Études in Homotopical Thinking: F₁-geometry, Concurrent Computing, and Motivic Measures

Author: Lieber, Joshua Franklin

Year: 2021

Degree: Dissertation (Ph.D.)

Advisor: Marcolli, Matilde

Committee Members: Flach, Matthias; Marcolli, Matilde; Mantovan, Elena; Graber, Thomas B.

Option: Mathematics

DOI: 10.7907/a4zm-1f28

Abstract

This thesis weaves together three papers, each of which provides a use of homotopical intuition in a different field of mathematics. The first applies it to the study of various models of F₁-geometry, focusing mainly on the Bost-Connes algebra. The second endeavors to compare two homotopical models for concurrent computing before introducing a new one as well. Finally, the last paper provides a construction for obtaining derived motivic measures from an abstract six functors formalism and, in particular, applies this idea to obtain a lift of the Gillet-Soulé motivic measure.

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