{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/98343"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/98343","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Towards a model theory of logarithmic transseries","abstract":"DSpace SAF Submission Ingestion Package generated from Vireo submission #11336 on 2017-09-29 at 11:28:25","abstract_html":"DSpace SAF Submission Ingestion Package generated from Vireo submission #11336 on 2017-09-29 at 11:28:25","abstract_has_math":false,"creators":["Gehret, Allen R"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["van den Dries, Lou","Hieronymi, Philipp","Aschenbrenner, Matthias","Nevins, Thomas"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-09-29T17:56:31Z","date_published":"2017-09-29T17:56:31Z","updated_at":"2026-07-22T22:24:35Z","subjects":["Logarithmic transseries","Model theory"],"languages":["en"],"rights":["Copyright 2017 Allen Gehret"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/98343","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["van den Dries, Lou","Hieronymi, Philipp","Aschenbrenner, Matthias","Nevins, Thomas"]},{"key":"dc:creator","label":"Author","values":["Gehret, Allen R"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-09-29T17:56:31Z","2017-07-09","2017-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Logarithmic transseries","Model theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Allen Gehret"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/98343"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["DSpace SAF Submission Ingestion Package generated from Vireo submission #11336 on 2017-09-29 at 11:28:25","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"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Towards a model theory of logarithmic transseries"]}]}],"canonical_facts":{"dc:contributor":["van den Dries, Lou","Hieronymi, Philipp","Aschenbrenner, Matthias","Nevins, Thomas"],"dc:creator":["Gehret, Allen R"],"dc:date":["2017-09-29T17:56:31Z","2017-07-09","2017-08"],"dc:description":["DSpace SAF Submission Ingestion Package generated from Vireo submission #11336 on 2017-09-29 at 11:28:25","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"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/98343"],"dc:language":["en"],"dc:rights":["Copyright 2017 Allen Gehret"],"dc:subject":["Logarithmic transseries","Model theory"],"dc:title":["Towards a model theory of logarithmic transseries"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:35Z"}