notes on mathematicians

7/28/2019 Notes on Mathematicians

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NOTES ON MATHEMATICIANS

7. Hermann Weyl (1885 - 1955). Y. K. Leong

Univers i ty of Singapore

Since the ou tburs t of in tense a c t i v i t i e s in mathematicsand physics dur ing the f i r s t two decades of t h i s c ~ n t u r y , withthe concomitant expansion o f perspec t ives in human knowledge,answers to old problems have only s t i r r ed up f re sh que r ie swhich demanded more answers to new problems. Specia l izede f f o r t s would be required to provide these answers which, int u rn , opened up new reg ions fo r sc i e n t i f i c endeavour. Aseach explorer maps out and s e t t l e s confor tab ly in to h is ownn i ~ h e of t e r r i t o r y , it would seem well-nigh impossible fo r amor ta l mind to pick i t s way dur ing one human l i f e t i m e f r o m ~ n e l abyr i n t h of knowledge to anothe r , l e t alone to s lay the Minotaurof each l abyr in th .

Throughout the ages , the grea t m ~ t h e m a t i c a l minds of eachepoch have pondered over a wide spectrum of knowledge - Archi medes [1] , Isaac Newton ( 2 ] , Joseph Lbuis Lagrange (3] , LeonhardEuler [4] , Car l Fr iedr ich Gauss ( 5 ] , Henr i Poincare [6], DavidHi lbe r t [7], Hermann Weyl, John von Neumann [ 8 ] , to name onlya few. The two gr ea t upholders of t h i s t r ad i t ion in the ninet een th century were P o i n c a r ~ and Hi lbe r t . The l a t t e r c rea tedthe Gott ingen school o f mathematics which inf luenced and di rec tedthe development of much o f modern mathematics and , to some ex ten t ,modern physics . Hilber t was Weyl's mentor. In c ha r a c t e r i s t i c a l l yHilbe r t ian fashion, Weyl devoted d i f f e r en t s tages of h is l i f eto the in tens ive study of d i f f e r en t sub jec t s ~ n a concer teda t tempt to i n j e c t new i deas , e f f e c t fundamental changes anddiscover underly ing connect ions .

As the t heo re t i c a l phy i i c i s t Freeman J . Dyson (g] wri te s1n the obi tuary publ ished in the sc i en t i f i c journa l Nature :"Among a l l the mathemat ic ians who began t h e i r working l i ve s in

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the twen t ie th cen tu ry , Hermann Weyl was th e one who made majorcon t r ibu t ions in th e g r e a t e s t number of d i f f e r e n t f i e l d s . Healone could s tand comparison with th e l a s t grea t un ive rsa l

/mathemat ic ians o f the n ine teen th cen tu ry , Hi lbe r t and Poincare .So long as he was a l i v e , he embodied a l i v ing co n tac t betweenth e main l i nes o f advance in pure mathematics and in t heo re t i ca lphys ics . Now he i s dead, the co n tac t i s broken, and our hopeso f comprehending th e phys ica l universe by a d i r e c t use o fc rea t ive mathemat ica l imaginat ion are fo r the t ime being ended ."(10]The making o f a mathemat ic ian Weyl was born on 9 November 1885in th e smal l town o f Elmshorn near Hamburg in Germany. Hisparents were Ludwig and Anna Weyl, and he spen t h is school daysin Altona . The d i rec to r of h is g y m n a ~ i u m (a school t h a t preparesi t s s tudents fo r th e un ive rs i ty ) happened to be a cousin o fH i lbe r t , then a p r o f e s s o ~ a t Gott ingen. So, 1n 1903, he went tofu r t h e r h is s tud ies a t the Univers i ty o f Gott ingen. Weylappeared as an e igh teen-year -o ld count ry l a d , shy and i n a r t i c u l a t ebut conf iden t o f h is own a b i l i t i e s . As he has w r i t t en , "In thefu l lness of my innocence and ignorance , I made bold to t ake th ecourse H i l b e r t had announced fo r t h a t t e rm, on th e not ion o fnumber and th e quadrature of th e c i r c l e . Most o f it went s t r a ighover my head. But the doors of a new world swung open fo r me,and I had not s a t long a t H i l b e r t ' s f e e t before th e r e so lu t ionformed i t s e l f in my young h e a r t t h a t I must by a l l means readand s tudy whatever t h i s man had w r i t t en . " [11]

Except fo r one year a t the Univers i ty o f Munich, Wey1'smathemat ica l educat ion was completed a t Gott ingen. During t h i sper iod , the two b r i g h t s t a r s t h a t lit up th e mathematicalf i rmament o f t h i s s c i e n t i f i c cen t re were H i l b e r t and HermannMinkowski [12] , both of them a t th e peak o f t h e i r ca ree r s . Theaging Fel ix Klein [13] , the grand old man of German mathematicswho himse l f embodied a l i v ing legend, contented him.s e l f 1nd i r ec t ing th e development o f Gott ingen as th e cen t re of th esc i en t i f i c w o ~ l d . The un ive r s i t i e s t h a t U b s e q ~ e n t l y sprangup in America toge the r with t h e i r sc i en t i f i c - t echno log ica l

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complexes were pa t t e rned a f t e r Got t ingen .At th e end o f th e f i r s t y ea r , Weyl re tu rned home with a

copy o f H i l b e r t ' s ZahZberioht (Die Theorie der aZgebraisohenZahZkorper), th e 1896 monumental r ep o r t t h a t l a i d th e founda-t i o n s o f th e modern theory o f a lgebra ic numbers. The fo l lowingmonths o f th e summer vaca t ion which Weyl, wi th o u t any prev iousknowledge o f elementary number theory o r Galo i s t h e o ry , spen tin going through th e r e p o r t , , w ~ r e , in h is own words , " thehapp ie s t months o f my l i f e , whose s h i n e , ac ross years burdenedwith our common share o f doubt and f a i l u r e , still comfor t s mysou' l . " [11]

