资源描述：
《Seven Famous Unsolved Math Puzzles》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
1、AboutahundredyearsagoDavidHilbert,aGermanmathematicianpresentedtwenty-threemathpuzzlestotheInternationalCongressofMathematicians.Today,onlythreeremainunsolved.Addedtothosewerefourmoreunsolvableproblems.Thesevenfamousunsolvedmathpuzzlesthathaveresistedallattemptstosol
2、vearelistedhere:TheBirchandSwinnerton-DyerConjecture,TheNavier-StokesEquation,ThePoincareConjecture,TheRiemannHypothesis(theoldestandmostfamous),ThePVersesNPProblem,TheHodgeConjecture,Yang-MillsExistenceandMassGap.Manyexpertsbelievethatsolvingtheseproblemswouldleadto
3、extraordinaryadvancesinphysics,medicineandmanyotherunknownareasintheworldofmath.ThePoincareconjectureIfyoustretcharubberbandaroundthesurfaceofanapple,thenwecanshrinkitdowntoapointbymovingitslowly,withouttearingitandwithoutallowingittoleavethesurface.Butifyouweretostr
4、etcharubberbandaroundthesurfaceofadoughnut,thenthereisnowayofshrinkingittoapointwithoutbreakingeithertherubberbandorthedoughnut.Thereforethesurfaceofanappleis“simplyconnected,”andtheoneofthedoughnutisnot.Aboutahundredyearsago,Poincareknewthatatwodimensionalsphereises
5、sentiallycharacterizedbythispropertyofsimpleconnectivity.Heaskedthecorrespondingquestionforthethreedimensionalsphere-thesetofpointsinfour-dimensionalspaceatunitdistancefromtheorigin.Asitturnsout,thisisanextraordinarilydifficultquestiontobeanswered.HenriPoincarepracti
6、callyinventedtopologywhiletryingtounderstandthesetofsolutionstoageneralalgebraicequationf(x,y,z)=0,wherex,y,zarecomplexnumbers.Aftertryingtheanalyticapproach,hebeganassigningalgebraicinvariantstogeometricobjectsasanapproachtoclassifyingtheobjects.TranslatedintoEnglis
7、h,Poincaresaidconsideracompact3-dimensionalmanifoldVwithoutboundary.IsitpossiblethatthefundamentalgroupofV-couldbetrivial,eventhoughVisnothomeomorphictothe3-dimensionalsphere?Since1904,thehypothesisthateverysimplyconnectedclosed3-manifoldishomeomorphictothe3-sphereha
8、sbeenknownasthePoincareconjecture.Fouryearsearlierhehadstatedthateverycompactpolyhedralmanifoldwiththehomologyofann-dimensionalsphe
