An Interview With Michael Atiyah

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An Interview With Michae l At iyah

Michael Atiyah was born in 1929 and received his B.A. and Ph.D. from Trinity College, Cambridge

(1952, 1955). Duri ng his career he has been Savilian Professor of Geometry at O xford (1963-69)

and Professor of Mathematics at the Institute for Advanc ed Stud y in Princeton (1969-72); he is

currently a Royal Society Research Professor of Mathematics at Oxford University.

Am on g other honors Professor Atiy ah is a Fellow of the Royal Society and a member of the National

Academies of France, Sweden, and the U SA. He received the Fields Medal at the International

Congress o f Mathematicians held in Mosco w, 196 6. His research interests span a bro ad area of

mathematics including topology, geometry, differential equations and mathematical physics.

The following is an edited version of an interview in Ox ford with Roberto Mini o, former editor

of The In te l l igencer .

M I N I O: I t h in k so m e in f o r m a t io n a b o u t y o u r b a c k-

g r o u n d m ig h t b e [ v a lu a b le ] . Wh e n d id y o u s t a r t g e t -

t i n g i n t e r e s t e d i n m a th e m a t i c s? Ho w e a r ly ?

ATIYAH: I t h in k I wa s a lwa y s i n t e r e s t e d i n m a th s

f r o m a v e r y y o u n g a g e . Bu t t h e r e wa s a s t a g e - - I wa s

a b o u t f i f t e e n - - w h e n I g o t v e r y i n t e r e s t e d i n c h e m -

is t ry , and I thought tha t would be a g rea t th ing ; a f te r

a b o u t a y e a r o f a d v a n c e d c h e m is t r y I d e c id e d th a t i t

wa sn ' t wh a t I wa n te d to d o a n d I we n t b a c k to m a th s .

I n e v e r se r io u s ly c o n s id e r e d d o in g a n y th in g e l se .

c a m e o v e r af t er t he w a r - - t h a t w a s a g o o d s c h oo l .

T h e r e we r e a l o t o f g o o d p u p i l s t h e re . T h e n I we n t t o

C a m b r i d g e , a n d t h e re w e r e m a n y g o o d s t u d e n t s

a r o u n d .

I don ' t th in k I was par t icu lar ly in f luenced by one

p e r so n . Bu t I h a d a g o o d e d u c a t io n , a n d I h a d p l e n ty

o f o p p o r t u n i ty t o m e e t g o o d m a th e m a t i c i a n s ; I h a d a

g o o d b a c k g r o u n d in t h a t s e n se .

M I N I O: An d th a t wa s a l r e a d y n o t i c ea b le v e r y e a rly ?

ATIYAH: Ye s , I t h in k so . M y p a r e n t s a lwa y s t h o u g h t

tha t I was cu t ou t to be a mathemat ic ian f rom a very

y o u n g a g e , a l l t h e wa y th r o u g h .

M I N I O : B u t t h e y w e r e n ' t m a t h e m at i c ia n s ?

ATIYAH: T h e y we r e n ' t m a th e m a t i c i a n s , n o .

M I N I O : A n d w e r e y o u h e l p e d at s c ho o l? W e r e y o u r

t e a c h e r s r e a so n a b ly g o o d w i th y o u ?

ATIYAH: We l l , I t h in k I h a d g o o d t e a c h e r s a n d m y

re la t ions wi th th em we re go od . I was a t schoo l fi r s t in

E g y p t a n d I we n t t o q u i t e a g o o d sc h o o l t h e r e .

M I N I O : W e r e y o u b o r n t h er e ?

ATIYAH: No , I wa s b o r n i n E n g la nd , b u t we l i v e d

i n t h e M i d dl e E a s t - - m y f a th e r w o r k e d i n th e S u d a n - -

so I we n t t o sc h o o l i n E g y p t f o r m y m a in se c o n d a r y

sc h o o lin g . I d id a c o u p le o f y e a rs i n E n g la n d wh e n I

THE MATHEM ATICAL INTELLIGENCER VOL. 6 , NO. 1 9 1984 Springer-Verlag New York 9



M I N I O : D i d y o u la r ge ly w o r k o n y o u r o w n a t C a m -

b r i d g e ?

A T I Y A H : W e l l , I c a m e t o C a m b r i d g e a f te r I h a d t w o

y e a r s o f N a t i o n a l S e r v ic e : it w a s a m a r v e l l o u s c o n t r a s t.

A c t u a l l y , I w e n t t o C a m b r i d g e a li tt le e a r l y i n t h e a c -

a d e m i c y e ar . I h a d a s u m m e r t e r m t h e r e a n d t h e l o v e l y

w e a t h e r a n d b e a u t i f u l s u r r o u n d i n g m a d e a tr e m e n -

d o u s i m p r e s s i o n o n m e . I u s e d t o e n j o y j u s t go i n g i nt ot h e l ib r a r y to r e a d , b e i n g s u r r o u n d e d b y a ll o f t h o s e

b o o k s . I t w a s a n i m p r e s s i v e a t m o s p h e r e ; i t c a p t u r e d

m y i m a g i n a t i o n .

T h e r e w e r e a l o t o f v e r y b r i gh t s t u d e n t s t h e r e a n d I

g o t r e a s o n a b l e h e l p f r o m t h e t e a c h e r s . I d o n ' t t h i n k

a n y o f t h e t e a c h e r s w a s p a r t i c u l a r ly i n s p i r in g ; s o m e o f

t h e le c t u re s w e r e g o o d a n d s o m e w e r e n o t s o g o o d .

M I N I O : O n e o f y o u r ea r ly p a p e r s w a s w i t h H o d g e ,

i s tha t r igh t?

