of a lon g history , finally reache d it s climacti c development s wit h th e wor k o f th e
number theorist s of the lat e 19th century (clas s field theory) an d th e currentl y ver y
active Langland s program . Part s o f thi s chapte r ma y b e regarde d a s a ver y brie f
introduction t o th e Langland s program .
When writin g an y book , th e questio n i s alway s this : fo r who m i s th e boo k
being written ? T o a larg e exten t I wa s guide d b y a desir e t o reac h th e beginnin g
graduate student s an d th e advance d undergraduates , an d t o communicat e t o the m
the marvelou s fac t tha t thing s lik e clas s field theory , Bore l summation , ellipti c
curves, an d s o on , di d no t com e ou t full y grow n fro m th e primeva l ocea n (o f th e
ancient Hindu s fo r example ) bu t gre w ou t o f smal l beginnings , an d man y o f thes e
go back t o Euler . I fee l tha t thi s historicall y motivate d metho d o f teachin g i s th e
best suite d t o conve y th e organi c structur e o f mathematics . Thi s o f cours e i s no t
the preferre d wa y of teaching nowadays , where the student s lear n coheren t sheave s
and adele s befor e learnin g th e Euler-Ferma t theore m o f prime s whic h ar e sum s o f
two squares . I t actuall y happene d i n a high-powere d cours e o n .D-module s I wa s
attending a fe w year s bac k tha t th e first differentia l equatio n tha t wa s written , a
few week s afte r som e very heav y stuf f o n D-modules , wa s Euler's, an d th e lecture r
got i t wron g becaus e h e forgo t tha t th e invarian t operato r o n C x i s no t d/dz
but z(d/dz)\ I a m convince d tha t th e onl y wa y t o produc e youn g mathematician s
who ar e no t imprisone d b y th e smal l numbe r o f ideas the y lear n i n a conventiona l
graduate educatio n i s to emphasiz e th e unit y o f mathematic s fro m th e beginning ,
and fo r this , the historica l method , calle d th e biogeneti c metho d b y Shafaraevitch ,
is the onl y possibl e one .
Thus this book is not a conventional historical essay on Euler—of these there are
many wonderfu l examples—bu t rathe r a discussion o f some of the Euleria n theme s
and ho w they fit into the modern perspective . Mor e than anyon e else, I know wha t
the shortcoming s o f my attemp t are . Fo r example , althoug h I hav e trie d t o kee p
the expositio n a s elementar y a s possible , her e an d ther e ar e place s wher e thi s ha s
been impossibl e t o maintain , an d I hav e ha d t o assum e familiarit y o n th e par t o f
the reade r o f mor e advance d material . Bu t I hav e trie d t o organiz e everythin g i n
such a manner tha t a beginning graduat e student , a s well as a mathematician wh o
does not alway s have a specialized knowledg e o f the topic s treated, wil l find thing s
understandable a s well as enjoyable .
It onl y remain s fo r m e t o giv e thank s t o th e peopl e wh o helpe d m e i n thi s
effort: t o Do n Babbit t an d Serge i Gel'fand , wh o kep t encouragin g m e throughou t
this enterprise ; t o Marin a Eskin , wh o attende d my cours e o n Eule r wit h a n en -
thusiasm tha t wa s infectious; t o Pierr e Deligne , wh o was extraordinarily generous ,
as h e alway s is , i n sharin g wit h m e hi s insight s o n man y aspect s o f Euler' s wor k
during hi s visit t o UCL A i n the sprin g of 2005; to Anit a Colby , science librarian a t
UCLA, who always gave her hel p freely , instantly , an d wit h a smile, for he r hel p i n
bringing man y item s o f Euleriana t o m y attention ; t o Richar d Tsa i fo r helpin g m e
with calculation s usin g MATHEMATICA ; t o Aaro n Pear l fo r hel p i n photograph -
ing page s fro m Euler' s papers ; t o Boya n Kostadinov , m y student , fo r hi s hel p i n
translating article s by and abou t Eule r from Russia n as well as in proofreading; an d
to my wife, Veda , whose understanding an d suppor t hav e been th e grea t steadyin g
influences i n my life .
V. S . Varadaraja n
Pacific Palisades
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