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1、OnrequestsforcareeradviceEverysooften,Ireceiveaqueryaskingforadviceonmathematicalcareerissues,suchasWhatfieldsinmathematicsshouldonestudy?Whatmathematicaltextsshouldonebuyorread?Whatproblemsshouldonetrytosolve?Howshouldoneapproachmathematicalproblems?Howshouldonewritemathemat
2、icalpapers?Whatuniversitiesshouldoneapplyto?Whatstrategiesshouldonepursuetoincreaseone’schancesofadmission(e.g.toUCLA)?Moregenerally,howshouldone“succeed”inmathematics?Theserequestsforadviceareofcourseveryflattering.Unfortunately,thesequestionsaretoogeneral,andtoodependentono
3、ne’sspecificcircumstances,interests,options,andcontextformetoofferanythingotherthangenericplatitudes(seebelow).Becauseofthis,andbecauseoflackofavailabletime,Iamregretfullyunabletomeaningfullyrespondtoanysuchqueries.Iwouldrecommendinsteadconsultingwithone’shighschool,undergrad
4、uateorgraduateadvisor,whoismoreattunedtoyourspecificsituationandwillbeabletooffermorerelevantadvice.Inparticular,IamunabletopersonallyadviseanyoneotherthanUCLAgraduatestudentswhohavealreadypassedtheirqualifyingexams.Regardingmathematicscompetitions:Ihavenotparticipatedinmathe
5、maticscompetitionssince1988,andamnotfamiliarwithhowtheyworknowadays.Foradviceonhowtosolvemathematicalproblems,youcantrymybookonthesubject.Also,Ishouldsaythatwhilemathematicscompetitionsarecertainlyalotoffun,theyareverydifferentactivitiesfrommathematicallearningormathematicalr
6、esearch;don’texpecttheproblemsyougetin,say,graduatestudy,tohavethesamecut-and-dried,neatflavourthatanOlympiadproblemdoes.(WhileindividualstepsinthesolutionmightbeabletobefinishedoffquicklybysomeonewithOlympiadtraining,themajorityofthesolutionislikelytorequireinsteadthemuchmor
7、epatientandlengthyprocessofreadingtheliterature,applyingknowntechniques,tryingmodelproblemsorspecialcases,lookingforcounterexamples,andsoforth.)Soenjoythesecompetitions,butdon’tneglectthemore“boring”aspectsofyourmathematicaleducation,asthoseturnouttobeultimatelymoreuseful.Gen
8、ericplatitudesAsIsaidabove,Ihaveno“secretformula”orotherone-size-fit
