Colloquium De Giorgi 2010–2012

80 Pages·2013·0.752 MB·English

by  Yves André (auth.), Umberto Zannier (eds.)

4
COLLOQUIA
YvesAndre´
De´partementdeMathe´matiquesetApplications
E´coleNormaleSupe´rieure
45rued’Ulm
75230ParisCedex05,France
NicholasM.Katz
DepartmentofMathematics
PrincetonUniversity
806FineHall
Princeton,NJ08544-1000,USA
EndreSzemere´di
Alfre´dRe´nyiInstituteofMathematics
HungarianAcademyofSciences
Rea´ltanodan.13-15
H-1053Budapest,Hungarian
DanAbramovich
DepartmentofMathematics
BrownUniversity
Box1917
Providence,RI02912,USA
EliasM.Stein
DepartmentofMathematics
PrincetonUniversity
802FineHall
Princeton,NJ08544-1000,USA
Shou-WuZhang
DepartmentofMathematics
PrincetonUniversity
505FineHall
Princeton,NJ08544-1000,USA
ColloquiumDeGiorgi2010-2012
Colloquium
De Giorgi
2010-2012
edited by
Umberto Zannier
EDIZIONI
DELLA
NORMALE
(cid:2)c 2013ScuolaNormaleSuperiorePisa
ISBN978-88-7642-455-7
ISBN978-88-7642-457-1(eBook)
Contents
Preface vii
YvesAndre´
Galoistheorybeyondalgebraicnumbersandalgebraicfunctions 1
NicholasM.Katz
Simplethingswedon’tknow 9
EndreSzemere´di
Islazinesspayingoff? (“Absorbing”method) 17
DanAbramovich
Moduliofalgebraicandtropicalcurves 35
EliasM.Stein
Threeprojectionoperatorsincomplexanalysis 49
Shou-WuZhang
CongruentnumbersandHeegnerpoints 61
Preface
Since2001theScuolaNormaleSuperiorediPisahasorganizedthe“Col-
loquio De Giorgi”, a series of colloquium talks named after Ennio De
Giorgi,theeminentanalystwhowasaProfessorattheScuolafrom1959
untilhisdeathin1996.
The Colloquio takes place once a month. It is addressed to a general
mathematicalaudience,andespeciallymeanttoattractgraduatestudents
andadvancedundergraduatestudents. Thelecturesareintendedtobenot
tootechnical,infieldsofwideinterest. Theyshouldprovideanoverview
of the general topic, possibly in a historical perspective, together with a
descriptionofmorerecentprogress.
Theideaofcollectingthematerialsfromtheselecturesandpublishing
theminannualvolumescameoutrecently,asarecognitionoftheirintrin-
sicmathematicalinterest,andalsowiththeaimofpreservingmemoryof
theseevents.
Forthispurpose,theinvitedspeakersarenowaskedtocontributewith
awrittenexpositionoftheirtalk,intheformofashortsurveyorextended
abstract. Thisserieshasbeencontinuedinacollectionthatwehopeshall
beincreasedinthefuture.
Thisvolumecontainsacompletelistofthetalksheldinthe“Colloquio
DeGiorgi”inrecentyearsbutalsointhepastandalsointhepastyears,
andatableofcontentsofallthevolumestoo.
Colloquiaheldin2001
PaulGauduchon
Weaklyself-dualKa¨hlersurfaces
TristanRivie`re
Topologicalsingularitiesformapsbetweenmanifolds
Fre´de´ricHe´lein
Integrable systems in differential geometry and Hamiltonian stationary
Lagrangiansurfaces
viii Preface
Jean-PierreDemailly
NumericalcharacterizationoftheKa¨hlerconeofacompactKa¨hlerman-
ifold
EliasStein
Discreteanaloguesinharmonicanalysis
JohnN.Mather
Differentiability of the stable norm in dimension less than or equal to
threeandoftheminimalaverageactionindimensionlessthanorequal
totwo
GuyDavid
AboutglobalMumford-Shahminimizers
JacobPalis
Aglobalviewofdynamics
AlexanderNagel
FundamentalsolutionsfortheKohn-Laplacian
AlanHuckleberry
Incidencegeometryandfunctiontheory
Colloquiaheldin2002
MichaelCowling
Generalizzazionidimappeconformi
FelixOtto
Thegeometryofdissipativeevolutionequations
CurtisMcMullen
Dynamicsoncomplexsurfaces
NicolaiKrylov
Some old and new relations between partial differential equations,
stochasticpartialdifferentialequations,andfinepropertiesoftheWiener
process
TobiasH.Colding
Disksthataredoublespiralstaircases
Ce´dricVillani
When statistical mechanics meets regularity theory: qualitative proper-
tiesoftheBoltzmannequationwithlong-rangeinteractions
ix Preface
Colloquiaheldin2003
JohnToland
Bernoullifreeboundaryproblems-progressandopenquestions
Jean-MichelMorel
Theaxiomaticmethodinperceptiontheoryandimageanalysis
JacquesFaraut
Randommatricesandinfinitedimensionalharmonicanalysis
AlbertFathi
C1 subsolutionsofHamilton-IacobiEquation
HakanEliasson
Quasi-periodicSchro¨dingeroperators-spectraltheoryanddynamics
YakovPesin
Ischaoticbehaviortypicalamongdynamicalsystems?
DonB.Zagier
Modularformsandtheirperiods
DavidElworthy
Functionsoffiniteenergyinfiniteandinfinitedimensions
Colloquiaheldin2004
Jean-ChristopheYoccoz
Hyperbolicityforproductsof 2×2matrices
GiovanniJona-Lasinio
Probabilita` emeccanicastatistica
JohnH.Hubbard
Thurston’stheoremoncombinatoricsofrationalfunctionsanditsgener-
alizationtoexponentials
MarceloViana
Equilibriumstates
BorisRozovsky
StochasticNavier-Stokesequationsforturbulentflows
MarcRosso
Braidsandshuffles
MichaelChrist
The d-bar Neumann problem, magnetic Schro¨dinger operators, and the
Aharonov-Bo¨hmphenomenon
x Preface
Colloquiaheldin2005
LouisNirenberg
Onethingleadstoanother
VivianeBaladi
DynamicalzetafunctionsandanisotropicSobolevandHo¨lderspaces
GiorgioVelo
Scatteringnonlineare
GerdFaltings
Diophantineequations
MartinNowak
Evolutionofcooperation
PeterSwinnerton-Dyer
Countingrationalpoints: Manin’sconjecture
Franc¸oisGolse
TheNavier-StokeslimitoftheBoltzmannequation
JosephJ.Kohn
Existenceandhypoellipticitywithlossofderivatives
DorianGoldfeld
OnGauss’classnumberproblem
Colloquiaheldin2006
YuriBilu
Diophantineequationswithseparatedvariables
CorradoDeConcini
Algebrecontracceerappresentazionidigruppiquantici
ZeevRudnick
Eigenvaluestatisticsandlatticepoints
LucienSzpiro
AlgebraicDynamics
SimonGindikin
Harmonicanalysisoncomplexsemisimplegroupsandsymmetricspaces
frompointofviewofcomplexanalysis
DavidMasser
From2topolarizationsonabelianvarieties