A T I Y A H : Y e s , t h a t w a s p a r t o f m y th e s i s r e a ll y . H e

w a s m y r e s e a r ch s u p e r v i s o r , a n d t h a t w a s v e r y im -

p o r t a nt f o r m e - - w o r k i n g w i t h h im . I 'd c o m e u p t o

C a m b r i d g e a t a t i m e w h e n t h e e m p h a s i s i n g e o m e t r y

w a s o n c l as s ic a l p r o j e c t iv e a l g e b r a i c g e o m e t r y o f th e

o l d - f a s h io n e d t y p e , w h i c h I t h o r o u g h l y e n j o y e d . I

w o u l d h a v e g o n e o n w o r k i n g i n th a t a r e a e x c e p t t h a t

H o d g e r e p r e s e n te d a m o r e m o d e r n p o i n t of v i e w - -

d i f f e re n t i a l g e o m e t r y i n r e l a ti o n t o t o p o l o g y ; I r ec o g -

n i z e d t h a t . I t w a s a v e r y i m p o r t a n t d e c i s i o n f o r m e . I

c o u l d h a v e w o r k e d i n m o r e t r a d it io n a l t h i n gs , b u t I

t h i n k t h a t it w a s a w i s e c h o i c e, a n d b y w o r k i n g w i t h

h i m I g o t m u c h m o r e i n v o l v e d w i t h m o d e r n i d e a s . H e

g a v e m e g o o d a d v i c e a n d a t o n e s t ag e w e c o l la b o r a t ed

t o g e th e r . T h e r e w a s s o m e r e c e n t w o r k i n F r a n c e a t t h e

t i m e o n s h e a f t h e o r y . I g o t i n t e r e s t e d i n i t , h e g o t in -

t e r e s t e d in it , a n d w e w o r k e d t o g e t h e r a n d w r o t e a

j o i nt p a p e r w h i c h w a s p a r t o f m y t h e s i s. T h a t w a s v e r y

b e n e f i c ia l f o r m e .

M I N I O : O n e t h i n g t h a t ' s n o t i c e a b l e i s t h a t y o u

w o r k e d q u i t e a lo t w i t h o t h e r p e o p l e - - w i t h S i ng e r,

w i t h H i r z e b r u c h , w i t h B o tt .

A T I Y A H : Y e s , t h a t ' s r ig h t , I w o r k a l o t w i t h p e o p l e ,

a n d I t h i n k t h a t t h a t ' s m y s t y l e . T h e r e a r e v a r i o u s r e a -s o n s , o n e o f w h i c h i s t h a t I d a b b l e i n a n u m b e r o f

d i f f e r e n t a r e a s. M y i n t e r e s t i s i n t h e f a c t t h a t t h i n g s i n

d i f f e r e n t s u b j e c t s i n t e r a c t ; i t' s v e r y h e l p f u l t o w o r k

w i t h o t h e r p e o p le w h o k n o w a bi t m o r e a b o u t s o m e -

t h i n g e l s e a n d c o m p l e m e n t y o u r i n t e re s t . I f i n d it v e r y

s t im u l a t in g t o e x c h a n g e i d e a s w i t h o t h e r p e o p l e .

I 'v e c o ll a b o r a te d w i t h m a n y p e o p l e , s o m e o f t h e m - -

m a n y o f t h e m - - o n a n e x t e n d e d b as is fo r m a n y y e a rs .

I t 's p a r t l y m y p e r s o n a l i t y , t h e w a y I t h i n k a n d i n t e r a c t

w i t h p e o p l e , a n d p a r t ly b e c a u s e o f th e k i n d o f m a t h -

e m a t i c s I l ik e d o i n g , w h i c h i s r a t h e r b r o a d a n d t h e r e -

f o r e h a r d t o b e c o m p l e t e l y e x p e r t i n . I t i s v e r y h e l p f u l

t o h a v e s o m e o n e e ls e w h o k n o w s a b it m o r e a b o u t

s o m e t h i n g d i ff e r en t . S o w h e n I w o r k w i t h S i n g e r, f o r

e x a m p l e , h e ' s m u c h s t r o n g e r o n t h e a n a ly s i s si d e,

w h e r e I w a s r a t h e r w e a k , a n d I k n o w m o r e a b o u t t he

a l g e br a ic g e o m e t r y a n d t o p o l o g y .

M I N I O : D o y o u t h e n s e p a r a t e o u t t h e p r o b l e m s .

A T I Y A H : N o , n o . T h e c o l l a b o r a t i o n i s c o m p l e t e l y i n -

t e r m i n g l e d ; w e m e r g e o u r i n t e r e s t s a n d l e a r n a b o u t

e a c h o t h e r ' s t e c h n i q u e s . A f t e r a w h i l e w e ' r e m o r e o n

a n e q u a l f o o t i n g i n m o s t p a r t s o f t h e s u b j e c t s . O u r

i n t e r e s t s a r e v e r y c lo s e ; i t 's j u s t t h a t o u r b a c k g r o u n d s

a r e a b i t d i f f e r e n t .

M I N I O : H o w d o y o u s e le c t a p r o b l e m to s t u dy ?

A T I Y A H : I t h i nk th a t p r e s u p p o s e s a n a n s w e r . I

d o n ' t t h i n k t h a t ' s t h e w a y I w o r k a t a ll . S o m e p e o p l e

m a y s i t b a c k a n d s a y , " I w a n t t o s o lv e th i s p r o b l e m "

a n d t h e y s i t d o w n a n d s a y , " H o w d o I s o lv e th is

p r o b l e m . ' " I d o n ' t . I j u s t m o v e a r o u n d i n t h e m a t h e -

m a t i c a l w a t e r s , t h i n k i n g a b o u t th i n g s , b e i n g c u r i o u s ,

i n t e r e s t e d , t a l k i n g to p e o p l e , s t ir r i ng u p i d e a s ; t h i n g s

e m e r g e a n d I f o l lo w t h e m u p . O r I s e e s o m e t h i n g

w h i c h c o n n e c t s u p w i t h s o m e t h i n g e l se I k n o w a b o u t ,

a n d I t r y t o p u t t h e m t o g e t h e r a n d t h i n g s d e v e l o p . I

h a v e p r a c ti c a ll y n e v e r s t a r te d o f f w i t h a n y i d e a o f w h a t

I ' m g o i n g t o b e d o i n g o r w h e r e i t ' s g o i n g t o g o . I ' m

i n t e r e s t e d i n m a t h e m a t i c s ; I t a l k , I l e a r n , I d i s c u s s a n d

t h e n i n t e r e s t in g q u e s t i o n s s i m p l y e m e r g e . I h a v e n e v e r

s t a r t e d o f f w i t h a p a r t i c u la r g o a l , e x c e p t t h e g o a l o f

u n d e r s t a n d i n g m a t h e m a t i c s .