It was indeed h i s g re a t fo r tune t h a t Weyl spen t h is form-a t ive years and came o f age as a mathemat ic ian dur ing th e goldenage o f th e h i s t o ry of Got t ingen . He obta ined h is d o c t o ra t e in1908 and became a Pr iva tdozen t [14) in 1910. One marvel s a tth e long list o f impress ive names, immorta l ized in th e r o l l - c a l lof p ioneers and c re a t o r s o f mathemat i ca l and s c i e n t i f i c t h o u g h t ,t h a t made t h e i r debu t as budding t a l e n t s in Got t ingen . In th ewords o f Weyl, "A un ive rs i ty such as Got t ingen , in th e halcyondays before 1914, was p a r t i c u l a r l y favourab le fo r th e developmento f a l i v ing s c i e n t i f i c schoo l . Once a band o f d i s c i p l e s hadgathered around H i l b e r t , i n t e n t upon re sea rch and little worr iedby th e chore o f t e ach ing , it was b ut n a t u ra l t h a t in j o i n t com-p e t i t i v e a sp i r a t i o n o f r e l a t e d aims each should s t imula te th eo th e r ; t h e r e was no need t h a t everyth ing come from t he mas te r . "[15]

Some o f Weyl ' s contemporar ies in Gott ingen were Max Born[ 1 6 ] , Richard Courant [ 1 7 ] , Harald Bohr [18] , Erich Heeke [19]and George Polya [20] . There were c l i q u es and in -groups t h a tru l ed over th e so c i a l l i f e of the mathemat i ca l world o f Got t ingen .But Weyl, even a f t e r becoming a Pr iva tdozen t , was still too shyfo r such in -g roups . It t h e r e f o r e came as a shock when he wonth e hand o f a l ady "whose charms were such t h a t when h er f a t h e rth rea tened to withdraw h er from the Univers i ty a p e t i t i o n begginghim to recons ide r was s igned even by p ro fe s s o r s " [ 2 ~ ] . InSeptember 1913 , he marr i ed Helene Joseph , the daugh ter o f a doc to r

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and th e t r a n s l a t o r in to German o f many Spanish wri t ings . Outo f t h i s marr iage two sons were born , one o f whom, Fr i t z Joachim,became a mathemat ic ian.

Weyl 's d o c to r a l d i s s e r t a t i o n d e a l t with s in g u la r i n t e g r a lequa t ions . The theory o f i n t e g r a l equa t ions was f i r s t i n i t i a t e dby Iva r Fredholm [22] in 1903, and , in the hands o f H i l b e r t ,bore f r u i t to r ev ea l t h e under ly ing r e l a t i onsh ip between i n t e g r a lequa t ions and l i n e a r a l g e b ra , l e ad ing to a gene ra l theory o f so ca l l ed H i lb e r t spaces . As a n a t u ra l con t inua t ion o f h is d i s s e r t a t i o n , Weyl appl ied the theory o f i n t e g r a l equa t ions to s in g u la re igenvalue problems o f ord ina ry d i f f e r e n t i a l equa t ions in a longse r i e s o f papers publ ished dur ing 1908 - 1915.

Appearing in p a r a l l e l with th e above s e r i e s was h is work onth e asympto t ic d i s t r i bu t i on o f n a t u r a l f requenc ies o f o s c i l l a t i n gcont inua such as membranes, e l a s t i c bodies and e lec t romagne t i cwaves. The r e s u l t had been conjec tured e a r l i e r by p h y s i c i s t sand the d iscovery o f i t s mathemat ica l proof was dec i s ive to Weyl.He r e c a l l e d how th e kerosene lamp began to fume as he workedf ever i sh ly on th e so lu t ion , and when he had f i n i sh e d , the re wasa th ick l a y e r o f soo t on h is pape rs , hands and face . [23]

In t e g ra l equa t ions occur f requen t ly in phys ic s , and th ec l a s s i c a l problems o f th e o s c i l l a t i o n s o f con t inua and o fp o t e n t i a l theory would have been i n t r a c t a b l e without a propertheory of t he se equa t ions . So it was from th e beginning t h a tWeyl 's f i r s t researches in a n a l y s i s , a branch o f pure mathemat ics ,had brought him face to face with problems o f th e n a t u r a l sc iences-- th e beginning too o f a l i f e - l o n g i n t e r e s t in app ly ing mathema-t i c s to the sc i ences .

As a Pr iva tdozen t 1n Got t ingen , he gave a course on th ec l a s s i c a l theory o f a lg eb r a i c and ana ly t i c func t ions on Riemannsu r faces . Out o f t h i s appeared in 1913 h is f i r s t book, DieIdee der Riemannschen Flache (The concept o f a Riemann sur face ) ,which was " to exer t ' a profound i n f luence on th e mathemat ica lthought o f h is age" . [23] For th e f i r s t t ime , it l a i d th erigol'\Ous founda t ions o f "geometr ica l" func t ion theory and a tth e same t ime l inked toge the r a n a l y s i s , geometry and topology

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( i t was only a few years before , in 1910 - 1911, t ha t thef i r s t bas ic topo log ica l not ions were given by L. E. J . Brouwer[ 24 ] ) . Forty- two years l a t e r , in 1955, Weyl publ i shed a newed i t ion o f th e book and ~ e c a s t it in the by now f a m i l i a r languageof topology .Controversy amidst t r a nqu i l l i t y The years 1913 - 1930 formed aper iod of comparat ive t r a nqu i l l i t y in Weyl 's l i f e . He went toZurich in 1913 a t the age o f twenty-e igh t to take up a professor sh ip of mathematics a t th e Swiss Federa l I n s t i t u t e o f TechnolOY(Eidgenoss ische Technische Hochschule) . The F i r s t World Warexploded, and he was en l i s t ed in to th e Swiss army and servedfo r one year as a pr iva t e with a gar r i son a t Saarbrucken. Atth e end of h is se rv ice , he re tu rned to l e c tu r e a t Zur ich . Hewas happi ly marr ied , with h is wife shar ing " to the f u l l h ist a s t e fo r phi losophy and fo r imaginat ive and poe t i ca l l i t e r a t u r e "[25] . The war came and p ~ s s e d by Swi tzer land , whose neu t r a l i t yin the con f l i c t ensured t ha t i t s peace and calm was no t breachedwhi le the r e s t o f Europe was t ea r ing i t s e l f apa r t .

