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1、RealAnalysis,QuantitativeTopology,andGeometricComplexityStephenSemmesThissurveyoriginatedwiththeJohnJ.GergenMemorialLecturesatDukeUniversityinJanuary,1998.TheauthorwouldliketothanktheMath-ematicsDepartmentatDukeUniversityfortheopportunitytogivetheselectures.See[Gro1
2、,Gro2,Gro3,Sem12]forrelatedtopics,insomewhatdi erentdirections.Contents1Mappingsanddistortion32Themathematicsofgoodbehaviormuchofthetime,andtheBMOframeofmind103Finitepolyhedraandcombinatorialparameterizationprob-lems17arXiv:math.MG/0010071v17Oct20004Quantitativetopo
3、logy,andcalculusonsingularspaces265Uniformrecti ability365.1SmoothnessofLipschitzandbilipschitzmappings.......425.2Smoothnessanduniformrecti ability..............475.3Aclassofvariationalproblems..................51AppendicesAFouriertransformcalculations54Theauthorwa
4、spartiallysupportedbytheNationalScienceFoundation.1BMappingswithbranching56CMoreonexistenceandbehaviorofhomeomorphisms59C.1Wildnessandtamenessphenomena...............59C.2Contractableopensets......................63C.2.1Somepositiveresults...................67C.2.2E
5、ndsofmanifolds.....................72C.3Interlude:lookingatin nity,orlookingnearapoint.....72C.4Decompositionspaces,1.....................75C.4.1Cellularity,andthecellularitycriterion.........81C.5Manifoldfactors..........................84C.6Decompositionspaces,2..
6、...................86C.7Geometricstructuresfordecompositionspaces.........89C.7.1Abasicclassofconstructions..............89C.7.2Comparisonsbetweengeometricandtopologicalprop-erties............................94C.7.3Quotientspacescanbetopologicallystandard,butge-omet
7、ricallytricky.....................96C.7.4Examplesthatareevensimplertopologically,butstillnontrivialgeometrically.................105C.8Geometricandanalyticresultsabouttheexistenceofgoodcoordinates............................109C.8.1Specialcoordinatesthatonemightcons
8、iderinotherdimensions.........................113C.9Nonlinearsimilarity:Anotherclassofexamples........118DDoingprettywellwithspaceswhichma