M I N I O : I s t h at h o w K - T h e o ry e m e r g e d ?

A T I Y A H : Y e s . I t w a s v e r y m u c h a n a c c id e n t i n s o m e

w a y s . I w a s i n t e re s t e d i n w h a t G r o t h e n d i e c k h a d b e e n

d o i n g i n a l g e b r a i c g e o m e t r y . H a v i n g g o n e t o B o n n I

w a s i n t e r e s t e d i n l e a r n i n g s o m e t o p o l o g y . I w a s i nt e r -

e s t e d i n s o m e o f th e q u e s t i o n s t h a t I o a n J a m e s h a d

b e e n s t u d y i n g o n t o p o l o gi c a l p r o b l e m s r e l a te d t o p r o -

j e c ti v e s p a c e s. I f o u n d t h a t b y u s i n g G r o t h e n d i e c k ' s

f o r m u l a s t h e s e t h i n g s c o u l d b e e x p l a i n e d , o n e g o t n i c e

r e s u l t s . T h e r e w a s B o t t 's w o r k o n t h e p e r i o d i c i t y t h e -o r e m ; I k n e w h i m a n d h i s w o r k . U s i n g t hi s I f o u n d

t h a t o n e c o u l d s o l v e s o m e i n t e r e s t in g p r o b l e m s . I t

s e e m e d n e c e s s a ry t o d e v e lo p s o m e m a c h in e r y t o m a k e

t h i s f o r m a l , a n d K - T h e o r y g r e w o u t o f t h a t .

Y o u c a n ' t d e v e l o p c o m p l e t e l y n e w i d e a s o r t h e o r ie s

b y p r e d i c ti n g t h e m i n a d v a n c e . I n h e r e n t ly , t h e y h a v e

t o e m e r g e b y i n t e l l i g e n t l y l o o k i n g a t a c o l l e c t i o n o f

p r o b l e m s . B u t d i ff e r e n t p e o p l e m u s t w o r k i n d if f e r e n t

w a y s . S o m e p e o p l e d e c i d e t h a t t h e r e i s a f u n d a m e n t a l

p r o b l e m t h a t t h e y w a n t t o s o l v e, s u c h a s t h e r e s o lu t i o n

o f s i n g u l a r i t i e s o r t h e c l a s s i f i c a t i o n o f f i n i te s i m p l e

1 0 T H E M A T H E M A T I C A L I N T E L L I G E N C E R V O L . 6, N O . 1 , 1 9 84



Pictured from left to right: Montgomery, Spencer, de Rham, Mrs. Garding, Lars Garding,Chandrasekharan, Bott, and Atiyah. Bombay, 1963.

g r oups . T he y sp e nd a l ar ge pa r t o f t he i r li fe de v o te d

to w or k ing tow a r ds t h i s e nd . I ' ve ne ve r d one tha t,

pa r t l y be c a use t ha t r e qu i r e s a s i ng l e - m inde d d e vo t ion

to one t op ic w hic h i s a t r e m e n dou s ga m ble .

I t a l so r e qu i r e s a s i ng l e - m inde d a ppr oa c h , by d i r e c t

o n s l a u g h t , w h i c h m e a n s y o u h a v e to b e t r e m e n d o u s l y

e xpe r t a t us ing t e c hn ic a l t oo l s . N ow som e pe op le a r e

ve r y good a t t ha t ; I ' m no t r e a l l y . M y e xpe r t i s e i s t o

sk i r t t he p r ob le m , t o go a r ound the p r ob le m , be h ind

t h e p r o b l e m . . , a n d s o t h e p r o b le m d i s ap p e a rs .

M I N I O : D o yo u fe e l t he r e a re m a ins t r e a m top i c s i n

m a the m a t i c s? A r e som e sub je c t s m or e im por t a n t t ha n

o the r s?

ATIYAH: Yes, wel l I th ink tha t i s t rue . I s t rongly

d i sa gr e e w i th t he v i e w tha t m a the m a t i c s i s s im ply a

c o l l e c t i on o f s e pa r a t e sub j e c t s , t ha t you c a n inve n t a

n e w b r a n c h o f m a t h e m a t i c s b y w r i t in g d o w n a x i o m s

1 , 2 , 3 a n d g o i n g a w a y a n d w o r k i n g o n y o u r o w n .

M a the m a t i c s is m u c h m or e o f a n o r ga n ic de ve lop m e nt .

I t ha s a l ong h i s to r y o f c onne c t ions w i th t he pa s t a nd

c onne c t ions w i th o the r sub j e c t s .

H a r dc or e m a the m a t i c s i s , i n som e se nse , t he sa m e

a s i t ha s a lw a ys be e n . I t i s c onc e r ne d w i th p r ob le m s

tha t ha ve a r i s e n f r om the a c tua l phys i c a l w or ld a ndo the r p r ob le m s ins ide m a the m a t i c s ha v ing to do w i th

n u m b e r s a n d b a s i c c a l c u la t i o n s , s o l v i n g e q u a t i o n s .

T h i s ha s a lw a ys be e n the m a in pa r t o f m a the m a t i c s .

A ny de ve lopm e nt t ha t she ds l i gh t on t he se t op i c s i s

a n im por t a n t pa r t o f m a the m a t i c s .