His s t a tu re in mathematics and phys ics grew. He took anac t ive i n t e r e s t in the mathemat ica l and sc i e n t i f i c r evo lu t ionsof t h a t e ra . The 1915 theory o f genera l r e l a t i v i t y o f Alber tEins te in (26], who d id a two-year spe l l from 1912 to 1914 a t th eFedera l I n s t i t u t e , a t t r ac t ed and s t imula ted him in to the searchfo r a "unif ied f i e ld theory" , an extens ion to cover gr av i t a t i onand elec t romagnet ism. A t t h e same t ime, he was swept as apar t i san in to the deep cur ren ts o f controversy t h a t lashed a t thefoundat ions o f mathemat ics . He found himsel f in the camp ofthe " In tu i t i on i s t s " led by Brouwer, the arch-enemy of Hi lbe r t inmathemat ics .

At the he igh t o f h is powers , he publ ished papers in numbertheory , d i f f e r e n t i a l geometry, harmonic ana lys i s and its applica+.t i ons to quantum mechanics. He was i n t e r e s t ed in p r a c t i c a l l ya l l f i e ld s o f crea t ive human endeavour , inc lud ing a r t , music ,phi losophy and l i t e r a t u r e . His crea t ive s p i r i t sought toexpress i t s e l f wherever it could . Even in mathemat ics , he thought

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of h imse l f as an a r t i s t . As he has sa id i n r e t ro sp e c t manyyears l a t e r in Zur ich , "Express ion and shape are almost moreto me than knowledge i t s e l f . But I be l ieve t h a t , ~ e a v i n gas ide my own p e c u l i a r n a t u re , the re i s in mathematics i t s e l f ,i n con t ras t t o th e exper imenta l d i sc ip l i nes , a ch a r ac t e rwhich i s neare r to t h a t o f f ree c rea t ive a r t . For t h i s reasonth e modern s c i e n t i f i c urge to found I n s t i t u t e s o f Science i snot so good fo r mathemat ics , where th e r e l a t i onsh ip betweent e ache r and pup i l should be milder and l o o s e r . In th e f inea r t s we do not normally seek to impose the sys temat ic t r a in ingof pupi ls upon c rea t ive a r t i s t s . " C25]

He was still in h is t h i r t i e s when h is fame and repu ta t ionemanated from Zurich. There were numerous o f fe r s of cha i r sby fore ign u n i v e r s i t i e s . In 1922, Klein and Hilber t askedhim to r e t u rn to the fo ld o f Gott ingen. Because o f h is r e s p ec tfo r them, Weyl pondered over the dec is ion fo r a long t ime . Hewas so worr ied t h a t he walked h is wife Hel la round and roundthe block o f t h e i r home u n t i l near ly midnight before he decidedto accept the o f f e r . He then rushed o f f to send a te legramo f acceptance , but re turned home a f t e r he had te legraphed tor e j e c t th e o f f e r i n s t ead . He was not prepared " to exchangethe t r an q u i l l i t y of l i f e in Zurich fo r the uncer t a in t i e s ofpost -war Germany". [21]

In th e summer o f 1917, Weyl gave a course of l ec tu res onthe theory o f genera l r e l a t i v i t y and publ ished it in 1918 asRaum-Zeit-Materie ( S p a c e ~ t i m e ~ matter) which went throughf ive ed i t ions in f ive years to become th e 1923 c l a s s i c . Onceh is i n t e r e s t was k ind led , he s e t out to gene ra l i ze Eins t e in ' stheory and produced in 1918 a "un i f i ed f i e l d theory" - thef i r s t o f many such t h e o r i e s , in an as y e t unf in ished ques tpioneered by Weyl, Arthur Eddington [27}, T. Kaluza [281 andEins te in h imse l f . By in t roduc ing the poss ib i l i t y o f a changeo f s i ze in th e geometry o f space- t ime , Weyl was able to i n t e r p re t electromagnet ism as an aspec t o f t h i s geometry, j u s t asEins te in i n t e rp re t e d grav i t a t i on as curva ture when he introduceda change o f d i r e c t i o n . Mathemat ica l ly and a e s t h e t i c a l l

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sa t i s fy ing though it was, Weyl 's theory had to be abandonedbecause of an objec t ion ra i sed by Eins te in -- t h a t the s izeo f an ob jec t would then depend on i t s h i s t o r y , with th er e s u l t t h a t chemical ly p u ~ e elements would not e x i s t andspec t r a l smears would be observed ins tead o f s p e c t r a l l i n e s .

Weyl 's a t tempt a t un i f i ed f i e ld theory had a grea t e rin f luence on d i f f e r e n t i a l geometry. In h is 1929 papers ongrav i t a t i on and the e l ec t ron , he a p ~ l i e d Eins t e in ' s not ion(o f a t t ach ing a l oca l s e t o f axes to e a c ~ po in t o f space-t ime) to quantum mechanics and in te rpre ted ' i t in t erms o fsp in . The desc r ibab i l i t y o f ce r t a in par t ic le ' s ca l l ed " f e r mions" by means o f a lgebra ic en t i t i e s ca l l ed sp inor s , f i r s tpointed ou t by Weyl in these papers , l s now v i n ~ i c a t e d by th emodern theory of neu t r inos . The wave equat ion obta ined byWeyl a r e , s t r ange ly enough, no t i n v a r i a ~ t under s p a t i a lr e f l e c t i o n ( "non-conserva t ion of pa r i ty" [2"9] ) , and one of themtu rns out to be a su i t ab l e d e sc r i p t i o n o f the so -ca l l ed "weaki n t e r ac t ions" ( the forces occurr ing in nature being the s t rongo r nuc lea r i n t e rac t i on , the e lec t romagne t ic i n t e r a c t i o n , th eweak i n t e rac t i on and the grav i t a t i ona l i n t e r ac t ion in decreas ing orde r o f s t r eng th ) . [30]

An of f - shoo t o f Weyl 's geometr ica l i nves t iga t ions ingenera l r e l a t i v i t y was h is subsequent i n t e r e s t in th e so-ca l led"space-problem" of ge t t i ng to the " root" of the s t ruc tu re of agenera l met r ic space . I t s so lu t ion was publ ished in 1923 asa book, Mathematische Analyse des Raumproblems (Mathematicalanalysis o f the space-problem). This led na tu ra l ly to th egenera l problem o f the rep resen ta t ion o f cont inuous groups.In h is t h ree c l a s s i c a l papers o f 1925 - 1926, he made thef i r s t s ign i f i can t con t r ibu t ions to the g loba l s tudy of Liegroups and blazed a t r a i l fo r modern re sea rch . One of thehigh po in t s o f h is inves t iga t ions i s th e well-known "Pe te r Weyl theorem" o f 1927.

