A parameterized adams spectral sequence for applied topology

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Given an environment that changes over time, under what conditions can it be navigated? From a prescient AI seeking strategies for Frogger to museum thieves trying to avoid the Night Watch (with The Night Watch safely in tow), this is a present and interesting mathematical question. In topological terms, we are interested in the space of sections of a map of spaces X -> I. More broadly, we are interested in describing and computing good invariants for parameterized spaces, which is to say, spaces X equipped with a map to a base space B. We record a number of invariants for such spaces, in the form of sheaves on the base space B, and discuss how these invariants can help us answer the original problem of navigating a changing space. Our main result is the construction of a Bousfield--Kan spectral sequence to aid in computing the "best" of our invariants, the parameterized homotopy groups of a parameterized space. We reinterpret this as an unstable Adams-type spectral sequence, and then give some computations which, in nice cases, computes the homotopy group of the space of sections of a map.


Type of resource text
Form electronic resource; remote; computer; online resource
Extent 1 online resource.
Place California
Place [Stanford, California]
Publisher [Stanford University]
Copyright date 2023; ©2023
Publication date 2023; 2023
Issuance monographic
Language English


Author Mackey, Wyatt Tyrrell
Degree supervisor Carlsson, G. (Gunnar), 1952-
Degree supervisor Vakil, Ravi
Thesis advisor Carlsson, G. (Gunnar), 1952-
Thesis advisor Vakil, Ravi
Thesis advisor Manolescu, Ciprian, 1978-
Degree committee member Manolescu, Ciprian, 1978-
Associated with Stanford University, School of Humanities and Sciences
Associated with Stanford University, Department of Mathematics


Genre Theses
Genre Text

Bibliographic information

Statement of responsibility Wyatt Mackey.
Note Submitted to the Department of Mathematics.
Thesis Thesis Ph.D. Stanford University 2023.
Location https://purl.stanford.edu/qp579zt0968

Access conditions

© 2023 by Wyatt Tyrrell Mackey
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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