P a r t s t h a t g o o f f a n d a r e v e r y f a r r e m o v e d f r o m

the se , a nd don ' t she d m uc h l i gh t on t he e s se n t i a l s o f

m a the m a t i c s , a r e un l ike ly to be im por t a n t . I t m a y be

t h a t a n e w b r a n c h g r o w s o n i ts o w n a n d c a n e v e n tu a l l y

cas t light on o ther things , bu t i f i t goe s too fa r aw ay

and ge ts cut of f , then i t r ea l ly i sn ' t very s igni f icant in

m a th e m a t i c a l t e r m s . T he r e a r e r e a l l y o r ig inal i de a s

w hic h m a y , f o r a w hi l e , ope n up ne w th ings , bu t s t il l

t he y a r e c onn e c t e d w i th o the r im por t a n t pa r t s o f m a th-

emat ics an d inte rac t . The im por t ance of a par t of math-

e m a t i c s i s som e th ing one c a n judge r ough ly by the

a m o un t o f in t e r a c ti on i t ha s w i th o the r pa r t s o f the

subjec t . I t i s a kind of se l f -consis tent def ini t ion of im-

por t a nc e .

M I N I O : I sn ' t i t poss ib l e t ho ugh tha t som e th ing in

f a c t ha s no im pa c t f o r qu i t e a w hi l e , bu t t he n m a ny

ye a r s l a t e r i s t a ke n up?

A T I Y A H ; We l l , I t h ink i t i s t r ue t ha t som e bo dy m a yha ve a m a the m a t i c a l i de a w hic h i s i n a dva nc e o f it s

t im e , a nd i t m a y be t ha t som e one m a ke s a c l e ve r sug-

ge s t i on a nd pe op le do n ' t s e e i ts s i gn if i ca nc e f or a l ong

pe r iod o f t im e . I t obv iou s ly doe s ha pp e n .

I w a sn ' t t h ink ing qu i t e so m uc h o f t h ings l i ke tha t .

I w a s t h i n k in g m o r e o f t h e t e n d e n c y t o d a y f o r p e o p l e

to de ve lop w h ole a r e a s o f m a the m a t i c s on t he ir ow n,

in a ra the r a bs t r a ct f a sh ion . T he y jus t go on be a ve r ing

away. I f you ask what i s i t a l l for , what i s i t s s igni f i -

c a nc e , w ha t doe s i t c onne c t w i th , you f i nd t ha t t he y

d o n ' t k n o w .

M I N I O " D o you fe e l l ike g iv ing a n e xa m ple?

A T I Y A H " T he r e a r e som e e xa m ple s i n a ll pa r t s o f

m o de r n m a the m a t i c s : som e pa r t s o f a bs t ra c t a lge br a ,

some par ts of func t iona l ana lys is , so me par ts of genera l

t o p o l o g y - - t h o s e p a r ts w h e r e o n e s e e s t h e ax i om a ti c

m e thod a t i t s w or s t .

A xiom s a r e de s igne d to t e m por a r i l y i so l a t e a c l a s s

o f p r o b l e m s f o r w h i c h y o u c a n t h e n d e v e l o p t e c h-

n ique s o f so lu t i on . So m e pe o p le t h ink o f a x iom s a s a

w a y o f de f in ing a w hole a r e a o f m a the m a t i c s t ha t i s

TH E M A TH EM A TI CA L I N T ELLI G EN CER V O L. 6 , N O . 1 , 1 9 84 11



s e l f -c o n t a i n e d . T h a t I t h i n k i s w r o n g . T h e n a r r o w e r

t h e a x io m s , t h e m o r e y o u c u t o u t .

W h e n y o u a b s tr a c t s o m e t h i n g i n m a t h e m a t i c s y o u

s e p a r a t e o u t w h a t y o u w a n t t o c o n c e n t r a t e o n a n d

w h a t y o u r e g a r d a s i r re l ev a n t. N o w t h a t m a y b e c o n -

v e n i e n t f o r a w h i l e ; i t c o n c e n t r a t e s t h e m i n d . B u t b y

d e f i n it i o n , y o u h a v e c u t a w a y a l ot o f t h i n g s y o u s a i d

y o u ' r e n o t i n t e r e s t e d i n a n d , i n th e l o n g r u n , t h a t h a s

c u t a lo t o f r o o t s . I f y o u c a n d e v e l o p s o m e t h i n g a x io -m a t i c a ll y , t h e n a t s o m e s t a g e y o u s h o u l d r e t u r n i t t o

i ts o ri g in , m e r g i n g a n d p r o d u c i n g c r o s s - f e rt i li z a ti o n .

T h a t ' s h e a l t h y .

Y o u w i ll f in d v i e w s l i ke t h e s e e x p r e s s e d b y v o n N e u -

m a n n a n d H e r m a n n W e y l , s o m e t h i rt y y e a r s a g o .

T h e y w o r r i e d a b o u t t h e w a y m a t h e m a t i c s m i g h t b e

g o i n g ; i f i t g o e s t o o f a r a w a y f r o m i ts s o u r c e s t h e n i t

m i g h t b e c o m e s t er i le . I th i n k t h a t i s f u n d a m e n t a l l y c o r -

rec t .

MIN IO : I t ' s c l e a r t h a t y o u h a v e a s t r o n g f e e l i n g f o r

t h e u n i ty o f m a t h e m a t i c s . H o w m u c h d o y o u t h in k

t h a t is a r e s u lt o f th e w a y y o u w o r k a n d y o u r o w n

p e r s o n a l i n v o l v e m e n t i n m a t h e m a t i c s ?

A T I Y A H : I t i s v e r y h a r d t o s e p a r a t e y o u r p e r s o n a l i ty

f r o m w h a t y o u t h i n k a b o u t m a t h e m a t i c s . I b e l i e v e t h a t

i t i s v e r y i m p o r t a n t t h a t m a t h e m a t i c s s h o u l d b e

t h o u g h t o f a s a u n i t y . A n d t h e w a y I w o r k r e f l e c t s t h a t;

w h i c h c o m e s f i r s t i s d i f f i c u l t t o s a y . I f i n d t h e i n t e r -

a c t io n s b e t w e e n t h e d i f f e r e n t p a r ts o f m a t h e m a t i c s i n -

t e r e s ti n g . T h e r i c h n e s s o f t h e s u b j e c t c o m e s f r o m t h i s

c o m p l e x i t y , n o t f r o m t h e p u r e s t r a n d a n d i s o l a t e d s p e -

c ia l i za t ion .

