CRM Series in Mathematical Physics Decio Levi Raphaël Rebelo Pavel Winternitz Editors Symmetries and Integrability of Diff erence Equations Lecture Notes of the Abecederian School of SIDE 12, Montreal 2016 CRM Series in Mathematical Physics EditorialBoard: JoelS.Fieldman DepartmentofMathematics UniversityofBritishColumbia Vancouver,BritishColumbiaV6T1Z2 Canada [email protected] DuongH.Phong DepartmentofMathematics ColumbiaUniversity NewYork,NY10027–0029 USA [email protected] YvanSaint-Aubin DépartementdeMathématiques etdeStatistique UniversitédeMontréal C.P.6128,SuccursaleCentre-ville Montréal,QuébecH3C3J7 Canada [email protected] LucVinet DépartementdeMathématiques etdeStatistique CRM,UniversitédeMontréal C.P.6128,SuccursaleCentre-ville Montréal,QuébecH3C3J7 Canada [email protected] Moreinformationaboutthisseriesathttp://www.springer.com/series/3872 Decio Levi (cid:129) Raphaël Rebelo Pavel Winternitz Editors Symmetries and Integrability of Difference Equations Lecture Notes of the Abecederian School of SIDE 12, Montreal 2016 123 Editors DecioLevi RaphaëlRebelo MathematicsandPhysicsDepartments, CollègeAhuntsic andINFN,SezioneRomaTre Montréal,Québec RomaTreUniversity Canada Roma,Italy PavelWinternitz CentredeRecherchesMathématiques andDépartementdeMathématiques etdeStatistique UniversitédeMontréal Montréal,Québec Canada CRMSeriesinMathematicalPhysics ISBN978-3-319-56665-8 ISBN978-3-319-56666-5 (eBook) DOI10.1007/978-3-319-56666-5 LibraryofCongressControlNumber:2017941640 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface ThisvolumeisbasedonnineseriesoflecturespresentedattheAbecedarianSchool onSymmetriesandIntegrabilityofDifferenceEquations(ASIDE).Theschooltook placeattheCentrederecherchesmathématiques,UniversitédeMontréal,June27– July 1, 2016. It was followed by an international meeting on the same subject, SIDE12,heldinSainte-Adèle,Québec,Canada,July33–9,2016. The SIDE meetings take place every two years since 1994 and it has become a tradition that a school precedes these meetings. The school is aimed at graduate students,postdoctoralfellowsandotherearlycareerresearchers(ECRs)toprepare themforfullparticipationintheactualmeeting. Lecturersatsummerschoolsareusuallyfamousoratleastdistinguishedsenior scientists; the participants are usually ECRs. The ASIDE schools break with this tradition. The lecturers and the students at the school were ECRs. The idea, originally proposed by ECRs at the SIDE 8 meeting in 2008, was that this would encourage a more informal atmosphere, make it easier to ask questions and have in-depthdiscussions.ApreviousASIDEschooltogetherwiththeSIDE10meeting took place near Shanghai in 2012. This is the first time that ASIDE lectures are published.Theywerecarefullyrefereedbyexpertsinthefield. The topics treated in each of the nine lecture series coincide with those treated intheninesessionsofSIDE12.TheASIDElecturerswereproposedbythesession organizers and were usually current or former graduate students or postdoctoral fellows of the session organizers. Thus they all had in-depth and up-to-date knowledge of the field they were lecturing on. The fact that their knowledge was recently acquired meant that the lecturers were well aware of the difficulties they hadencountered.Thiswasagreatadvantagefromthepedagogicalpointofview. TheconsensuswasthattheASIDEexperimentwasasuccessfulone.Mostofthe participantsstayedfortheSIDE12meetingandcontributedtoandbenefittedfrom bothevents. Therewere95participantsatSIDE12,56atASIDEwithalargeoverlapbetween thetwo.BothSIDE12andASIDEweremadepossiblebygeneroussupportfrom theNSF,theCRMandtheUMI/CRNS. v vi Preface EachoftheselecturesisanintroductiontothecorrespondingSIDEsessionand wearesurethattheyareofpermanentvalue.Thevolumeshouldbeusefulbothto beginnersenteringintothefieldandtoseasonedpractitioners. Roma,Italy DecioLevi Montréal,QC,Canada RaphaëlRebelo Montréal,QC,Canada PavelWinternitz February2017 Contents Continuous,DiscreteandUltradiscretePainlevéEquations................ 1 NobutakaNakazono,YangShi,andMasatakaKanki EllipticHypergeometricFunctions............................................. 43 FokkoJ.vandeBult IntegrabilityofDifferenceEquationsThroughAlgebraicEntropy andGeneralizedSymmetries ................................................... 75 GiorgioGubbiotti Introduction to Linear and Nonlinear Integrable Theories inDiscreteComplexAnalysis................................................... 153 UlrikeBücking Discrete Integrable Systems, Darboux Transformations, andYang–BaxterMaps.......................................................... 195 DenizBilmanandSotirisKonstantinou-Rizos Symmetry-PreservingNumericalSchemes.................................... 261 AlexanderBihloandFrancisValiquette IntroductiontoClusterAlgebras............................................... 325 MaxGlickandDylanRupel AnIntroductiontoDifferenceGaloisTheory................................. 359 JulienRoques LecturesonQuantumIntegrability:Lattices,SymmetriesandPhysics... 391 MiłoszPanfil Index............................................................................... 431 vii Contributors AlexanderBihlo DepartmentofMathematicsandStatistics,MemorialUniversity ofNewfoundland,St.John’s,NL,Canada Deniz Bilman Department of Mathematics, University of Michigan, Ann Arbor, MI,USA Ulrike Bücking Institut für Mathematik, Technische Universität Berlin, Berlin, Germany Max Glick Department of Mathematics, University of Connecticut, Storrs, CT, USA Giorgio Gubbiotti Dipartimento di Matematica e Fisica, Universitàa˘ degli Studi RomaTre,Roma,Italy MasatakaKanki FacultyofEngineeringScience,KansaiUniversity,Suita,Osaka, Japan Sotiris Konstantinou-Rizos Institute of Mathematical Physics & Seismodynam- ics,Grozny,Russia Nobutaka Nakazono Department of Physics and Mathematics, Aoyama Gakuin University,Sagamihara,Kanagawa,Japan MiłoszPanfil FacultyofPhysics,UniversityofWarsaw,Warszawa,Poland Julien Roques Institut Fourier, Université Grenoble Alpes, UMR 5582 CNRS- UGA,Gières,France DylanRupel DepartmentofMathematics,UniversityofNotreDame,NotreDame, IN,USA YangShi SchoolofMathematicsandStatistics,TheUniversityofSydney,Sydney, NSW,Australia ix x Contributors FrancisValiquette DepartmentofMathematics,StateUniversityofNewYorkat NewPaltz,NewPaltz,NY,USA Fokko J. van de Bult Department of Applied Mathematics, Delft University of Technology,CDDelft,TheNetherlands