{"id":"https://openalex.org/W2104375826","doi":"https://doi.org/10.2178/bsl/1102083758","title":"Computability Theory and Differential Geometry","display_name":"Computability Theory and Differential Geometry","publication_year":2004,"publication_date":"2004-12-01","ids":{"openalex":"https://openalex.org/W2104375826","doi":"https://doi.org/10.2178/bsl/1102083758","mag":"2104375826"},"language":"en","primary_location":{"id":"doi:10.2178/bsl/1102083758","is_oa":true,"landing_page_url":"https://doi.org/10.2178/bsl/1102083758","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4743592E2DCEA0CD3AB8D20372F10C6D/S1079898600003589a.pdf/div-class-title-computability-theory-and-differential-geometry-div.pdf","source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bulletin of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4743592E2DCEA0CD3AB8D20372F10C6D/S1079898600003589a.pdf/div-class-title-computability-theory-and-differential-geometry-div.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023422267","display_name":"Robert I. Soare","orcid":null},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]},{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Robert I. Soare","raw_affiliation_strings":["Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1546, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1546, E-mail:","institution_ids":["https://openalex.org/I40347166","https://openalex.org/I39422238"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5023422267"],"corresponding_institution_ids":["https://openalex.org/I39422238","https://openalex.org/I40347166"],"apc_list":null,"apc_paid":null,"fwci":5.0876,"has_fulltext":false,"cited_by_count":41,"citation_normalized_percentile":{"value":0.96218227,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","issue":"4","first_page":"457","last_page":"486"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9994000196456909,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7267410159111023},{"id":"https://openalex.org/keywords/diffeomorphism","display_name":"Diffeomorphism","score":0.6101878881454468},{"id":"https://openalex.org/keywords/riemannian-manifold","display_name":"Riemannian manifold","score":0.596373438835144},{"id":"https://openalex.org/keywords/sectional-curvature","display_name":"Sectional curvature","score":0.5590704679489136},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.4592359960079193},{"id":"https://openalex.org/keywords/submanifold","display_name":"Submanifold","score":0.4574209749698639},{"id":"https://openalex.org/keywords/homotopy","display_name":"Homotopy","score":0.4535267949104309},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4357134699821472},{"id":"https://openalex.org/keywords/homology","display_name":"Homology (biology)","score":0.4304737150669098},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.39502251148223877},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.39139482378959656},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.35263514518737793},{"id":"https://openalex.org/keywords/scalar-curvature","display_name":"Scalar curvature","score":0.3103278875350952}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7267410159111023},{"id":"https://openalex.org/C47556283","wikidata":"https://www.wikidata.org/wiki/Q1058314","display_name":"Diffeomorphism","level":2,"score":0.6101878881454468},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.596373438835144},{"id":"https://openalex.org/C42448751","wikidata":"https://www.wikidata.org/wiki/Q855622","display_name":"Sectional curvature","level":4,"score":0.5590704679489136},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.4592359960079193},{"id":"https://openalex.org/C151300846","wikidata":"https://www.wikidata.org/wiki/Q125539","display_name":"Submanifold","level":2,"score":0.4574209749698639},{"id":"https://openalex.org/C5961521","wikidata":"https://www.wikidata.org/wiki/Q746083","display_name":"Homotopy","level":2,"score":0.4535267949104309},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4357134699821472},{"id":"https://openalex.org/C165525559","wikidata":"https://www.wikidata.org/wiki/Q224180","display_name":"Homology (biology)","level":3,"score":0.4304737150669098},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.39502251148223877},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39139482378959656},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.35263514518737793},{"id":"https://openalex.org/C12520029","wikidata":"https://www.wikidata.org/wiki/Q1147161","display_name":"Scalar curvature","level":3,"score":0.3103278875350952},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.2178/bsl/1102083758","is_oa":true,"landing_page_url":"https://doi.org/10.2178/bsl/1102083758","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4743592E2DCEA0CD3AB8D20372F10C6D/S1079898600003589a.pdf/div-class-title-computability-theory-and-differential-geometry-div.pdf","source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bulletin of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.bsl/1102083758","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.bsl/1102083758","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.110.3793","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.3793","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.uchicago.edu/~soare/Publications/geom.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.119.515","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.119.515","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.cs.uchicago.edu/~soare/Publications/geom.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.120.6753","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.120.6753","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://people.cs.uchicago.edu/~soare/References/geom.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.77.359","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.77.359","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.people.cs.uchicago.edu/~soare/res/Geometry/geom.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.2178/bsl/1102083758","is_oa":true,"landing_page_url":"https://doi.org/10.2178/bsl/1102083758","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/4743592E2DCEA0CD3AB8D20372F10C6D/S1079898600003589a.pdf/div-class-title-computability-theory-and-differential-geometry-div.pdf","source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Bulletin of Symbolic Logic","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2104375826.pdf","grobid_xml":"https://content.openalex.org/works/W2104375826.grobid-xml"},"referenced_works_count":60,"referenced_works":["https://openalex.org/W182031189","https://openalex.org/W958315670","https://openalex.org/W1481268094","https://openalex.org/W1495768073","https://openalex.org/W1504831268","https://openalex.org/W1505284964","https://openalex.org/W1538946872","https://openalex.org/W1541824250","https://openalex.org/W1556167844","https://openalex.org/W1574266655","https://openalex.org/W1577258119","https://openalex.org/W1581416963","https://openalex.org/W1587069887","https://openalex.org/W1949813140","https://openalex.org/W1980124684","https://openalex.org/W1994737398","https://openalex.org/W1996294702","https://openalex.org/W1996511490","https://openalex.org/W2004193076","https://openalex.org/W2016474213","https://openalex.org/W2024071496","https://openalex.org/W2026119659","https://openalex.org/W2026429431","https://openalex.org/W2037469541","https://openalex.org/W2041603746","https://openalex.org/W2043554419","https://openalex.org/W2045292456","https://openalex.org/W2048519990","https://openalex.org/W2066635892","https://openalex.org/W2071250698","https://openalex.org/W2078200291","https://openalex.org/W2086448011","https://openalex.org/W2126160338","https://openalex.org/W2136003379","https://openalex.org/W2136111664","https://openalex.org/W2221528423","https://openalex.org/W2316365877","https://openalex.org/W2319243709","https://openalex.org/W2323777246","https://openalex.org/W2328824725","https://openalex.org/W2333555861","https://openalex.org/W2487893106","https://openalex.org/W2494377738","https://openalex.org/W2503148187","https://openalex.org/W2578936313","https://openalex.org/W2914873732","https://openalex.org/W2963452980","https://openalex.org/W3038420293","https://openalex.org/W3038588914","https://openalex.org/W3200414303","https://openalex.org/W4206033904","https://openalex.org/W4213077439","https://openalex.org/W4213199372","https://openalex.org/W4230191592","https://openalex.org/W4232277851","https://openalex.org/W4239662749","https://openalex.org/W4240542879","https://openalex.org/W4247266993","https://openalex.org/W4254784218","https://openalex.org/W6636667896"],"related_works":["https://openalex.org/W2302344580","https://openalex.org/W3023374613","https://openalex.org/W92362892","https://openalex.org/W2082355153","https://openalex.org/W1563501152","https://openalex.org/W4385061167","https://openalex.org/W3021699816","https://openalex.org/W4381586607","https://openalex.org/W1544477353","https://openalex.org/W2005999811"],"abstract_inverted_index":{"Abstract":[0],".":[1,163],"Let":[2,22],"M":[3,24,28,30,39,60,139],"be":[4,32],"a":[5,81,130,169,225,267],"smooth,":[6],"compact":[7],"manifold":[8],"of":[9,35,49,88,133,178,189,260,263,276,278,298,346],"dimension":[10],"n":[11,90,103,106],"\u2265":[12],"5":[13,372],"and":[14,43,116,126,218,233,241,248,280,301,349],"sectional":[15],"curvature":[16],"\u2223":[17,19],"K":[18,345],"\u2264":[20],"1.":[21],"Met(":[23,59,138],")":[25,31,61,87,140],"=":[26],"Riem(":[27],")/Diff(":[29],"the":[33,46,64,101,120,134,147,186,190,206,222,261,274,282,299,309,329,347,369],"space":[34,262],"Riemannian":[36,264],"metrics":[37,265],"on":[38,58,112,137,158,254,266],"modulo":[40],"isometries.":[41],"Nabutovsky":[42,217,247],"Weinberger":[44,219,249],"studied":[45],"connected":[47,121],"components":[48],"sublevel":[50],"sets":[51,180,293],"(and":[52],"local":[53,131,361],"minima)":[54],"for":[55,69,183,194,199,312,365],"certain":[56],"functions":[57],"such":[62,96],"as":[63,256,307],"diameter.":[65],"They":[66],"showed":[67,220],"that":[68,97,182,198,221,360],"every":[70],"Turing":[71,192],"machine":[72,193],"T":[73,109,156,371],"e":[74,76,86,110,152,157],",":[75,115,123,125,203],"\u03f5":[77],"\u03c9,":[78],"there":[79,355],"is":[80,98,127,143,208,356],"sequence":[82,172,331],"(uniformly":[83],"effective":[84],"in":[85,117,251,308],"homology":[89],"-spheres":[91],"which":[92],"are":[93,285],"also":[94],"hypersurfaces,":[95],"diffeomorphic":[99],"to":[100,146,271,294],"standard":[102],"-sphere":[104],"S":[105],"(denoted":[107],")iff":[108],"halts":[111],"input":[113],"k":[114,154,162],"this":[118],"case":[119],"sum":[122],"so":[124,181,242],"associated":[128,191],"with":[129,210,229],"minimum":[132],"diameter":[135],"function":[136],"whose":[141],"depth":[142],"roughly":[144],"equal":[145],"settling":[148,187],"time":[149,188],"ae":[150],"\u03c3":[151],"(":[153],")of":[155],"inputs":[159],"y":[160],"&lt;":[161],"At":[164],"their":[165,252],"request":[166],"Soare":[167,330],"constructed":[168],"particular":[170],"infinite":[171],"{":[173,332],"A":[174,195,200,333],"i":[175,185,196,201,334,336],"}":[176,335],"\u03f5\u03c9":[177,337],"c.e.":[179,292,327,342],"all":[184],"dominates":[197],"+1":[202],"even":[204,364],"when":[205],"latter":[207],"composed":[209],"an":[211],"arbitrary":[212],"computable":[213],"function.":[214],"From":[215,273],"this,":[216],"basins":[223,237],"exhibit":[224],"\u201cfractal\u201d":[226],"like":[227,368],"behavior":[228],"extremely":[230],"big":[231],"basins,":[232],"very":[234],"much":[235],"smaller":[236],"coming":[238],"off":[239],"them,":[240],"on.":[243],"This":[244],"reveals":[245],"what":[246],"describe":[250],"paper":[253],"fractals":[255],"\u201cthe":[257],"astonishing":[258],"richness":[259],"smooth":[268],"manifold,":[269],"up":[270],"reparametrization.\u201d":[272],"point":[275],"view":[277],"logic":[279],"computability,":[281],"Nabutovsky-Weinberger":[283],"results":[284,306],"especially":[286],"interesting":[287],"because:":[288],"(1)":[289],"they":[290,322],"use":[291,323],"prove":[295],"structural":[296],"complexity":[297],"geometry":[300],"topology,":[302],"not":[303,339],"merely":[304,340],"undecidability":[305],"word":[310],"problem":[311],"groups,":[313],"Hilbert's":[314],"Tenth":[315],"Problem,":[316],"or":[317],"most":[318],"other":[319],"applications;":[320],"(2)":[321],"nontrivial":[324],"information":[325],"about":[326],"sets,":[328],"above,":[338],"G\u00f6del's":[341],"noncomputable":[343],"set":[344],"1930's;":[348],"(3)":[350],"without":[351],"using":[352],"computability":[353],"theory":[354],"no":[357],"known":[358],"proof":[359],"minima":[362],"exist":[363],"simple":[366],"manifolds":[367],"torus":[370],"(see":[373],"\u00a79.5).":[374]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":4}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