At the same t ime , quantum theory was tak ing long s t r idesfo l lowing th e i n i t i a l breakthroughs o f the previous decade.An unders tanding o f th e microcospic world o f the atom gradual ly


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emerged with the profound c on t r ibu t ions o f phys i c i s t s l ikeBorn, Werner Heisenberg [31) , Wolfgang Pau l i (32] , P.A.M.Dirac [3 3 ] , Eugene Wigner [34] and Erwin Schrodinger [3 5] .With h is discern ing e y e ~ o r r e l a t i o n s h i p s , Weyl saw t h a th is newly developed theory of cont inuous groups could p laya v i t a l ro l e in the mathemat ical formula t ion o f quantumt heo ry . Thus appeared in 1928 h is c l a s s i c a l work Gruppen-theorie und Quantenmechanik (The theory o f groups andquantum mechanics). I t s in f luence may be summed up in thesewords o f Dyson : "By br ing ing group theory in to quantummechanics he led the way to our modern s ty l e o f th ink ingin phys ics . Today th e i n s t i nc t i ve r e a c t ion o f every t he o re t i c a l p h y s i c i s t , conf ronted with an unexplained r egu la r i tyin the behav iour o f elementary p a r t i c l e s , i s to pos tu la tean under ly ing symmetry-group." [101

Another founda t ion-shaking event in th e f i r s t decadeo f t h i s century was Brouwer 's r e v o l t aga ins t th e abso lu teva l id i t y o f c l a s s i c a l ( Ar i s t o t e l i an ) l og i c . One o f th etwo "laws" of the c l a s s i c a l l og ic , the so-ca l led "law o fthe excluded midd le" , s t a t e s t h a t a given s ta tement i se i the r t rue or f a l s e . Clas s i ca l mathemat ics , or mathemat ics up to th e l a t e n ine teen th cen tu ry , has always assumedthe unive rsa l va l id ty o f t h i s law when appl ied to a givenmathemat ica l system ( in p a r t i c u l a r , number t he o ry ) . Theex i s t ence o f mathemat ical ob je c t s i s deemed to be e s t a b l i s he donce t h e i r non-ex is tence could be shown to lead to a cont r a d i c t i o n with in the system even though the proof i t s e l fgives no i nk l ing as to what the objec t s a r e . E a r l i e r , L.Kronecker [36) had objec ted s t rong ly to the use o f th e"a c tua l ly i n f i n i t e " in mathematics and would only acceptcons t ruc t ive de f in i t ions and proofs . He merc i le s s ly criti-cized the works o f Richard Dedekind [37] in ana lys i s and ofGeorg Cantor [38] in se t t heo ry . This seemingly i c onoc la s t i ca t t i t ude found little suppor t or sympathy from th e r e s t ofth e mathemat ical community. However, in h is 1907 t he s i s ont h ~ l imi t a t ions o f th e law o f th e excluded middle , Brouwer

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reopened th e old wounds o f Kronecker ' s a t t ack and s t a r t ed asys temat ic programme to r ebu i ld mathematics on th e " in tu i t i on -i s t i c " phi losophy t h a t a l l mathematics may be based on thei n t ege rs and developed by means of " in tu i t i ve ly c l e a r " con-s t ruc t ive methods. [39]

These ideas propagated to Zurich and found in the youngWeyl a r ecep t ive mind. "Brouwer opened our eyes and made ussee how f a r c l a s s i c a l mathemat ics , nourished by a b e l i e f inth e ' ab so lu t e ' t h a t t ranscends a l l human p o ss ib i l i t i e s ofr e a l i z a t i o n , goes beyond such s ta tements as can cla im r e a lmeaning and t ru th founded on evidence ." (40] With youthfu lzes t and missionary z e a l , Weyl championed th e cause o f th eI n s t u i t i o n i s t school , much to th e dismay o f h is o ld t eacher ,Hi lbe r t . Fear ing t h a t much o f what was dear and va luable tohim would be j e t t i soned from th e bulk o f mathemat ics , Hi lbe r tcountered with h is own "fo. rmalis t" programme of es t ab l i sh ingthe cons is tency o f mathemat ics by reducing mathemat ics to aformal game o f meaningless symbols -- an approach which wasfore ign and repugnant to Weyl.

A monograph, Das Kontinuum (The continuum) appeared in1918 in which Weyl l a id down some of h is own i deas , and anexpos i t ion o f h is s tand appeared in h is bookFhiZosophie derMathematik und Naturwissenschaft ( t r ans l a t ed and expanded in1949 as Philosophy o f mathematics and natural science) , whichwas publ ished in th e 1927 Handbuch der Philosophie. Althoughh is con t r ibu t ions to th e founda t ions o f mathemat ics were no tas i nc i s ive as h is o th e r e f f o r t s , h is wri t ings gained fo r th ei n t u i t i o n i s t s a wider audience and " turned th e r evo lu t ionarydoc t r ine o f h is t ime in to th e orthodoxy of today . " [25]

Among some o f h is work done dur ing the Zurich p e r io d , butwhich do not fit in to any of the above gene ra l themes, are h ispapers on "uniform O.is t r ibut ion modulo 1 " , convex surfaces andalmost per iod ic func t ions . A sequence o f r e a l numbers a 1 ,a 2 , i s sa id to be uniformly d i s t r i bu ted modulo 1 if, fo r eachpos i t i ve i n t ege r n a n d each pos i t ive r e a l number c , A(n ,c ) /nt ends to c as n tends to i n f i n i t y , where A(n ,c) i s th e numbero f those o f a 1 , ,a which have f r a c t i o n a l p a r t s between 0nand c . In h is c l a s s i c a l papers o f 1916, Weyl gave a necessary

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and su f f i c i en t condi t ion fo r a sequence to be uniformly d i s t r i bu t ed modulo 1 , namely t h a t , fo r each non-zero i n t ege r h ,1 n 2rrihakE e t ends to zero as h t ends to i n f i n i t y . He a lson k=ldeveloped a method of es t imat ing ce r t a in exponent ia l sumse s s e n t i a l to the s tudy o f Waring 's Problem [41] and of thees t imat ion o f th e Riemann ze ta - func t ion [42) . This work cameclose to the theory of almost per iod ic funct ions , whosefoundat ions were l a id by Bohr in 1924 and on which Weyl l a t e rpubl ished some r e s u l t s .