B u t t h e r e a r e p h i l o s o p h i c a l a n d s o c ia l a r g u m e n t s a sw e l l. W h y d o w e d o m a t h e m a t i c s ? W e m a i n l y d o

m a t h e m a ti c s b e c a u s e w e en joy d o i n g m a t h e m a t i c s . B u t

i n a d e e p e r s e n s e , w h y s h o u l d w e b e p a i d t o d o m a t h -

e m a t i c s ? I f o n e a s k s f o r t h e j u s t if i c a t i o n f o r t h a t , t h e n

I t h i n k o n e h a s t o t a k e t h e v i e w t h a t m a t h e m a t i c s

i s p a r t o f t h e g e n e r a l s c i e n t if i c c u l t u r e . W e a r e c o n -

t r i b u t i n g t o a w h o l e , o r g a n i c c o l l e c t i o n o f i d e a s ,

e v e n i f t h e p a r t o f m a t h e m a t i c s w h i c h I ' m d o i n g n o w

i s n o t o f d i r e c t r e l e v a n c e a n d u s e f u l n e s s t o o t h e r

p e o p l e . I f m a t h e m a t i c s i s a n i n t e g r a t e d b o d y o f

t h o u g h t , a n d e v e r y p a r t is p o t e n t i a l l y u s e f u l to e v e r y

o t h e r p a r t , t h e n w e a r e a l l c o n t r i b u t i n g t o a c o m m o no b j e c t i v e .

I f m a t h e m a t i c s i s to b e t h o u g h t o f a s f r a g m e n t e d

s p e c i a li z a t io n s , a l l g o i n g o f f i n d e p e n d e n t l y a n d j u s t i -

f y i n g t h e m s e l v e s , t h e n i t i s v e r y h a r d t o a r g u e w h y

p e o p l e s h o u l d b e p a i d t o d o t h i s . W e a r e n o t e n t e r -

t a i n e r s , l i k e t e n n i s p l a y e r s . T h e o n l y j u s t i f i c a t i o n i s

t h a t it i s a r e a l c o n t r i b u t i o n t o h u m a n t h o u g h t . E v e n

i f I ' m n o t d i r e c t ly w o r k i n g i n a p p l i e d m a t h e m a t i c s , I

f e e l t h a t I ' m c o n t r i b u t i n g t o t h e s o r t o f m a t h e m a t i c s

t h a t c a n a n d w i l l b e u s e f u l f o r p e o p l e w h o a r e i nt e r -

e s t e d i n a p p l y i n g m a t h e m a t i c s t o o t h e r th i n g s.

E v e r y b o d y h a s t o t r y t o j u s t i f y h is l i fe p h i lo s p h i c a l ly ,

t o h i m s e l f a t l e a st . I f y o u a r e t e a c h i n g y o u c a n s a y ,

" W e l l , m y j o b is to t ea c h , I t u r n o u t e d u c a t e d y o u n g

p e o p l e a n d I a m p a i d f o r t h a t . R e s e a r c h I d o i n m y

s p a r e t i m e a n d t h e y a l l o w m e t o d o t h a t o u t o f g e n -

e r o s i t y . " B u t i f y o u ' r e a f u l l -t i m e r e s e a r c h e r , t h e n

y o u ' v e g o t t o t h i n k m u c h h a r d e r a b o u t j u st i fy i n g y o u r

w o r k .

I n s o m e s e n s e , I s ti ll d o m a t h e m a t i c s b e c a u s e I e n j o yd o i n g i t. I ' m g l a d t h a t p e o p l e p a y m e t o d o w h a t I

e n j o y . B u t I t r y t o f e e l t h a t t h e r e i s a s e r i o u s s i d e t o i t

w h i c h p r o v i d e s a j u s t i f i c a t i o n .

M IN I O : W h a t d o y o u t h i n k a b o u t s t a t e m e n t s l i ke

" p u r e m a t h e m a t i c s i s n ' t v e r y u s e f u l a n d w i t h in f i v e

y e a r s e v e r y b o d y w il l b e j u s t c o m p u t i n g " ?

A T I Y A H : T h e r e ' s a lw a y s a d a n g e r i n s u c h a p o i n t o f

v i e w . I f p u r e m a t h e m a t i c i a n s t a k e a n i v o r y t o w e r a t -

t i t u d e , d o n ' t t h i n k a b o u t th e i r r e l a ti o n s h i p w i t h o t h e r

s u b j e c t s , t h e r e i s t h e d a n g e r t h a t p e o p l e w i l l e v e n t u -

a l ly t u r n a r o u n d a n d s a y , " W e d o n ' t r e a ll y n e e d y o u - -

y o u a r e a l u x u r y - - a n d w e w i ll e m p l o y p e o pl e d o in g

m u c h m o r e p r a c t i c a l t h i n g s . " I t h i n k t h a t i s a d a n g e r

t h a t i s a l w a y s t h e r e a n d b e c o m e s m u c h m o r e s e r i o u s

i n t i m e s o f f in a n c i a l d i f fi c u lt y , s u c h a s w e a r e g o i n g

t h r o u g h n o w . A n d I t h i n k t h e m e s s a g e i s b e g i n n i n g

t o c o m e t h r o u g h .

C e r t a i n l y i n t h e l a s t f i v e o r t e n y e a r s t h e r e h a s b e e n

q u i t e a g r o w i n g a p p r e c i a t i o n a m o n g p u r e m a t h e m a t i -

c i a n s t h a t t h e y h a v e t o j u s t i f y t h e m s e l v e s a b it m o r e .

B u t I s ti ll th i n k w i t h m a n y p e o p l e t h i s h a s n ' t c o m e

n a t ur a l ly ; t h e y h a v e o n l y d o n e i t u n d e r p r e s s u r e . I

t h i n k i t w o u l d b e h e a l t h y if p u r e m a t h e m a t i c i a n s i ng e n e r a l w e r e m o r e s e l f - c r i t i c a l .

