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Wayne State University

1. Zabka, Matthew John. Cohomology Operations On Random Spaces.

Degree: PhD, Mathematics, 2016, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/1500

Topology has recently received more attention from statisticians as some its tools have been applied to understanding the shape of data. In particular, a data set can generate a topological space, and this space’s topological structure can give us insight into some properties of the data. This framework has made it necessary to study random spaces generated by data. For example, without an understanding of the probabilistic properties of random spaces, one cannot conclude with any degree of confidence what the tools of topology tell us about a data set. While some results are known about the cohomological structure of a random space, not much is known about how cohomology operations behave on random spaces. This dissertation proves some results about the asymptotic properties of cohomology operations on random spaces and discusses the idea of a random Bockstein operation in a related purely algebraic context.
*Advisors/Committee Members: Daniel C. Isaksen.*

Subjects/Keywords: Bockstein; Cohomology operations; Mathematics; Statistics and Probability

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Zabka, M. J. (2016). Cohomology Operations On Random Spaces. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/1500

Chicago Manual of Style (16^{th} Edition):

Zabka, Matthew John. “Cohomology Operations On Random Spaces.” 2016. Doctoral Dissertation, Wayne State University. Accessed October 24, 2020. https://digitalcommons.wayne.edu/oa_dissertations/1500.

MLA Handbook (7^{th} Edition):

Zabka, Matthew John. “Cohomology Operations On Random Spaces.” 2016. Web. 24 Oct 2020.

Vancouver:

Zabka MJ. Cohomology Operations On Random Spaces. [Internet] [Doctoral dissertation]. Wayne State University; 2016. [cited 2020 Oct 24]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1500.

Council of Science Editors:

Zabka MJ. Cohomology Operations On Random Spaces. [Doctoral Dissertation]. Wayne State University; 2016. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1500

2. Gheorghe, Bogdan. The Motivic Cofiber Of Τ And Exotic Periodicities.

Degree: PhD, Mathematics, 2017, Wayne State University

URL: https://digitalcommons.wayne.edu/oa_dissertations/1804

Consider the Tate twist τ ∈ H 0,1 (S 0,0 ) in the mod 2 cohomology of the motivic sphere.
After 2-completion, the motivic Adams spectral sequence realizes this element as a map
τ : S 0,−1 GGA S 0,0 . This thesis begins with the study of its cofiber, that we denote by Cτ.
We first show that this motivic 2-cell complex can be endowed with a unique E ∞ ring
structure. This promotes the known isomorphism π ∗,∗ Cτ ∼= Ext ∗,∗ BP ∗ BP (BP ∗ ,BP ∗ )
to an isomorphism of rings which also preserves higher products.
This structure allows us to consider its closed symmetric monoidal category of modules
( Cτ Mod,− ∧ Cτ −), which happens to live in the kernel of Betti realization. This category
has surprising applications, and moreover contains many interesting motivic spectra. In
particular, we construct exotic motivic fields K(w n ), detecting motivic w n -periodicity. This
theory of motivic w n -periodicity can be roughly seen as perpendicular to the v n -periodicity
story, detected by the motivic Morava K-theories K(n). Finally, we also explain why the category
Cτ Mod is so computable. The above isomor phism comes in a more structured version.
In work that is joint with Zhouli Xu and Guozhen Wang, we show that there is an equivalence
of ∞-categories D b ( MGL ∗,∗ MGL Comod ev ) ∼= GGGA Cτ Cell comp
between an algebraic derived category, and the subcategory Cτ Cell comp of cellular Cτ-
modules that are complete with respect to a version of the algebraic cobordism spectrum MGL.
*Advisors/Committee Members: Daniel C. Isaksen.*

Subjects/Keywords: chromatic; homotopy; motivic; periodic; tau; triangulated; Mathematics

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Gheorghe, B. (2017). The Motivic Cofiber Of Τ And Exotic Periodicities. (Doctoral Dissertation). Wayne State University. Retrieved from https://digitalcommons.wayne.edu/oa_dissertations/1804

Chicago Manual of Style (16^{th} Edition):

Gheorghe, Bogdan. “The Motivic Cofiber Of Τ And Exotic Periodicities.” 2017. Doctoral Dissertation, Wayne State University. Accessed October 24, 2020. https://digitalcommons.wayne.edu/oa_dissertations/1804.

MLA Handbook (7^{th} Edition):

Gheorghe, Bogdan. “The Motivic Cofiber Of Τ And Exotic Periodicities.” 2017. Web. 24 Oct 2020.

Vancouver:

Gheorghe B. The Motivic Cofiber Of Τ And Exotic Periodicities. [Internet] [Doctoral dissertation]. Wayne State University; 2017. [cited 2020 Oct 24]. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1804.

Council of Science Editors:

Gheorghe B. The Motivic Cofiber Of Τ And Exotic Periodicities. [Doctoral Dissertation]. Wayne State University; 2017. Available from: https://digitalcommons.wayne.edu/oa_dissertations/1804