Ear l i e r in Gott ingen, Weyl had col labora ted with Hi lbe r tin the publ ica t ion o f Minkowski 's co l l ec ted works and wass t imula ted by Minkowski 's theory o f convex bodies . On h isre tu rn to Zurich, he publ ished in 1916 and 1917 some workin t h i s f i e l d , but h is i n t e r e s t by then had sh i f t ed to gene ra lr e l a t i v i t y .Uncertainty The Zurich period 1913 - 1930 was punctuated bya shor t s t i n t (1928 - 1929) as Jones r esea rch p ro fes so r inmathematical phys ics a t Prince ton Univers i ty . His re turn toZurich was confronted by an o f f e r from Gott ingen to succeedHi lbe r t a t th e Mathematical I n s t i t u t e , whose pos t -war development had progressed s t ead i l y under th e e f f o r t s o f Courantu n t i l i t s abrupt cessa t ion by th e Nazi pol icy of prese rv ing th eAryan "pur i ty" of German un ive rs i t i e s . . Weyl had re fused anea r l i e r oppor tuni ty but now, in h is f o r t i e s , he could notr e s i s t the c a l l o f the g re a t Gat t ingen t r a d j t i o n t h a t hadseemingly survived i n t ac t from the pr iva t ions of the ear lypost-war per iod . Af ter some h e s i t a t i o n , he f ina l ly made th ec ruc i a l decis ion of accept ing th e of fe r . And in the spr ingof 1930, he a r r ived in Gott ingen with optimism and expecta t ion .What he had hoped fo r tu rned ou t to be a b r i e f sojourn o f t e n ~sion and uncer t a in ty . "The t h ree years t h a t fol lowed werethe most pa in fu l t h a t Hel la and I have known." [23]

In 1932, German mathemat ic ians ce lebra ted the seven t i e thb i r ~ h d a y o f H i lbe r t . Weyl wrote a bir thday gree t ing in DieNaturwissenschaf ten . Fr iends , former col leagues and s tuden ts

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converged a t Gott ingen fo r the occas1on which would be thel a s t supper o f Gott ingen mathemat ic ians . In th e same y ea r ,th e Nat ional S o c i a l i s t Par ty came to power and a sys temat iccampaign was soon s e t in motion to purge a l l Jews from th ei n t e l l e c t u a l , p o l i t i c a l and economic spheres o f German l i f e .The axe f e l l on the Mathemat ical I n s t i t u t e . Those mathema-t i c i a n s who were c la s s i f i ed as Jews were dismissed and forcedto migra te . Those who were n o t dismissed but who unders toodth e meaning o f t h i s a l so l e f t Germany in d i sgus t .

Meanwhile, Weyl took over as th e head o f th e Mathemat icalI n s t i t u t e and t r i ed hard to sa lvage th e s i t ua t i on . By l a t esummer o f 1933, it was evident t h a t nothing could be sa lvaged.Weyl and h is family were back in Switzer land on vaca t ion .Friends wrote to him from America pleading with him to leaveGermany be fore it was too l a t e . The newly crea ted In s t i t u t efo r Advanced Study a t Pr ince ton of fe red him a p o s i t i o n . F i n a l l y ,a f t e r an agonizing per iod of h e s i t a t i o n , he took E i n s t e i n ' sadvice to jo in him a t th e I n s t i t u t e .Pr ince ton and t h e r e a f t e r . From Zur ich , Weyl sen t to Got t ingenh is l e t t e r of re s igna t ion and a r r ived a t Pr ince ton towards th eend o f 1933 to take up a professorsh ip of mathematics a t th eI n s t i t u t e fo r Advanced Study. He he ld the pos i t ion u n t i l h isr e t i r emen t in 1952. The i n i t i a l years a t th e In s t i t u t e wereyears o f pa in fu l adjus tment to an unfami l ia r enviroment anda fore ign tongue. But he responded to the cha l lenge and ove r -came it. His newly gained happiness was fo r a while sha t t e redby th e dea th o f h is wife Hel la in 1948. This was a c ru e l blowto him and h is usua l s e l f was only re s to red by a second marr iagein 1950 to El len Baer Lohnste in o f Zurich. He would hencefor thdiv ide h is t ime between Pr ince ton and ZUrich. I t seemed t h a t[4 3 ] , as a r e s ~ l t of l iv ing in Switzer land fo r a t ime , he v io -l a t ed the permis s ib i l i t y fo r a na tu ra l i zed American c i t i z e n tos tay abroad and r e t a in h is c i t i zensh ip . His l o ss o f Americanc i t i zensh ip by negl igence caused an uproar among American mathe-mat ic ians . Effo r t s were made to r e s to r e h is c i t i z e n s h i p . But

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on 8 December 1955) one month a f t e r h is sevent ie th b i r thdayce leb ra t ion ) Weyl went out to pos t a l e t t e r in Zurich andsuf fered a f a t a l hear t a t t ack .

When Weyl migrated to America, he was in h is l a t e f o r t i e s ,and he knew t h a t th e bulk o f h is achievements lay behind him.But he was no t content to r e s t on h is l au re l s . He could y e thelp to bui ld a t th e In s t i t u t e a grea t cen t re fo r world mathemat ics . Most of h is t ime was now devoted to expanding on ands impl i fy ing th e i n t r i cac i e s o f the g re a t themes o f h is mathemat ica l work. Thus in h is book The oZassioaZ g r o u p s ~ theirinvariants and representat ions publ ished in 1939) he gave apurely a lgebra ic t rea tment o f h is well-known r e s u l t s of 1926and, i n add i t i on , s e t ou t to r e l a t e these r e su l t s with thetheory of i nva r i an t s whose c e n t r a l problems had been " f in i shedonce and fo r a l l " ( in h is own words) by H i lbe r t . With t h i snew synthes i s , "one can as_k whether Weyl had no t , in f ac t )del ivered i t s aoup de grace". [ 2 3]

He l ec tu red and wrote in Engl ish with the f ine s t rokesof a maste r . In 1952 a book Symmetry was publ ished from th el ec tures he had given a t Prince ton Univer s i ty . I t mani fes t s ,not without awe- inspi r ing f ee l i ngs , th e depth and breadth o fh is Promethean un ive r sa l i t y . Up to h is f i n a l years , he d i s played a keen i n t e r e s t in and sharp awareness o f th e achievementsof the modern genera t ion o f young mathemat ic ians . This i sexempl i f ied by h is address on the occas ion of th e presen ta t iono f the Fie lds Medal Award [44] a t the 1954 In t e rna t iona l Congress o f Mathematicians a t Amsterdam. His own growth anddevelopment as a mathematician had run p a r a l l e l to the evolu t iono f modern mathematics . "But when Weyl was asked by a pub l i she rto wr i te a h i s to ry o f mathematics in the twent ie th century heturned it down because he f e l t t h a t no one person could do it."[43] A sober ing thought a t the presen t s tage o f div i s ion andpro l i f e r a t ion o f mathematical knowledge.

