LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute, UniversityofOxford,24–29StGiles,OxfordOX13LB,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressatwww.cambridge.org/mathematics. 182 GeometricgrouptheoryII, G.A.NIBLO&M.A.ROLLER(eds) 183 Shintanizetafunctions, A.YUKIE 184 Arithmeticalfunctions, W.SCHWARZ&J.SPILKER 185 Representationsofsolvablegroups, O.MANZ&T.R.WOLF 186 Complexity:knots,colouringsandcounting, D.J.A.WELSH 187 Surveysincombinatorics,1993, K.WALKER(ed) 188 Localanalysisfortheoddordertheorem, H.BENDER&G.GLAUBERMAN 189 Locallypresentableandaccessiblecategories, J.ADAMEK&J.ROSICKY 190 Polynomialinvariantsoffinitegroups, D.J.BENSON 191 Finitegeometryandcombinatorics, F.DECLERCKetal 192 Symplecticgeometry, D.SALAMON(ed) 194 Independentrandomvariablesandrearrangementinvariantspaces, M.BRAVERMAN 195 Arithmeticofblowupalgebras, W.VASCONCELOS 196 Microlocalanalysisfordifferentialoperators, A.GRIGIS&J.SJO¨STRAND 197 Two-dimensionalhomotopyandcombinatorialgrouptheory, C.HOG-ANGELONIetal 198 Thealgebraiccharacterizationofgeometric4-manifolds, J.A.HILLMAN 199 InvariantpotentialtheoryintheunitballofCn, M.STOLL 200 TheGrothendiecktheoryofdessinsd’enfant, L.SCHNEPS(ed) 201 Singularities, J.-P.BRASSELET(ed) 202 Thetechniqueofpseudodifferentialoperators, H.O.CORDES 203 HochschildcohomologyofvonNeumannalgebras, A.SINCLAIR&R.SMITH 204 Combinatorialandgeometricgrouptheory, A.J.DUNCAN,N.D.GILBERT&J.HOWIE(eds) 205 Ergodictheoryanditsconnectionswithharmonicanalysis, K.PETERSEN&I.SALAMA(eds) 207 GroupsofLietypeandtheirgeometries, W.M.KANTOR&L.DIMARTINO(eds) 208 Vectorbundlesinalgebraicgeometry, N.J.HITCHIN,P.NEWSTEAD&W.M.OXBURY(eds) 209 Arithmeticofdiagonalhypersurfacesoverinfitefields, F.Q.GOUVE´A&N.YUI 210 HilbertC∗-modules, E.C.LANCE 211 Groups93Galway/StAndrewsI, C.M.CAMPBELLetal(eds) 212 Groups93Galway/StAndrewsII, C.M.CAMPBELLetal(eds) 214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders, V.KOWALENKOetal 215 Numbertheory1992–93, S.DAVID(ed) 216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed) 217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER 218 Surveysincombinatorics,1995, P.ROWLINSON(ed) 220 Algebraicsettheory, A.JOYAL&I.MOERDIJK 221 Harmonicapproximation, S.J.GARDINER 222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds) 223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, KAZUAKITAIRA 224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds) 225 Amathematicalintroductiontostringtheory, S.ALBEVERIOetal 226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 227 Novikovconjectures,indextheoremsandrigidityII, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds) 229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK 230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN 231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds) 232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS 233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds) 234 Introductiontosubfactors, V.JONES&V.S.SUNDER 235 Numbertheory1993–94, S.DAVID(ed) 236 TheJamesforest, H.FETTER&B.G.DEBUEN 237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal 238 Representationtheoryandalgebraicgeometry, A.MARTSINKOVSKY&G.TODOROV(eds) 240 Stablegroups, F.O.WAGNER 241 Surveysincombinatorics,1997, R.A.BAILEY(ed) 242 GeometricGaloisactionsI, L.SCHNEPS&P.LOCHAK(eds) 243 GeometricGaloisactionsII, L.SCHNEPS&P.LOCHAK(eds) 244 Modeltheoryofgroupsandautomorphismgroups, D.EVANS(ed) 245 Geometry,combinatorialdesignsandrelatedstructures, J.W.P.HIRSCHFELDetal 