M I N I O : B a c k t o t h e m a t h e m a t i c s t h a t y o u d o . I s

t h e r e a t h e o r e m t h a t y o u a r e m o s t h a p p y t o h a v e

p r o v e d ?

A T I Y A H : I t h i n k so . T h e I n d e x T h e o r e m I p r o v e d

w i t h S i n g e r i s i n m a n y w a y s t h e c l e a r e s t s i n g l e t h i n g

I ' v e d o n e . I r e a l l y t h i n k t h e I n d e x T h e o r e m i s a n i c e ,

c le a r th e o r e m w h i c h o n e c a n p o i n t t o . M o s t o f m y

w o r k c e n t e r s a r o u n d i t i n o n e f o r m o r an o t h e r.

I t s t a r t e d f r o m w o r k i n t o p o l o g y a n d a l g e b r a i c g e -o m e t r y , b u t t h e n i t h a s h a d q u i t e a n i m p a c t o n f u n c -

t i o n a l a n a l y s i s ; o v e r t h e l a s t t e n y e a r s t h i s a s p e c t h a s

b e e n d e v e l o p e d b y m a n y p e o p l e . A n d a l so it i s n o w

b e i n g r e a l i z e d t h a t i t h a s i n t e r e s t i n g c o n n e c t i o n s w i t h

m a t h e m a t i c a l p h y s i c s . S o i t' s s ti ll d e v e l o p i n g a n d a c -

t iv e in m a n y w a y s . I t s y m b o l i z e s , i n a w a y , m y m a i n

i n t e r e s t w h i c h i s t h e i n t e r a c t i o n s a n d c o n n e c t i o n s b e -

t w e e n a ll p a r t s o f m a t h e m a t i c s . I t i s a n a r e a i n w h i c h

a l g e b ra i c t o p o l o g y a n d a n a l y s i s c o m e t o g e t h e r i n a

v e r y n a t u r a l w a y , a l o n g w i t h d i f f e r e n ti a l e q u a t i o n s i n

v a r i o u s f o r m s .

12 THE MATHEMATICAL 1NTELLIGENCER VOL. 6, NO. 1, 1984



M I N I O : D i d y o u a n t ic i p a te t h e r e c e n t r e v i v e d i n-

t e r e s t in m a t h e m a t i c a l p h y s i c s a m o n g m a t h e m a t i -

c i a n s ?

t h e h e a l t h o f m a t h e m a t i c s a t it s r e s e a r c h l e v e l , I t h i n k

i t i s v e r y i m p o r t a n t t o m a i n t a i n t h a t l i n k a s m u c h a s

p o s s i b l e .

A T I Y A H : N o t r e a ll y . I h a v e h a d a n i n t e r e s t i n m a t h -

e m a t ic a l p h y s i c s fo r q u it e a l o n g ti m e . N o t v e r y d e e p - -

I t r ie d t o u n d e r s t a n d q u a n t u m m e c h a n i c s a n d r e l a t e d

t o p ic s . B u t w h a t h a s h a p p e n e d i n th e l a st f iv e y e a r s - -

t h e in t e r e s t o f m a t h e m a t i c i a n s i n G a u g e t h e o r i e s - -w a s u n e x p e c t e d f o r m e . I d i d n ' t k n o w e n o u g h p h y s i c s

t o k n o w i t w a s l i k el y. Q u a n t u m F i el d T h e o r y w a s o n e

o f t h o s e b i g m y s t i c p h r a s e s a s f ar a s I w a s c o n c e r n e d .

I t h i n k t h e p h y s i c i s t s t h e m s e l v e s w e r e s u r p r i s e d .

T h e f a c t th a t t h e g e o m e t r i c a l a s p e c t b e c a m e s i g n i fi c a n t

a n d d o m i n a n t w a s n ' t p r e d i c te d b y m a n y o f t h e m ( a nd

i s s t i l l d i s p u t e d b y s o m e ! ) . T h e m a i n p r o b l e m s l o o k e d

a s t h o u g h t h e y w e r e d i f f e r e n t - - a n a l y t i c a l q u e s t i o n s ,

a l g e b r a i c p r o b l e m s . S o m e p e o p l e l i k e R o g e r P e n r o s e

w e r e n ' t r e a ll y s u rp r i s e d . T h e y h a d b e e n w o r k i n g i n i t

f r o m t h e ir o w n p o i n t o f v i e w f or a l o n g t i m e a n y w a y .

B u t I t h i n k i t ' s a n i c e e x a m p l e : i f y o u d o i n t e r e s t i n g ,

b a s ic m a t h e m a t ic a l w o r k i n th e m a i n s t r e a m , t h e n y o u

s h o u l d n ' t b e s u r p r i s e d w h e n o t h e r s f in d i t a u s e f u l

t o o l. I t j u s t if i e s o n e ' s b e l i e f i n t h e u n i t y o f m a t h e -

m a t i c s , i n c l u d i n g p h y s i c s .

M I N I O : H o w f ar w o u l d y o u g o w i t h t h at st a te m e n t ?

A T I Y A H : T h e m o r e I ' v e l e a rn e d a b o u t p h y s i c s , t h e

m o r e c o n v i n c e d I a m t h a t p h y s i c s p r o v i d e s , i n a s e n s e ,

t h e d e e p e s t a p p l i c at i o n s o f m a t h e m a t i c s . T h e m a t h e -

m a t ic a l p r o b l e m s t h a t h a v e b e e n s o l v e d o r te c h n i q u e s

t h a t h a v e a r i s e n o u t o f p h y s i c s in t h e p a s t h a v e b e e n

t h e l i f e b l o o d o f m a t h e m a t i c s . A n d i t ' s s ti ll t r u e . T h ep r o b l e m s t h a t p h y s i c i s t s t a c k l e a r e e x t r e m e l y i n t e r -

e s t i n g , d i f f i c u l t , c h a l l e n g i n g p r o b l e m s f r o m a m a t h e -

m a t i c a l p o i n t o f v i e w . I t h i n k m o r e m a t h e m a t i c i a n s

o u g h t t o b e i n v o l v e d i n a n d t r y t o le a r n a b o u t s o m e

p a r t s o f ph y s i c s; t h e y s h o u l d t r y t o b r i n g n e w m a t h -

e m a t i c a l t e c h n i q u e s i n t o c o n j u n c t i o n w i t h p h y s i c a l

p r o b l e m s .

