Koszul duality and the bar spectral sequence
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zhang-adelayyz-phd-math-2023-thesis.pdf
Description
Thesis PDF
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793.76 KB
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cb3b4b38cf2706d22373d68241abb632
Author(s)
Zhang, Adela YiYu
Advisor(s)
Miller, Haynes R.
Date Issued
September 2023
Publisher
Massachusetts Institute of Technology
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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