246 p-Automorphismsoffinitep-groups, E.I.KHUKHRO 247 Analyticnumbertheory, Y.MOTOHASHI(ed) 248 Tametopologyando-minimalstructures, L.VANDENDRIES 249 Theatlasoffinitegroups:tenyearson, R.CURTIS&R.WILSON(eds) 250 Charactersandblocksoffinitegroups, G.NAVARRO 251 Gro¨bnerbasesandapplications, B.BUCHBERGER&F.WINKLER(eds) 252 Geometryandcohomologyingrouptheory, P.KROPHOLLER,G.NIBLO,R.STO¨HR(eds) 253 Theq-Schuralgebra, S.DONKIN 254 Galoisrepresentationsinarithmeticalgebraicgeometry, A.J.SCHOLL&R.L.TAYLOR(eds) 255 Symmetriesandintegrabilityofdifferenceequations, P.A.CLARKSON&F.W.NIJHOFF(eds) 256 AspectsofGaloistheory, H.VO¨LKLEINetal 257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE 258 Setsandproofs, S.B.COOPER&J.TRUSS(eds) 259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds) 260 GroupsStAndrews1997inBath,I, C.M.CAMPBELLetal 261 GroupsStAndrews1997inBath,II, C.M.CAMPBELLetal 262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL 263 Singularitytheory, B.BRUCE&D.MOND(eds) 264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds) 265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART 267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds) 268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND 269 Ergodictheoryandtopologicaldynamics, M.B.BEKKA&M.MAYER 270 AnalysisonLieGroups, N.T.VAROPOULOS&S.MUSTAPHA 271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV 272 Charactertheoryfortheoddorderfunction, T.PETERFALVI 273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds) 274 TheMandelbrotset,themeandvariations, TANLEI(ed) 275 Descriptivesettheoryanddynamicalsystems, M.FOREMANetal 276 Singularitiesofplanecurves, E.CASAS-ALVERO 277 Computationalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal 278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO 279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds) 280 CharactersandautomorphismgroupsofcompactRiemannsurfaces, T.BREUER 281 Explicitbirationalgeometryof3-folds, A.CORTI&M.REID(eds) 282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO 283 Nonlinearelasticity, Y.FU&R.OGDEN(eds) 284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds) 285 Rationalpointsoncurvesoverfinitefields, H.NIEDERREITER&C.XING 286 Cliffordalgebrasandspinors2ed, P.LOUNESTO 287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCEetal 288 Surveysincombinatorics,2001, J.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE 290 QuantumgroupsandLietheory, A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed) 292 Aquantumgroupsprimer, S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces, G.DAPRATO&J.ZABCZYK 294 Introductiontooperatorspacetheory, G.PISIER 295 GeometryandIntegrability, L.MASON&YAVUZNUTKU(eds) 296 Lecturesoninvarianttheory, I.DOLGACHEV 297 Thehomotopycategoryofsimplyconnected4-manifolds, H.-J.BAUES 298 Higheroperads,highercategories, T.LEINSTER 299 KleinianGroupsandHyperbolic3-Manifolds, Y.KOMORI,V.MARKOVIC&C.SERIES(eds) 300 IntroductiontoMo¨biusDifferentialGeometry, U.HERTRICH-JEROMIN 301 StableModulesandtheD(2)-Problem, F.E.A.JOHNSON 302 DiscreteandContinuousNonlinearSchro¨dingerSystems, M.J.ABLOWITZ,B.PRINARI&A.D.TRUBATCH 303 NumberTheoryandAlgebraicGeometry, M.REID&A.SKOROBOOATOV(eds) 304 GroupsStAndrews2001inOxfordVol.1, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 305 GroupsStAndrews2001inOxfordVol.2, C.M.CAMPBELL,E.F.ROBERTSON&G.C.SMITH(eds) 306 Peyresqlecturesongeometricmechanicsandsymmetry, J.MONTALDI&T.RATIU(eds) 307 