The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein

665 Pages·2004·5.8 MB·German

by  Jerrold E. Marsden

Progress in Mathematics
Volume232
SeriesEditors
HymanBass
JosephOesterle´
AlanWeinstein
The Breadth of Symplectic
and Poisson Geometry
Festschrift in Honor of Alan Weinstein
Jerrold E. Marsden
Tudor S. Ratiu
Editors
Birkha¨user
Boston • Basel • Berlin
JerroldE.Marsden TudorS.Ratiu
CaliforniaInstituteofTechnology EcolePolytechniqueFe´de´raledeLausanne
DepartmentofEngineering De´partementdeMathe´matiques
andAppliedScience CH-1015Lausanne
ControlandDynamicalSystems Switzerland
Pasadena,CA91125
U.S.A.
AMSSubjectClassifications:53Dxx,17Bxx,22Exx,53Dxx,81Sxx
LibraryofCongressCataloging-in-PublicationData
ThebreadthofsymplecticandPoissongeometry:festschriftinhonorofAlanWeinstein/
JerroldE.Marsden,TudorS.Ratiu,editors.
p.cm.–(Progressinmathematics;v.232)
Includesbibliographicalreferencesandindex.
ISBN0-8176-3565-3(acid-freepaper)
1.Symplecticgeometry.2.Geometricquantization.3.Poissonmanifolds.I.Weinstein,
Alan,1943-II.Marsden,JerroldE.III.Ratiu,TudorS.IV.Progressinmathematics
(Boston,Mass.);v.232.
QA665.B742004
516.3’.6-dc22 2004046202
ISBN0-8176-3565-3 Printedonacid-freepaper.
(cid:1)c2005Birkha¨userBoston
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Contents
Preface ............................................................ ix
AcademicgenealogyofAlanWeinstein ................................. xiii
AboutAlanWeinstein................................................ xv
StudentsofAlanWeinstein............................................ xv
AlanWeinstein’spublications ......................................... xvi
Diracstructures,momentummaps,andquasi-Poissonmanifolds
HenriqueBursztyn,MariusCrainic ................................... 1
ConstructionofRicci-typeconnectionsbyreductionandinduction
MichelCahen,SimoneGutt,LorenzSchwachhöfer....................... 41
Amathematicalmodelforgeomagneticreversals
J.J.Duistermaat.................................................. 59
Nonholonomicsystemsviamovingframes:CartanequivalenceandChaplygin
Hamiltonization
KurtEhlers,JairKoiller,RichardMontgomery,PedroM.Rios............. 75
Thompson’sconjectureforrealsemisimpleLiegroups
SamEvens,Jiang-HuaLu .......................................... 121
TheWeinsteinconjectureandtheoremsofnearbyandalmostexistence
ViktorL.Ginzburg ................................................ 139
Simplesingularitiesandintegrablehierarchies
AlexanderB.Givental,TodorE.Milanov .............................. 173
Momentummapsandmeasure-valuedsolutions(peakons,filaments,and
sheets)fortheEPDiffequation
DarrylD.Holm,JerroldE.Marsden.................................. 203
viii Contents
Higher homotopies and Maurer–Cartan algebras: Quasi-Lie–Rinehart,
Gerstenhaber,andBatalin–Vilkoviskyalgebras
JohannesHuebschmann ............................................ 237
Localizationtheoremsbysymplecticcuts
LisaJeffrey,MikhailKogan ......................................... 303
RefinementsoftheMorsestratificationofthenormsquareofthemomentmap
FrancesKirwan .................................................. 327
Quasi,twisted,andallthat…inPoissongeometryandLiealgebroidtheory
YvetteKosmann-Schwarzbach ....................................... 363
Minimalcoadjointorbitsandsymplecticinduction
BertramKostant .................................................. 391
Quantizationofpre-quasi-symplecticgroupoidsandtheirHamiltonianspaces
CamilleLaurent-Gengoux,PingXu................................... 423
Dualityandtriplestructures
KirillC.H.Mackenzie ............................................. 455
Starexponentialfunctionsastwo-valuedelements
Y.Maeda,N.Miyazaki,H.Omori,A.Yoshioka .......................... 483
FrommomentummapsanddualpairstosymplecticandPoissongroupoids
Charles-MichelMarle ............................................. 493
ConstructionofspectralinvariantsofHamiltonianpathsonclosedsymplectic
manifolds