P h y s i c s is v e r y s o p h i s t i c a t e d . I t i s t r e m e n d o u s l y

m a t h e m a t i c a l a n d t h e c o m b i n a t i o n o f p h y s i c a l in s i g h t,

o n t h e o n e h a n d , a n d m a t h e m a t i c a l t e c h n iq u e , o n t he

o t h e r, i s a v e r y d e e p c o n n e c t i o n b e t w e e n t h e s u b je c t s.

N e w e r a p p l i c a t i o n s o f m a t h e m a t i c s , s a y i n th e s o c ia ls c i e n c e s , e c o n o m i c s , c o m p u t i n g , a r e i m p o r t a n t . I t i s

i m p o r t a n t t h a t w e t u r n o u t s t u d e n t s w i t h t hi s v i e w of

a p p l i e d m a t h e m a t i c s b e c a u s e th i s is w h a t i s r e q u i r e d

i n t h e c o m m e r c i a l w o r l d ; t h o u s a n d s a n d t h o u s a n d s o f

s t u d e n t s r e q u i r e t h i s .

O n t h e o t h e r h a n d , f r o m t h e p o i n t o f v i e w o f th e

d e p t h o f m a t h e m a t i c s i n v o l v e d , t h e r e i s n o c o m p a r -

i s o n . A l t h o u g h t h e r e a r e i n te r e s t i n g q u e s t i o n s i n s a y

e c o n o m i c s a n d s t a ti s ti c s , b r o a d l y s p e a k i n g , t h e d e p t h

o f m a t h e m a t i c s i n v o l v e d i s v e r y s h a l l o w . T h e r e a l ly

d e e p q u e s t i o n s a r e s t i l l i n t h e p h y s i c a l s c i e n c e s . F o r

M I N I O : Y o u a r e o b v i o u s l y i n t e r e s te d i n e d u c a t io n .

O n t h e o t h e r h a n d , a s fa r a s y o u r j o b is c o n c e r ne d ,

y o u a r e v e r y c le a r l y a r e s e a r c h m a t h e m a t i c i a n . H o w

d o y o u e x p l a i n t h i s ?

A T I Y A H : M y r e a s o n s f o r a n i n t e r e s t i n e d u c a t i o n a r e

t h e s a m e a s m y r e a s o n s f o r a n i n t e r e s t i n t h e u n i t y o f

m a t h e m a t i c s . U n i v e r s i t i e s a r e i n s t i t u t io n s t h a t ar e e d -

u c a t i o n a l and i n v o l v e d i n r e s e a r c h . I t h i n k t h a t i s v e r y

i m p o r t a n t - - t h e r e s h o u l d b e u n i t y in th e u n i ve r s it y

a n d u n i t y i n t h e w h o l e s o c i a l s t r u c t u r e t h a t a t t e m p t s

t o k e e p a b r o a d b a l a n c e b e t w e e n m a t h e m a t i c a l r e-

s e a r c h a n d m a t h e m a t ic a l e d u c a t i o n . A n d w h e n u n i v -

e r s i t ie s g iv e c o u r s e s f o r e d u c a t i o n a l p u r p o s e s , t h e y

s h o u l d b e s u r e t h a t t h e y a r e p e r f o r m i n g t h e r i g h t t a s k

f o r t h e s t u d e n t s , n o t j u s t g i v i n g c o u r s e s i n ( sa y ) a d -

v a n c e d t o p o l o g y b e c a u s e t h e y a r e i n t e r e s te d i n t u r n i n g

o u t r e s e a r c h s t u d e n t s . T h a t ' s a d i s a s t r o u s m i s t a k e .

U n i v e r s i t i e s m u s t t r y t o b a l a n c e t w o a c t i v i t i e s . T h e y

o u g h t t o k n o w w h a t ' s u s e f u l f o r s t u d e n t s t o l e a rn ,

b e a r i n g i n m i n d w h a t t h e y a r e g o i n g t o b e d o in g l a te r

o n . A t t h e s a m e t i m e , t h e y o u g h t t o f o s t e r r e s e a r c h .

S o m e p e o p l e w i l l b e d o i n g a l l r e s e a r c h a n d s o m e

p e o p l e w i l l b e m o s t l y te a c h in g , a n d m a i n l y p e o p l e w i l l

b e i n b e t w e e n . A l t h o u g h I a m o n l y i n v o lv e d w i t h t h e

r e s e a r c h e n d o f i t, I l i v e i n t h e u n i v e r s i t y , I h a v e c o l -

l e a g u e s i n t h e u n i v e rs i t y , I k n o w w h a t t h e y a r e i n-

v o l v e d w i t h , s o I a m c o n c e r n e d t o s e e t h a t a p r o p e r

b a l a n c e i s s t r u c k b e t w e e n t h e d i f f e r e n t f u n c t i o n s o f t h e

u n i v e r s i t y .

M I N I O : D o y o u t h i n k t h e r e w a s t o o m u c h e x p a n s i o n

o f t h e u n i v e r s i t i e s i n B r it a in d u r i n g t h e l a s t t w e n t y

y e a r s ?

A T I Y A H : I d o n ' t t h in k t h e r e w a s t o o m u c h e x p a n -

s i o n . B y c o m p a r i s o n w i t h o t h e r c o u n t r i e s , p a r t i c u l a r ly

A m e r i c a , i t i s c l e a r t h a t t h e n u m b e r o f p e o p l e g o i n g

o n t o h i g h e r e d u c a t i o n w a s r e a ll y v e r y s m a l l a n d o u g h t

t o h a v e b e e n i n c re a s e d ; f u n d a m e n t a l l y i t o u g h t t o c o n -

t i n u e t o e x p a n d . I c a n ' t b e l i e v e t h a t f o r t h e n e x t c e n -

t u r y t h e p r o p o r t i o n o f p e o p l e g e t t i n g h i g h e r e d u c a t i o nw i l l s t a y t h e s a m e a s it i s n o w . I t i s b o u n d t o c h a n g e .

