{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,4,18]],"date-time":"2024-04-18T21:11:54Z","timestamp":1713474714196},"reference-count":19,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":6529,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1996,3]]},"abstract":"<jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field \u211d (or any real closed field) where algebra alone determines the ordering and hence the topology of the field:<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S1079898600007939_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>In the case of the field \u2102, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but this will be too coarse to give a diferentiable structure.<\/jats:p><jats:p>A celebrated example of how partial algebraic and topological data (<jats:italic>G<\/jats:italic> a locally euclidean group) determines a differentiable structure (<jats:italic>G<\/jats:italic> is a Lie group) is Hilbert's 5th problem and its solution by Montgomery-Zippin and Gleason.<\/jats:p><jats:p>The main result which we discuss here (see [13] for the full version) is of a similar flavor: we recover an algebraic and later differentiable structure from a topological data. We begin with a linearly ordered set \u27e8<jats:italic>M<\/jats:italic>, &lt;\u27e9, equipped with the order topology, and its cartesian products with the product topologies. We then consider the collection of definable subsets of <jats:italic>M<jats:sup>n<\/jats:sup><\/jats:italic>, <jats:italic>n<\/jats:italic> = 1, 2, \u2026, in some first order expansion \u2133 of \u27e8<jats:italic>M<\/jats:italic>, &lt;\u27e9.<\/jats:p>","DOI":"10.2307\/421047","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:08:40Z","timestamp":1146985720000},"page":"72-83","source":"Crossref","is-referenced-by-count":4,"title":["Geometry, Calculus and Zil'ber's Conjecture"],"prefix":"10.1017","volume":"2","author":[{"given":"Ya'acov","family":"Peterzil","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergei","family":"Starchenko","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007939_ref014","volume-title":"An introduction to stability theory","author":"Pillay","year":"1983"},{"key":"S1079898600007939_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90171-9"},{"key":"S1079898600007939_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(94)90007-8"},{"key":"S1079898600007939_ref016","volume-title":"Definability of a field in sufficiently rich incidence systems","author":"Rabinovich","year":"1986"},{"key":"S1079898600007939_ref006","unstructured":"Hrushovski E. and Zil'ber B. , Zariski geometries, to appear."},{"key":"S1079898600007939_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761295"},{"key":"S1079898600007939_ref018","unstructured":"Wilkie A. , Model completeness results for expansions of the real field II: The exponential function, to appear."},{"key":"S1079898600007939_ref004","article-title":"Weakly normal groups","volume":"85","author":"Hrushovski","journal-title":"Logic Colloquium"},{"key":"S1079898600007939_ref017","unstructured":"van den Dries L. , Tame topology and o-minimal structures, preliminary version, 1991."},{"key":"S1079898600007939_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02758643"},{"key":"S1079898600007939_ref002","unstructured":"Hrushovski E. , Contributions to stable model theory, Ph.D. thesis , Berkeley, 1986."},{"key":"S1079898600007939_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833697-X"},{"key":"S1079898600007939_ref019","first-page":"115","volume-title":"Logic methodology and philosophy of science VII","author":"Zil'ber","year":"1986"},{"key":"S1079898600007939_ref001","unstructured":"Hrushovski E. , The Mordell-Lang conjecture for function fields, preprint."},{"key":"S1079898600007939_ref013","unstructured":"Peterzil Y. and Starchenko S. , A trichotomy theorem for o-minimal structures, preprint."},{"key":"S1079898600007939_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/2275179"},{"key":"S1079898600007939_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1986-0833698-1"},{"key":"S1079898600007939_ref012","unstructured":"Peterzil Y. , Pillay A. , and Starchenko S. , A classification of simple groups in o-minimal structures, in preparation."},{"key":"S1079898600007939_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-1993-00380-X"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600007939","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T22:13:27Z","timestamp":1557699207000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600007939\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,3]]},"references-count":19,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1996,3]]}},"alternative-id":["S1079898600007939"],"URL":"https:\/\/doi.org\/10.2307\/421047","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,3]]}}}