SurveysinCombinatorics2003, C.D.WENSLEY(ed) 308 Topology,geometryandquantumfieldtheory, U.L.TILLMANN(ed) 309 CoringsandComdules, T.BRZEZINSKI&R.WISBAUER 310 TopicsinDynamicsandErgodicTheory, S.BEZUGLYI&S.KOLYADA(eds) 311 Groups:topological,combinatorialandarithmeticaspects, T.W.MU¨LLER(eds) 312 FoundationsofComputationalMathematics,Minneapolis2002, FELIPECUCKERetal(eds) 313 Transcendantalaspectsofalgebraiccycles, S.MU¨LLER-STACH&C.PETERS(eds) 314 Spectralgeneralizationsoflinegraphs, D.CVETKOVIC,P.ROWLINSON&S.SIMIC 315 Structuredringspectra, A.BAKER&B.RICHTER(eds) 317 Advancesinellipticcurvecryptography, I.F.BLAKE,G.SEROUSSI&N.SMART 318 Perturbationoftheboundaryinboundary-valueproblemsofPartialDifferentialEquations, DANHENRY 319 DoubleAffineHeckeAlgebras, I.CHEREDNIK 320 L-FunctionsandGaloisRepresentations, D.BURNS,K.BUZZARD&J.NEKOVA´Rˇ(eds) 321 SurveysinModernMathematics, V.PRASOLOV&Y.ILYASHENKO(eds) 322 Recentperspectivesinrandommatrixtheoryandnumbertheory, F.MEZZADRI,N.C.SNAITH(eds) 323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds) 324 SingularitiesandComputerAlgebra, C.LOSSEN&G.PFISTER(eds) 325 LecturesontheRicciFlow, P.TOPPING 326 ModularRepresentationsofFiniteGroupsofLieType, J.E.HUMPHREYS 328 FundamentalsofHyperbolicManifolds, R.D.CANARY,A.MARDEN&D.B.A.EPSTEIN(eds) 329 SpacesofKleinianGroups, Y.MINSKY,M.SAKUMA&C.SERIES(eds) 330 NoncommutativeLocalizationinAlgebraandTopology, A.RANICKI(ed) 331 FoundationsofComputationalMathematics,Santander2005, L.PARDO,A.PINKUS,E.SULI&M.TODD(eds) 332 HandbookofTiltingTheory, L.ANGELERIHU¨GEL,D.HAPPEL&H.KRAUSE(eds) 333 SyntheticDifferentialGeometry2ed, A.KOCK 334 TheNavier-StokesEquations, P.G.DRAZIN&N.RILEY 335 LecturesontheCombinatoricsofFreeProbability, A.NICA&R.SPEICHER 336 IntegralClosureofIdeals,Rings,andModules, I.SWANSON&C.HUNEKE 337 MethodsinBanachSpaceTheory, J.M.F.CASTILLO&W.B.JOHNSON(eds) 338 SurveysinGeometryandNumberTheory, N.YOUNG(ed) 339 GroupsStAndrews2005Vol.1, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 340 GroupsStAndrews2005Vol.2, C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 341 RanksofEllipticCurvesandRandomMatrixTheory, J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C.SNAITH(eds) 342 EllipticCohomology, H.R.MILLER&D.C.RAVENEL(eds) 343 AlgebraicCyclesandMotivesVol.1, J.NAGEL&C.PETERS(eds) 344 AlgebraicCyclesandMotivesVol.2, J.NAGEL&C.PETERS(eds) 345 AlgebraicandAnalyticGeometry, A.NEEMAN 346 SurveysinCombinatorics,2007, A.HILTON&J.TALBOT(eds) 347 SurveysinContemporaryMathematics, N.YOUNG&Y.CHOI(eds) LondonMathematicalSocietyLectureNoteSeries:349 Model Theory with Applications to Algebra and Analysis Volume 1 ZOE´ CHATZIDAKIS CNRS–Universite´ Paris7 DUGALD MACPHERSON UniversityofLeeds ANAND PILLAY UniversityofLeeds ALEX WILKIE UniversityofManchester CAMBRIDGE UNIVERSITY PRESS Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521694841 (cid:3)C CambridgeUniversityPress2008 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout 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Table of contents for Volume 1 Preface ix Contributors xiii Model theory and stability theory, with applications in differential algebra and algebraic geometry 1 Anand Pillay Differential algebra and generalizations of Grothendieck’s conjecture on the arithmetic of linear differential equations 25 Anand Pillay Schanuel’s conjecture for non-isoconstant elliptic curves over function fields 41 Daniel Bertrand An afterthought on the generalized Mordell-Lang conjecture 63 Damian R¨ossler On the