Yong-GeunOh.................................................... 525
TheuniversalcoveringandcoveredspacesofasymplecticLiealgebraaction
Juan-PabloOrtega,TudorS.Ratiu ................................... 571
Poissonhomotopyalgebra:Anidiosyncraticsurveyofhomotopyalgebraic
topicsrelatedtoAlan’sinterests
JimStasheff...................................................... 583
DiracsubmanifoldsofJacobimanifolds
IzuVaisman...................................................... 603
Quantummapsandautomorphisms
SteveZelditch .................................................... 623
Preface
Alan Weinstein is one of the top mathematicians in the world working in the area
of symplectic and differential geometry. His research on symplectic reduction, La-
grangiansubmanifolds,groupoids,applicationstomechanics,andrelatedareashas
had a profound influence on the field.This area of research remains active and vi-
brant today and this volume is intended to be a reflection of that vigor. In addition
to reflecting the vitality of the field, this is a celebratory volume to honor Alan’s
60th birthday. His birthday was also celebrated inAugust, 2003 with a wonderful
week-longconferenceheldattheESI:theErwinSchrödingerInternationalInstitute
forMathematicalPhysicsinVienna.
Alan was born in New York in 1943. He was an undergraduate at MIT and a
graduatestudentatUCBerkeley,wherehewasawardedhisPh.D.in1967underthe
directionofS.S.Chern.AfterspendingpostdoctoralyearsatIHESnearParis,MIT,
andtheUniversityofBonn,hejoinedthefacultyatUCBerkeleyin1969,becoming
afullProfessorin1976.
Alan has received many honors, including anAlfred P. Sloan Foundation Fel-
lowship, a Miller Professorship (twice), a Guggenheim Fellowship, election to the
American Academy of Arts and Sciences in 1992, and an honorary degree at the
UniversityofUtrechtin2003.
AttheESIconference,S.S.Chern,Alan’sadvisor,sentthefollowingwordsto
celebratetheoccasion:
“IamgladaboutthiscelebrationandIthinkAlanrichlydeservesit.Alan
came to me in the early sixties as a graduate student at the University of
California at Berkeley.At that time, a prevailing problem in our geometry
group,andthegeometrycommunityatlarge,waswhetheronaRiemannian
manifold the cut locus and the conjugate locus of a point can be disjoint.
Alanimmediatelyshowedthatthiswaspossible.Theresultbecamepartof
his Ph.D. thesis, which was published in the Annals of Mathematics. He
received his Ph.D. degree in a short period of two years. I introduced him
toIHESandtheFrenchmathematicalcommunity.Hestaysclosewiththem
andwiththemathematicalideasofCharlesEhresmann.Heisoriginaland
x Preface
often came up with ingenious ideas.An example is his contribution to the
solutionoftheBlaschkeconjecture.Iamveryproudtocounthimasoneof
mystudentsandIhopehewillremaininterestedinmathematicsuptomy
age,whichisnow91.’’
Alan’stechnicalcontributionsarewideranginganddeep.Asmanyofhisearly
papersinhispublicationlistillustrate,hestartedoffinhisthesisandtheyearsim-
mediatelyfollowinginpuredifferentialgeometry,atopichehascomebacktofrom
timetotimethroughouthiscareer.
AlreadystartingwithhispostdocyearsandhisearlycareeratBerkeley,hebecame
interestedinsymplecticgeometryandmechanics.Inthisareaherapidlyestablished
himselfasoneoftheworld’sauthorities,producingimportantanddeepresultsranging
from reduction theory to Lagrangian and Poisson manifolds to studies of periodic
orbits in Hamiltonian systems. He also did important work in fluid mechanics and
plasmaphysicsandthroughthiswork,heestablishedwarmrelationswiththeBerkeley
physicistsAllanKaufmanandRobertLittlejohn.
Alan’simportantworkonperiodicorbitsinHamiltoniansystemsledhimeven-
tuallytoformulatethe“Weinsteinconjecture,’’namelythatforagivenHamiltonian
flow on a symplectic manifold, there must be at least one closed orbit on a regular
compactcontacttypelevelsetoftheHamiltonian.AlongwithArnold’sconjecture,
theWeinsteinconjecturehasbeenoneofthedrivingforcesinsymplectictopology
overthelasttwodecades.
