Table Of ContentContributions in Mathematical
and Computational Sciences Volume 2
•
Editors
HansGeorg Bock
WilliJäger
OtmarVenjakob
Forothertitlespublishedinthisseries,goto
http://www.springer.com/series/8861
•
Jakob Stix
Editor
The Arithmetic
of Fundamental Groups
PIA 2010
123
Editor
JakobStix
MATCH-MathematicsCenterHeidelberg
DepartmentofMathematics
HeidelbergUniversity
ImNeuenheimerFeld288
69120Heidelberg
Germany
[email protected]
ISBN978-3-642-23904-5 e-ISBN978-3-642-23905-2
DOI10.1007/978-3-642-23905-2
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011943310
MathematicsSubjectClassification(2010):14H30,14F32,14F35,14F30,11G55,14G30,14L15
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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toLucieEllaRoseStix
•
Preface to the Series
Contributions toMathematical and Computational Sciences
Mathematicaltheoriesandmethodsandeffectivecomputationalalgorithmsarecru-
cialincopingwiththechallengesarisinginthesciencesandinmanyareasoftheir
application. Newconceptsandapproachesarenecessaryinordertoovercomethe
complexitybarriersparticularlycreatedbynonlinearity,high-dimensionality,mul-
tiplescalesanduncertainty. Combiningadvancedmathematicalandcomputational
methodsand computertechnologyis an essential keyto achievingprogress, often
eveninpurelytheoreticalresearch.
Thetermmathematicalsciencesreferstomathematicsanditsgenuinesub-fields,
aswellastoscientificdisciplinesthatarebasedonmathematicalconceptsandmeth-
ods, including sub-fields of the natural and life sciences, the engineering and so-
cial sciences and recently also of the humanities. It is a major aim of this series
to integratethe differentsub-fieldswithin mathematicsand the computationalsci-
ences,andtobuildbridgestoallacademicdisciplines,toindustryandotherfields
ofsociety,wheremathematicalandcomputationalmethodsarenecessarytoolsfor
progress. Fundamentalandapplication-orientedresearchwillbecoveredinproper
balance.
The series will further offer contributions on areas at the frontier of research,
providingboth detailed informationon topical research, as well as surveys of the
state-of-the-artinamannernotusuallypossibleinstandardjournalpublications.Its
volumes are intended to cover themes involving more than just a single “spectral
line”oftherichspectrumofmathematicalandcomputationalresearch.
TheMathematicsCenterHeidelberg(MATCH)andtheInterdisciplinaryCenter
forScientificComputing(IWR)withitsHeidelbergGraduateSchoolofMathemat-
icalandComputationalMethodsfortheSciences(HGS)areinchargeofproviding
andpreparingthematerialforpublication.Asubstantialpartofthematerialwillbe
acquiredinworkshopsandsymposiaorganizedbytheseinstitutionsintopicalareas
ofresearch. Theresultingvolumesshouldbemorethanjustproceedingscollecting
vii
viii PrefacetotheSeries
paperssubmittedinadvance.Theexchangeofinformationandthediscussionsdur-
ingthemeetingsshouldalsohaveasubstantialinfluenceonthecontributions.
This series is a venture posing challenges to all partners involved. A unique
styleattractingalargeraudiencebeyondthegroupofexpertsinthesubjectareasof
specificvolumeswillhavetobedeveloped.
SpringerVerlagdeservesourspecialappreciationforitsmostefficientsupportin
structuringandinitiatingthisseries.
HeidelbergUniversity, HansGeorgBock
Germany WilliJäger
OtmarVenjakob
Preface
Duringthemorethan100yearsofitsexistence,thenotionofthefundamentalgroup
hasundergoneaconsiderableevolution.ItstartedbyHenriPoincaréwhentopology
asasubjectwasstillinitsinfancy. Thefundamentalgroupinthissetupmeasures
thecomplexityofapointedtopologicalspacebymeansofanalgebraicinvariant,a
discrete group, composedof deformationclasses of based closed loopswithin the
space. In this way, for example, the monodromyof a holomorphicfunction on a
Riemannsurfacecouldbecapturedinasystematicway.
It was throughthe workof AlexanderGrothendieckthat, raising into the focus
the roleplayedby thefundamentalgroupingoverningcoveringspaces, so spaces
overthegivenspace,aunificationofthetopologicalfundamentalgroupwithGalois
theory of algebra and arithmetic could be achieved. In some sense the roles have
been reversed in this discrete Tannakian approach of abstract Galois categories:
first,wedescribeasuitableclassofobjectsthatcapturesmonodromy,andthen,by
abstractpropertiesofthisclassaloneandmoreoveruniquelydeterminedbyit, we
find a pro-finite group that describes this category completely as the category of
discreteobjectscontinuouslyacteduponbythatgroup.
Butthedifferentincarnationsofafundamentalgroupdonotstophere.Thecon-
ceptofdescribingafundamentalgroupthroughitscategoryofobjectsuponwhich
thegroupnaturallyactsfindsitspro-algebraicrealisationinthetheoryofTannakian
categoriesthat,whenappliedtovectorbundleswithflatconnections,ortosmooth
(cid:2)-adicétalesheaves,ortoiso-crystalsor...,givesrisetothecorrespondingfunda-
mentalgroup,eachwithinitsnaturalcategoryasahabitat.
In more recent years, the influence of the fundamental group on the geometry
of Kähler manifolds or algebraic varieties has become apparent. Moreover, the
program of anabelian geometry as initiated by Alexander Grothendieck realised
somespectacularachievementsthroughtheworkoftheJapaneseschoolofHiroaki
Nakamura, AkioTamagawaandShinichiMochizukiculminatinginthe proofthat
hyperboliccurvesoverp-adicfieldsaredeterminedbytheouterGaloisactionofthe
absoluteGaloisgroupofthebasefieldontheétalefundamentalgroupofthecurve.
Anaturalnexttargetforpiecesofarithmeticcapturedbythefundamentalgroup
arerationalpoints,thegenuineobjectofstudyofDiophantinegeometry.Herethere
ix
Description:In the more than 100 years since the fundamental group was first introduced by Henri Poincaré it has evolved to play an important role in different areas of mathematics. Originally conceived as part of algebraic topology, this essential concept and its analogies have found numerous applications in