When H i l b e r t d ied , only Weyl could have wri t t en by h imse l fa comprehensive survey (11] o f Hi lbe r t ' s work. But when an

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appr a i s a l o f Weyl 's accomplishments had to be wri t t en a f t e rWeyl 's dea th , it was done through the combined e f fo r t s of f ivemathematicians (see (25] ) . Another appra i sa l [23] was wri t t enby two of the wor ld ' s lead ing mathematicians . In the l a t t e r ,Weyl 's papers a re c l a s s i f i e d under eleven broad sub jec t headings :ana lys i s , geometry, invar ian t s and Lie groups , r e l a t i v i t y ,quantum t heo ry , theory o f a lge b ra s , geometry o f numbers , l og ic ,phi losophy , h i s to ry and b iog raph ies , and o the r s . The paperson ana lys i s are d i s t r ibu ted fu r ther in to twelve spec ia l izedt op ic s ; geometry in to four ; invar ian t s and Lie groups in toth r ee .

In the ob i tuary [15] , Weyl wri te s o f Hilber t : "The methodica lun i ty o f mathematics was fo r him a mat ter o f be l i e f and exper ience,I t appeared to him es sen t i a l t ha t -- in the face o f the manifoldi n t e r r e l a t i ons and fo r th e sake of the f e r t i l i t y o f research -th e product ive mathemat ic ian should make himsel f a t home in a l lf i e ld s ... A cha rac t e r i s t i c f ea tu re o f H i l b e r t ' s method i sa pecu l ia r ly d i r e c t a t t ack on problems, unfe t te red by any a lgo -

~ i t h m s ; he always goes back to th e ques t ions in t h e i r o r ~ g i n a ls imp l i c i t y . " Perhaps these words could wel l have been wr i t t eno f Weyl himsel f .

Mathematics fo r him i s not j u s t i f i e d by i t s success o ruse fu lness . The ~ a i s o n d'e tre o f mathematics l i e s in the veryhe a r t o f human exis tence i t s e l f . " I bel ieve t h a t mathematizing,l i ke music , i s a crea t ive a b i l i t y deeply grounded in man'sna ture . Not as an i so la ted t echn ica l accomplishment , but onlyas pa r t o f human ex i s t ence in i t s t o t a l i t y can it f ind i t sj u s t i f i c a t i on . ' ' [45] The ul t ima te c r i t e r i on o f h is own work i sbeauty . "My work always t r i ed to uni te the t rue with thebeau t i fu l ; but when I had to choose one o r the o the r , Iusua l ly chose th e beau t i fu l . " [10]


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Notes and re fe rences

1. Archimedes (c.287B.C. - 212B.C.) , Greek mathematician,phys i c i s t and engineer ; probably s tud ied with successors ofEucl id (c.300B.C.) a t Alexandria ; grea tes t mathemat ica l geniuso f an t iqu i ty ; cont r ibu ted t o a r i t hme t i c , geometry, hydros ta t i c s ,mechanics; developed proofs by methods of exhaust ion andreduct ion to absurd i ty .2. Isaac Newton ( 1 6 4 2 - 1727) , Engl ish mathematician,phys i c i s t and as t ronomer; invented d i f f e r e n t i a l and i n t eg ra lca lcu lus ; founded t heo re t i c a l mechanics; discovered law ofgrav i ta t ion .3. Joseph Louis Lagrange ( 1 7 3 6 - 1813) , French mathemat ic ianand mathemat ica l phys ic i s t ; contr ibuted to number theory , ca lcu lusof va r i a t ions , p a r t i a l d i f f e r e n t i a l ~ q u a t i o n s , hydrodynamics,c e l e s t i a l mechanics .4. Leonhard Euler (1707 - 1783) , Swiss mathematician andphys ic i s t ; s tud ied under Johann Bernoul l i (1667 - 1748) a tBasle (Swi tzer land) ; spent most o f h is t ime a t S t. Petersburg(now Leningrad, USSR); most pr o l i f i c mathematician in h i s to ry ;contr ibuted to ana lys i s , geometry, number theory , c e l e s t i a lmechanics , hydrodynamics, op t i c s , accous t i cs , nav iga t ion .5. Car l Fre ide r ich Gauss (1777 - 1855) German mathemat ic ian ,astronomer and phys i c i s t ; es tab l i shed the t r ad i t i on of Gott ingen;exer ted a unive rsa l in f luence on mathematics and physics throughh is work on number theory , ana lys i s , geometry, c e l e s t i a l mechanics ,geodesy, t e legraphy , elect romagnet ism. See C. T. Chong, "Noteson mathema,t i c i a n s 1. Car l Fr iedr ich Gauss", This Medley Vol. 3 ,No.1 (1975), 6- 10.6. Henri Poincare ( 1 8 5 4 - 1912) , French mathematician and

/ /phys ic i s t ; s tudied a t Ecole Polytechniqae Ecole Superieur desMines; worked a t Caen (France) , Par i s ; contr ibuted to ana lys i s ,c e l e s t i a l mechanics , number theory , geometry, f lu id mechanics ,elect romagnet ism, phi losophy o f sc ience . See C. T. Chong,"Notes on mathematicians 4. Henri Poincare" , This Medley Vol .4 ,