Alankeptuphisinterestinsymplecticreductiontheorythroughouthislaterwork.
For instance, he laid some important foundation stones in the theory of semidirect
product reduction as well as in singular reduction through his work on Satake’s
V-manifolds, along with finding important links with singular structures in moduli
spaces.
Intertwinedwithhisworkonsymplecticgeometryandmechanics,hedidexten-
siveworkongeometricPDE,eigenvalues,theSchrödingeroperatorandgeometric
quantization. Alan took the point of view of microlocal analysis and phase space
structuresinhisworkinthisarea,emphasizingthelinkswithquantummechanics.
Preface xi
HisworkonthelimitdistributionofeigenvalueclustersintermsofthegeodesicRadon
transformofthepotentialinspiredalargenumberofrelatedarticles.Heshowedthat
thegeodesicflowofaZollsurfacewassymplecticallyequivalenttothatofaround
sphere,andhencethatitsLaplaciancouldbeconjugatedgloballytotheroundLapla-
cian plus a pseudodifferential potential. This work inspired many other results on
conjugacies.
OneofAlan’sfundamentalcontributionstoPoissongeometrywastheintroduc-
tionofsymplecticgroupoidsin1987,whichmarkstheofficialbeginningofhis“oids’’
period. In these works, he makes sweeping generalizations about a wide variety of
constructionsinsymplecticgeometry,including(withCourant)theimportantnotion
of Dirac structures. During this period of generalizations he constantly returned to
specific topics in symplectic and Poisson geometry, such as geometric phases and
Poisson Lie groups, in addition to making other key links. For instance, symplec-
tic groupoids are used to link Poisson geometry to noncommutative geometry, and
groupoidsarealsointimatelyrelatedtomanyotherareas,includingsymmetriesand
reduction,dualpairs,quantizationandthetheoryofsigmamodels.Oneofthecentral
ideas is that the usual theory of Hamiltonian actions, momentum maps, and sym-
plectic reduction makes sense in the more general context of actions of symplectic
groupoids;inthissetting,momentummapsarePoissonmapstakingvaluesingeneral
Poissonmanifolds,ratherthanjustLie–Poissonmanifolds(thatis,dualsofLiealge-
bras).Alanhasraisedthequestionofwhetherthisframeworkcanbefurtherextended
to include new notions of momentum maps such as quasi-Poisson manifolds with
group-valuedmomentummapsaswellasoptimalmomentummaps.
Alan is well known not only for his brilliant papers and conjectures, but also
forhisgeneralphilosophy,suchasthesymplecticcreed:EverythingisaLagrangian
submanifold.Thoseofuswhoknowhimwellalsoappreciatehisveryspecialinsight.
For example, in the middle of a discussion (for instance, as we both had in our
joint works on semidirect product reduction as well as stability theory) he will say
somethinglikewhatyouarereallydoingis... andthengiveussomeusuallyvery
specialinsightthatinvariablysubstantiallyimprovesthewholeproject.
Alan also has a very interesting and charming sense of humor that even makes
its way into his papers from time to time. For instance,Alan had great fun in his
paperswiththe“EastCoast–WestCoast’’discussionsofwhetheroneshouldusethe
term momentum map or moment map. He also gave us a good laugh with the term
symplecticbonesasitrelatestotheFrenchtranslationofPoissonasFish.
Alanisagreateducator.Hislectures,evenonCalculus,arealwaysatreatandare
veryinspiringfortheirspecialinsight,theirwitandlivelypresentation.Hisenthu-
siasmformathematicsisinfectious.Onestorythatcomestomindontheeducation
front is this: during the days when he was exceptionally keen about groupoids, he
waspreparingalectureforundergraduatesonthesubject.Someofusconvincedhim
topresentitasacolloquiumlectureforfaculty,keepinginmindtheoldadvice“no
colloquiumtalkcanbetoosimple.’’Itwas,infact,notonlyabeautifulcolloquium
talk,butwasperfectlypitchedforthefaculty,anditbecameapopulararticleinthe
NoticesoftheAmericanMathematicalSociety.Partofbeingagoodeducatorisbeing