{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/152683"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/152683","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Koszul duality and the bar spectral sequence","abstract":"The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript ∞-algebras yields the structure of operations on mod p Topological André-Quillen cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record an attempt with Andrew Senger on detecting these differentials via deformation of the bar comonad.","abstract_html":"The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript ∞-algebras yields the structure of operations on mod p Topological André-Quillen cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record an attempt with Andrew Senger on detecting these differentials via deformation of the bar comonad.","abstract_has_math":false,"creators":["Zhang, Adela YiYu"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Mathematics","school":null,"contributors":[],"advisors":["Miller, Haynes R."],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-09","date_published":"2023-09","updated_at":"2026-07-22T22:21:22Z","subjects":[],"languages":[],"rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"rights_urls":["https://rightsstatements.org/page/InC-EDU/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1721.1/152683","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Miller, Haynes R."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript ∞-algebras yields the structure of operations on mod p Topological André-Quillen cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record an attempt with Andrew Senger on detecting these differentials via deformation of the bar comonad."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Koszul duality and the bar spectral sequence"]}]}],"canonical_facts":{"dc:contributor.advisor":["Miller, Haynes R."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Zhang, Adela YiYu"],"dc:date.accessioned":["2023-11-02T20:08:15Z"],"dc:date.available":["2023-11-02T20:08:15Z"],"dc:date.issued":["2023-09"],"dc:description.abstract":["The bar spectral sequence for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript ∞-algebras yields the structure of operations on mod p Topological André-Quillen cohomology and the homotopy groups of spectral partition Lie algebras, building on the work of Brantner-Mathew. In the colimit, the unary operations are Koszul dual to the Dyer-Lashof algebra. On the other hand, the bar construction against certain spectral Lie algebras models labeled configuration spaces by a theorem of Knudsen. The associated bar spectral sequence yields new results on their mod p homology at low weights, as well as interesting patterns of universal differentials. We also record an attempt with Andrew Senger on detecting these differentials via deformation of the bar comonad."],"dc:description.degree":["Ph.D."],"dc:identifier.uri":["https://hdl.handle.net/1721.1/152683"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["In Copyright - Educational Use Permitted","Copyright retained by author(s)"],"dc:rights.uri":["https://rightsstatements.org/page/InC-EDU/1.0/"],"dc:title":["Koszul duality and the bar spectral sequence"],"dc:type":["Thesis"],"thesis:degree_name":["Doctoral","Doctor of Philosophy"]},"updated_at":"2026-07-22T22:21:22Z"}