2026-09-19T20:25:11Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1269412026-06-16T18:52:57Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Jacob Lurie.
Yuan, Allen,Ph. D.Massachusetts Institute of Technology.
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2020-09-03T16:42:34Z
2020-09-03T16:42:34Z
2020
2020
https://hdl.handle.net/1721.1/126941
1191268053
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 107-110).
Given a homotopy invariant of a space, one can ask how much of the space can be recovered from that invariant. This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960's and in work of Dwyer-Hopkins and Mandell on p-adic homotopy theory in the 1990's. In this thesis, we describe a way to unify these ideas and recover a space in its entirety, rather than up to an approximation. The approach is centered around the study of the higher Frobenius map. First defined by Nikolaus and Scholze, the higher Frobenius map generalizes to E[subscript infinity]-ring spectra the classical Frobenius endomorphism for rings in characteristic p. Our main result is that there is an action of the circle group on (a certain subcategory of) p-complete [subscript infinity]-rings whose monodromy is the higher Frobenius. Using this circle action, we give a fully faithful model for a simply connected finite complex X in terms of Frobenius-fixed [subscript infinity]-rings.
by Allen Yuan.
Ph.D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
110 pages
eng
Massachusetts Institute of Technology
MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
On the higher Frobenius
Thesis
application/pdf
2020-09-03T16:42:33Z
Publication
Doctoral
Math
unknown
unknown
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<Abstract>Given a homotopy invariant of a space, one can ask how much of the space can be recovered from that invariant. This question was first addressed in work of Quillen and Sullivan on rational homotopy theory in the 1960's and in work of Dwyer-Hopkins and Mandell on p-adic homotopy theory in the 1990's. In this thesis, we describe a way to unify these ideas and recover a space in its entirety, rather than up to an approximation. The approach is centered around the study of the higher Frobenius map. First defined by Nikolaus and Scholze, the higher Frobenius map generalizes to E[subscript infinity]-ring spectra the classical Frobenius endomorphism for rings in characteristic p. Our main result is that there is an action of the circle group on (a certain subcategory of) p-complete [subscript infinity]-rings whose monodromy is the higher Frobenius. Using this circle action, we give a fully faithful model for a simply connected finite complex X in terms of Frobenius-fixed [subscript infinity]-rings.</Abstract>
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