definitions of difference Galois groups 73 Zo´e Chatzidakis, Charlotte Hardouin and Michael F. Singer Differentially valued fields are not differentially closed 111 Thomas Scanlon Complex analytic geometry in a nonstandard setting 117 Ya’acov Peterzil and Sergei Starchenko Model theory and K¨ahler geometry 167 Rahim Moosa and Anand Pillay v vi Some local definability theory for holomorphic functions 197 A.J. Wilkie Some observations about the real and imaginary parts of complex Pfaffian functions 215 Angus Macintyre Fusion of structures of finite Morley rank 225 Martin Ziegler Establishing the o-minimality for expansions of the real field 249 Jean-Philippe Rolin On the tomography theorem by P. Schapira 283 Sergei Starchenko A class of quantum Zariski geometries 293 Boris Zilber Model theory guidance in number theory? 327 Ivan Fesenko Table of contents for Volume 2 Preface ix Contributors xiii Conjugacy in groups of finite Morley rank 1 Olivier Fr´econ and Eric Jaligot Permutation groups of finite Morley rank 59 Alexandre Borovik and Gregory Cherlin A survey of asymptotic classes and measurable structures 125 Richard Elwes and Dugald Macpherson Counting and dimensions 161 Ehud Hrushovski and Frank Wagner A survey on groups definable in o-minimal structures 177 Margarita Otero vii viii Decision problems in Algebra and analogues of Hilbert’s tenth problem 207 Thanases Pheidas and Karim Zahidi Hilbert’s Tenth Problem for function fields of characteristic zero 237 Kirsten Eisentr¨ager First-order characterization of function field invariants over large fields 255 Bjorn Poonen and Florian Pop Nonnegative solvability of linear equations in ordered Abelian groups 273 Philip Scowcroft Model theory for metric structures 315 Ita¨ı Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov Preface These two volumes contain both expository and research papers in the generalareaofmodeltheoryanditsapplicationstoalgebraandanalysis. The volumes grew out of the semester on “Model Theory and Applica- tions to Algebra and Analysis” which took place at the Isaac Newton Institute(INI),Cambridge,fromJanuarytoJuly2005. We,theeditors, were also the organizers of the programme. The contributors have been selected from among the participants and their papers reflect many of the achievements and advances obtained during the programme. Also someoftheexpositorypapersarebasedontutorialsgivenattheMarch- April2005trainingworkshop. Wetakethisopportunity,bothaseditors of these volumes and organizers of the MAA programme, to thank the Isaac Newton Institute and its staff for supporting our programme and providing a perfect environment for mathematical research and collabo- ration. The INI semester saw activity and progress in essentially all areas on the “applied” side of model theory: o-minimality, motivic integration, groups of finite Morley rank, and connections with number theory and geometry. With the exception of motivic integration and valued fields, these topics are well represented in the two volumes. The collection of papers is more or less divided into (overlapping) themes, together with a few singularities. Aspects of the interaction betweenstabilitytheory, differentialanddifferenceequations, andnum- ber theory, appear in the first six papers of volume I. The first paper, basedonPillay’sworkshoptutorial,canalsoserveasafastintroduction to model theory for the general reader, although it quickly moves to an account of Mordell-Lang for function fields in characteristic 0. The ix