences of elements of the space $\\textrm{FIN}_k$, Ramsey properties of such families, and Ramsey properties localized on selective or semiselective coideals and ultrafilters."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Forcing in Analysis and Combinatorics"]}]}],"canonical_facts":{"dc:contributor.advisor":["Todorcevic, Stevo","Farah, Ilijas"],"dc:contributor.department":["Mathematics"],"dc:creator":["Calderon Wilches, Daniel"],"dc:date":["2024-11"],"dc:date.accessioned":["2024-11-13T18:12:20Z"],"dc:date.available":["2024-11-13T18:12:20Z"],"dc:date.issued":["2024-11"],"dc:description.abstract":["In this thesis, we explore some new applications of the forcing technique in the context of Analysis and Combinatorics by: (1) Constructing a model of Set Theory in which strong measure zero subsets of the real line are meager-additive while Borel’s conjecture fails, answering a long-standing question due to Bartoszy\\'nski and Judah. 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