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No.1 (1976), 1 3 - 34.7. David Hi lbe r t ( 1 8 6 1 - 1943) , German mathemat ic ian andmathemat ica l phys ic i s t ; b u i l t up the school o f Gott ingen;contr ibu ted to a lgebra , number theory , geometry, ana lys i s ,foundat ions o f mathematics , mathematical methods of phys ics .See C. T. Chong, "Notes on mathemat ic ians 6. David Hi lbe r t " ,

r ~ i s Medley Vol .4 , No.3 (1976) , 1 3 4 - 159.8. John von Neumann ( 1 9 0 3 - 1957) , Hungarian-born mathemat i c i an and t heo re t i ca l phys ic i s t ; s tudied a t Budapes t , Ber l in ,Zur ich , Got t ingen; worked a t Ber l in , Hamburg, I n s t i t u t e fo rAdvanced Study (USA); con t r ibu ted to se t t heo ry , a lgebra ,ana lys i s , game theory , computer sc ience , mathematical phys ics .See Y. K. Leong, "Notes on mathemat ic ians 3. John von Neumann",This Medley Vol .3 , No.3 (1975) , 90 - 106.9. Freeman John Dyson ( 1 9 2 3 - ) , Bri t i sh -born Americant h e o r e t i c a l phys ic i s t ; s tudied a t Cambridge Univers i ty ;worked a t Cambridge, Cornel l , Pr inceton; professor a t I n s t i t u t e fo r Advanced Study (USA); con t r ibu ted to quantum e l e c t r o dynamics and theory o f e lec t romagne t ic r ad ia t ion .10. Freeman J . Dyson, "Obi tuary" , Nature 177 (1956) , 457 -458.11. Hermann Weyl, "David Hilber t and h is mathematical work",Bulle t in o f the American Mathematical Society 50 (1944) , 6 1 2 -654. Repr inted in Gesammelte Abhandlungen, Band IV, Spr inger ,Ber l in , 1968.12. Hermann Minkowski ( 1 8 6 4 - 1909), Russian-born Germanmathematician and t heo re t i ca l p h y s i c i s t ; s tudied a t Ber l in ,Konigsberg (now Kal in ingrad , U.S.S .R. ) ; worked a t Konigsberg,Zurich , Federa l I n s t i t u t e o f Technology (Swi tzer land) , Got t ingen;con t r ibu ted to geometry o f numbers, a lgebra , genera l r e l a t i v i t y ,elect rodynamics .13. Fel ix Klein ( 1 8 4 9 - 1925) , German mathematician; s tud ieda t Bonn, Leipz ig , Got t ingen , Vienna; worked a t Bonn, Got t ingen ,Erlangen (Germany); con t r ibu ted to geometry, a n a l y s i s , d i f f e ren t i a lequat ions .

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14. In th e pre-war German academic sys tem, a doctor o fphi losophy was requ i red to produce another or ig ina l pieceo f work fo r the so-ca l led "Hab i l i t a t i on" . He would then beawarded the title o f Pr iva tdozen t and the pr iv i l ege tol ec tu re in the univer s i ty without pay. He could , however ,c o l l e c t fees from s tuden t s a t t end ing h is l e c tu re s . Uponr ecogn i t ion o f h is work and a b i l i t i e s , he would r ece ivethe s a l a r i ed pos t o f an Ext rao rd ina r iu s ( a s s i s t a n t pro fe s s o r ) .The f i n a l goal would be an Ord ina r i a t (p rofessor ) .15. Hermann Weyl, "Obi tuary : David Hi lbe r t 1862 - 1943",Obituary Notices o f FeZZows o f the RoyaZ Society 4 (1944),5 4 7 - 553; American PhiZosophicaZ Society Year Book (1944),387 - 395. Repr inted in GesammeZte A b h a n d Z u n g e n ~ Band IV.16. Max Born (1882 - ) , German t heo re t i ca l phys ic i s t ;Nobel l a u re a t e (1954); s tud ied a t Breslau (Poland) , Heide l berg , Zur ich , Got t ingen; worked a t Got t ingen , Be r l in , Frankf u r t , Edinburgh; con t r ibu ted to quantum theory and genera lr e l a t i v i t y .17. Richard Courant (1888 - 1972), Pol i sh-born Americanmathemat ic ian; s tud ied a t Bres lau , Zur ich , Got t ingen;worked a t Got t ingen , Munster (Germany), Cambridge, New YorkUnivers i ty ; d i r e c t o r o f Courant I n s t i t u t e o f Mathematics(New York); cont r ibu ted to appl ied mathemat ics , ana lys i s andi t s appl ica t ions to physics .18. Harald Bohr (1887 - 1951) , Danish mathemat ic ian; bro the ro f phys i c i s t Niels Bohr (1885 - 1962); s tud ied a t Copenhagen and Got t ingen; worked a t Copenhagen and College of Technology(Denmark); con t r ibu ted to ana lys i s and theory o f a lmos t per iod icfunct ions .19. Erich Heeke


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IIIIil

Federa l I n s t i t u t e o f Technology (Swi tzer land) , Brown, Stanf or d ; con t r ibu ted to ana lys i s , number t heo ry , pr obab i l i t y ,appl ied mathemat ics , mathemat ica l pedagogy.21. Constance Reid, HiZbert , Allen & Unwin, Spr inger , London,Be r l in , 1970.22. Er ik Iva r Fredholm (1866 - 1927), Swedish mathematician;s tud ied a t Stockholm Poly t echn ic , Uppsala , Stockholm; workeda t Stockholm; con t r ibu ted to theory o f i n t e g r a l equat ions .23 . C . Cheval ley and A. Weil , "Hermann Weyl (1885 - 1955)"( in Fr ench ) , L J Enseignement Mathematique, tome I I I , f a s c . 3( 1957).24. L.E . J . Brouwer (1881 - 1966), Dutch mathemat ic ian ; s tud iedand wor ked a t Amsterdam; con t r ibu ted to founda t ions o f mathematics 1l og ic , t o pology.25. M.H.A. Newman, "Hermann Weyl", BiographicaZ memoirs o fFel low s o f the Royal Society 3 (1957), 3 0 5 - 328.26 . Alber t Eins te in ( 1 8 7 9 - 1955), German-born American phy

~ ; ; i c i s t ; Nobel l au r ea t e (1922); s tud ied a t Federa l I n s t i t u t e o fTe chnology (Swi t ze r l and) ; worked a t Bern, Zur ich , Prague, Leyden,Ber l i n , I n s t i t u t e fo r Advanced Study (USA); con t r ibu ted toBrownia n motion, p h o t o e l e c t r i c i t y , s pe c ia l and genera l r e l a t i v i t y .27. Arthur Stanley Eddington ( 1 8 8 2 - 1944), Engl i sh a s t r o phys i c i s t ; educa t ed a t Cambridge; worked a t Royal Observatory(Greenw i c h ) , Cambridge; cont r ibu ted to genera l r e l a t i v i t y ands t e l l a r as t rophys ic s .28. Theodor Kaluza ( 1 8 8 5 - 1954) , German mathemat ica l p h y s i c i s t ;s tud ied a t Konigsberg (now Kal in ingrad , USSR); worked a t Konigsb er g , Kie l , Got t ingen; con t r ibu ted to genera l r e l a t i v i t y ( thefir s t to in t roduce a f i f t h dimension in to a uni f ied f i e ld theory) .29 . The concept of par i ty in phys ics i s t h a t of mi r ro r symmetryo f phys i c a l phenomena. For example, a charge moving p a r a l l e lto an e l e c t r i c cur ren t i s def lec ted toward the c u r r e n t by th einduced elec t romagnet ic f i e l d . In the mi r ror image o f t h i s

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phenomenon, the current i s reversed and the charge i s s t i l ldeflected toward the current . Pari ty is conserved in e lec t romagnetic interact ions . In 1957, T.D. Lee (1926- ) andC.N. Yang (1922 - ) suggested tha t pari ty i s not conservedfor weak interact ions . This was confirmed by C.S. Wu (1913 - )and others for beta-decay processes.30. See, for example, B.L. van der Waerden Group theory andquantum meahanias, Springer, Berlin, 1974.31. Werner Heisenberg (1901 - 1976), German theore t ica l phys ic i s t ; Nobel laureate (1932); studied a t Munich, Gottingen;worked a t Leipzig, Berlin, Gottingen, Munich; contributed toquantum theory.32. Wolfgang Pauli (1900- 1958), Austrian theore t ica l phys i c i s t ; Nobel laureate (1945); studied a t Munich; worked a tGottingen, Copenhagen, Hampurg, Federal Ins t i tu te of Technology (Switzerland); contr ibuted to quantum theory; predictedexistence of neutr ino.33, Paul A. M. Dirac (1902 - ) , English theore t ica l phy-s i c i s t ; Nobel laureate (1933); studied a t Bris to l , Cambridge;worked a t Cambridge, Ins t i tu te for Advanced Study (USA); nowa t Cambridge; contr ibuted to quantum theory; predicted existenceof posi t ron.34. Eugene Paul Wigner (1902 - 1975); Hungarian-born Americanphysic is t ; Nobel laureate (1963); studied a t Ins t i tu te of Technology (Berl in); worked a t Princeton, Wisconsin, Chicago, ClintonLaboratories , Oak Ridge; contr ibuted to quantum theory, nuclearphysics.35. Erwin Schrodinger (1887 - 1961), Austrian physic is t ; Nobellaureate (1933); studied a t Vienna; worked a t Vienna, Stut tgar t(Germany), Zurich, Berl in , Ins t i tu te of Advanced Study ( I ~ e l a n d ) ;returned to Vienna af te r re t i rement in 1955; contributed toquantum mechanics, theory of colour .36. Leopold Kronecker (1823 - 1891), German mathematician;studied a t Berl in; had independent source of income from own

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II"I

bus iness ; did no t hold any univer s i ty pos i t ion u n t i l 1883 (asprofessor a t Ber l in ) ; contr ibuted to number theory , a lgebra .37. Richard Dedekind ( 1 8 3 1 - 1916) , German mathematician;s tud ied a t Gott ingen (under Gauss); worked a t Got t ingen ,Federa l I n s t i t u t e o f Technology (Swi tqer land) , TechnicalI n s t i t u t e (Brunswick); contr ibuted to number t he o ry , ana lys i s .38. Georg Cantor ( 1 8 4 5 - 1918) , German mathematician; s tudieda t Zurich , Got t ingen , Frankfur t , Be r l in ; worked a t Halle (Germany); contr ibuted to se t theory , theory of t r a n s f i n i t e numbers.39. See, fo r example, Raymond L. Wilder , Introduction to thefoundations o f mathematics, 2nd ed i t i on , Wiley, New York, 1965.40. Hermann Weyl, "Mathematics and l og i c . A b r i e f surveyserving as a preface to a review of ' T ~ e philosophy of BertrandRusse l l ' " ,Amer ican Mathematical Monthly 53 (1946) , 2- 13.Repr inted in GesammeZte AbhandZungen, Band IV.41. Waring ' s Problem concerns the determinat ion of the numberg( k ) , which i s the sma l le s t value of r fo r which every pos i t ivein teger i s the sum o f r non-negat ive k th powers. For i n s t ance ,Lagrange has shown t ha t every pos i t i ve i n t ege r i s the sum o f 4squares . The ex is tence o f g(k) fo r each k has been proved byHi lbe r t in 1909. The value o f g(k) i s now known fo r a l l kexcept 4 and 5.42. The Riemann ze ta - func t ion . def ined by the t;;(s)S ser1es =1 + _!_+ 1 + l+ fo r complex it ' 1 '+ ... . . s = (J + (J >2s 3s snwhose ana ly t i c cont inua t ion gives a meromorphic func t ion . Thishas been in t roduced in the i nves t i ga t i on of th e number of primesl e ss than a given number.43. S. M. Ulam, Adventures o f a mathematician, Char les Scr ibne r ' sSons, New York, 1976.44. Once in four yea r s , th e In te rna t iona l Mathematical Unionholds an In t e rna t iona l Mathematical Congress a t which th e pr e s t ig ious Fie lds Medal i s awarded to one or more mathematicians(below the age of 40) fo r s ign i f i c an t cont r ibu t ions to mathematics,

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The r ec ip i en t s of the 1974 Congress a t Vancouver were DavidMumford of America and Enrico Bombieri of I t a l y .45. Quoted on f ron t i sp iece of Band I o f GesammeZte Abhand-Zungen.

* * * * * * * * * *

ANNOUNCEMENT

The In t e rn a t i o n a l Congress o f Mathematicians w i l l be he ld inHels inki , F in land , dur ing August 15-23 , 1978. Correspondenceconcerning the Congress should be addressed to

In t e rna t iona l Congress of Mathemat ic ians ,ICM 78Department o f MathematicsUnive rs i ty o f Hels ink iHal l i tu ska tu 15SF-00100 Hels ink i 10Fin land

notes on mathematicians

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