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1、AfewwordsonresearchforgraduatestudentsForgraduatestudentswhowishtoexploretheworldofmathematics,hereareafewthingsthatyouprobablywouldliketoknowatthebeginningofthejourney.(Iwishsomeonewouldhavetoldmeso!)Whatisresearch?Whatdoesresearchmean?Thepursuitoftruth?Aq
2、uestintotheunknown?Well,tostart,itmeansfindingaproblemandsolvingit.Itcouldbeatinystepofasmallpuzzle,(bystandingattheshoulderofagiant!).Or,itcouldalsobeaglimpseatabiggerpicture(eventually).Sinceanyworthwhileproblemwon'tbeeasytosolve,theroadcanbelongandwindyw
3、ithtwistsandturns.Nevertheless,herearesomefactorsinyourfavor:Itisnotarealtimegame.Itisnotanexamthatyouneedtoturninwithinanhour.(Ofcourse,thereisanupperboundforthelengthofyourgraduatestudy.HereatUCSD,theboundis5years!)Youareallowedtomakemistakes,plentyofthem
4、.Wealldo.Thepercentageofsuccessisnotrequiredtobehigh.Infact,ifyoucandooneresearchproblemoutofsomelargenumber(10,100,...)ofproblems,itisgoodenough.Youaregoingtogethelpfromyouradvisor,friends,professors,otherresearchers,....Youarenotalone.Therearemanywaysthat
5、canhelpincreasingtheoddsifyoukeepreading....Fortheloveofit!Itispossibletodowellonlyifyoureallyenjoytheprocess.Findaproblemthatyouhavesomefeelingabout.Gettingstarted.Ioftenheard,"Mathistoobig.WaituntilIhavelearnedenough."Thefactisthatnobodyknowseverything.Do
6、n'twait.Justgetstarted.Trytopickupsomeproblembytalkingtoyourprofessors.Itmightalsohelpifyougotoseminarsordosomereading.Findafavoriteproblemandsinkintoitdeeplysothatitwillrequireyoutolearnmore.Incombinatorics,wehavetheadvantagethatalotofproblemsareeasytostat
7、e(butoftennotsoeasytosolve).Selectionandtaste.Asajuniorresearcher,justtrytoproveanytheorem.Ifasmallresultisrelevanttoseveraldifferentproblems,ithasachancetogrowintoabiggerresult.Ifyouhaveseveralinterrelatedresults,itispossibletodevelopintoatheory.Overall,if
8、aproblemisfundamental,itwillcomethroughinmanydifferentareas.Ontheotherhand,goodapplicationshelpguidingtheresearchtowardcentralproblems.Thejudgementforselectionandtastewillcomelaterinanaturalway.Depthve
