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DSpace SAF Submission Ingestion Package generated from Vireo submission #11336 on 2017-09-29 at 11:28:25 van den Dries, Lou Hieronymi, Philipp Aschenbrenner, Matthias Nevins, Thomas Gehret, Allen R 2017-09-29T17:56:31Z 2017-09-29T17:56:31Z 2017-07-09 2017-08 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 $\Gamma_{\log}$ a certain map $\psi$; together forming the pair $(\Gamma_{\log},\psi)$, the \emph{asymptotic couple of $\mathbb{T}_{\log}$}. We study the asymptotic couple $(\Gamma_{\log},\psi)$ and show that it has a nice model theory. Among other things, we prove that $\Th(\Gamma_{\log},\psi)$ 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{$\Psi$-closed}. Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2017-09-29 without embargo terms The student, Allen Gehret, accepted the attached license on 2017-07-07 at 13:21. The student, Allen Gehret, submitted this Dissertation for approval on 2017-07-07 at 13:29. This Dissertation was approved for publication on 2017-07-09 at 14:05. Made available in DSpace on 2017-09-29T17:56:31Z (GMT). No. of bitstreams: 3 GEHRET-DISSERTATION-2017.pdf: 3034183 bytes, checksum: ebdd59fb46f86f1857b7a0a1688b85e0 (MD5) LICENSE.txt: 4209 bytes, checksum: b377477866b869fc4f8cc867290ce27d (MD5) PROQUEST_LICENSE.txt: 4555 bytes, checksum: 8651cd385cda9a676478aab82fd58996 (MD5) Previous issue date: 2017-07-09 application/pdf http://hdl.handle.net/2142/98343 en Copyright 2017 Allen Gehret Logarithmic transseries Model theory Towards a model theory of logarithmic transseries text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.