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University of Illinois – Urbana-Champaign

1. Gehret, Allen R. Towards a model theory of logarithmic transseries.

Degree: PhD, Mathematics, 2017, University of Illinois – Urbana-Champaign

URL: http://hdl.handle.net/2142/98343

The ordered valued differential field \mathbb{T}_{log} of logarithmic transseries is conjectured to have good model theoretic properties. This thesis records our progress in this direction and describes a strategy moving forward. As a first step, we turn our attention to the value group of \mathbb{T}_{log}. The derivation on \mathbb{T}_{log} induces on its value group Γ_{log} a certain map ψ; together forming the pair (Γ_{log},ψ), the \emph{asymptotic couple of \mathbb{T}_{log}}. We study the asymptotic couple (Γ_{log},ψ) and show that it has a nice model theory. Among other things, we prove that \Th(Γ_{log},ψ) has elimination of quantifiers in a natural language, is model complete, and has the non-independence property (NIP). As a byproduct of our work, we also give a complete characterization of when an H-field has exactly one or exactly two Liouville closures. Finally, we present an outline for proving a model completeness result for \mathbb{T}_{log} in a reasonable language. In particular, we introduce and study the notion of \emph{\LD-fields} and also the property of a differentially-valued field being \emph{Ψ-closed}.
*Advisors/Committee Members: van den Dries, Lou (advisor), Hieronymi, Philipp (Committee Chair), Aschenbrenner, Matthias (committee member), Nevins, Thomas (committee member).*

Subjects/Keywords: Logarithmic transseries; Model theory

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APA (6^{th} Edition):

Gehret, A. R. (2017). Towards a model theory of logarithmic transseries. (Doctoral Dissertation). University of Illinois – Urbana-Champaign. Retrieved from http://hdl.handle.net/2142/98343

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

Gehret, Allen R. “Towards a model theory of logarithmic transseries.” 2017. Doctoral Dissertation, University of Illinois – Urbana-Champaign. Accessed July 02, 2020. http://hdl.handle.net/2142/98343.

MLA Handbook (7^{th} Edition):

Gehret, Allen R. “Towards a model theory of logarithmic transseries.” 2017. Web. 02 Jul 2020.

Vancouver:

Gehret AR. Towards a model theory of logarithmic transseries. [Internet] [Doctoral dissertation]. University of Illinois – Urbana-Champaign; 2017. [cited 2020 Jul 02]. Available from: http://hdl.handle.net/2142/98343.

Council of Science Editors:

Gehret AR. Towards a model theory of logarithmic transseries. [Doctoral Dissertation]. University of Illinois – Urbana-Champaign; 2017. Available from: http://hdl.handle.net/2142/98343