W h e n y o u h a v e a p e r i o d o f r a p i d e x p a n s io n , w h i c h

w a s n e c e s s a r y a f t e r t h e w a r , i t d o e s p r o d u c e s o m e

p r o b l e m s . I t p r o d u c e s a d i s c o n t i n u i ty . Y o u r e c r u i t a l o t

o f p e o p l e t o t ea c h i n th e u n i v e r s it i es , a n d w h e n t h e

e x p a n s i o n s t o p s y o u h a v e f i ll e d u p a l l o f t h e p o s i -

t i o n s - y o u c a n 't a p p o i n t a n y y o u n g p e o p le . O n e c a n

c r it ic i ze t h e o v e r e n t h u s i a s m o r l a ck o f c a u t i o n w h i c h

t h e u n i v e r s i t i e s h a d i n n o t f o r e s e e i n g s o m e o f t h e u l -

t i m a t e d i f f ic u l t ie s w h i c h w e r e g o i n g t o e m e r g e . F o r

e x a m p l e , u n l i k e A m e r i c a n u n i v e r s i t i e s , E n g l i s h u n i -

THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 1, 1984 13



v e r s if i e s i n th i s p e r i o d o f e x p a n s i o n g a v e t e n u r e a p -

p o i n t m e n t s i m m e d i a t e l y a f t e r t h e P h . D . b e c a u s e t h e

u n i v e r s i ti e s w e r e c o m p e t i n g w i t h e a c h o t h e r . I t h i n k

t h a t w a s a m i s t a k e a n d t h e y a r e p a y i n g f o r it n o w .

I t w o u l d h a v e b e e n w i s e r n o t t o h a v e g i v e n p e o p l e

l i f e t i m e p o s i t i o n s f r o m t h e m o m e n t t h e y a r r i v e d b u t

t o h a v e h a d s o m e m o r e f l ex ib le s y s t e m w h i c h e n a b l e d

t h e m t o a d a p t . N o w w e a r e g e t t i n g a v e r y s h a r p d i s -

c o n t i n u i t y a n d i t ' s g o i n g t o p r o d u c e a c r i s i s a n d c o n -f r o n ta t i o n s . P e r h a p s p e o p l e in th e u n i v e r si t i e s o u g h t

t o b e a b i t m o r e c a u t i o u s .

MIN IO : B a c k f o r a m o m e n t t o t e a c h i n g a n d r e s e a r c h .

Y o u t a l k e d a b o u t b o t h b e i n g e s s e n t i a l p a r t s o f t h e u n i -

v e r s i t y l i f e , b u t y o u t a l k a b o u t t h e m s t i l l q u i t e s e p a -

r a t el y . T h e r e h a s b e e n a g r o w t h o f r e s e a r c h i n st i-

t u t e s - B o n n , W a r w i c k , P r i n c e t o n - - a t w h i c h t h er e is

n o t e a ch i n g. D o y o u f i n d t h a t a h e a l t h y d e v e l o p m e n t ?

AT IYAH : I t h i n k t h e f i r s t t h i n g t o s a y a b o u t t h e s e

k i n d s o f i n s t it u t i o n s i s t h a t t h e y h a v e e i t h e r n o p e r -

m a n e n t s ta f f o r t h e y h a v e a v e ry s m a ll p e r m a n e n t

s t af f. M o s t o f t h e p e o p l e w h o g o t h e r e a r e g o i n g t h e r e

o n r e f r e s h e r c o u r s e s . T h e y a r e g o i n g f r o m t h e u n i -

v e r s if i e s t o s p en d a t e r m o r a y ea r t h e r e an d t h en t h ey

g o b a c k . S o t h e s e a r e a k i n d o f g e n e r a l i z e d c o n f e r e n c e

c e n t r e , w h e r e p e o p l e g e t t o g e t h e r t o e x c h a n g e i d e a s

a n d t h e n g o b a c k t o c a rr y o n t h e i r w o r k . T h e y a re

s i m p l y h e l p i n g t o k e e p p e o p l e i n u n i v e r s i t i e s a c t i v e l y

i n t e r e s t e d i n t h e i r r e s e a r c h - - t h a t i s t h e i r m a i n f u n c -

t ion .

O n t h e o t h e r h a n d , i f y o u h a d a s y s t e m l i ke E a s te r n

E u r o p e , w h e r e t h e y h a v e b i g re s e a r c h i n s ti t u ti o n s e m -

p l o y i n g la r ge n u m b e r s o f p e o p l e o n a p e r m a n e n t b a s is ,s i p h o n i n g o f f f r o m t h e u n i v e r s i ti e s a s i g n i fi c a n t p e r -

c e n t a g e o f t h e i r s t a f f - - t h a t r a i se s d i f f e re n t p r o b l e m s .

T h e n y o u r e a l l y a r e s e p a r a t i n g t h e u n i v e r s i t y f r o m r e -

s e a rc h i n a b i g w a y . B u t i n m a t h e m a t i c s t h e n u m b e r

o f t h e s e c e n t r e s i s s o s m a l l , t h e n u m b e r o f s t a f f i s

m i n u t e , a n d t h e p e o p l e g o i n g a n d c o m i n g b a c k ar e

s i m p l y s t r e n g t h e n i n g t h e u n i v e r s i t y s y s t e m . I t h i n k

t h a t i s q u i t e h e a l t h y .

T h e y c a n a l s o s e r v e a n o t h e r p u r p o s e w h i c h i s t o

h e l p t o o r i e n t o r s t e e r o r g u i d e p e o p l e i n t o p r o f i t a b l e

a r e a s o f m a t h e m a t i c a l t h o u g h t . I n a d d i t i o n t o g o i n g t o

a c e n tr e t o b o o s t y o u r o w n c u r r e n t w o r k , y o u c a n g ot o a c e n t r e o f t h i s t y p e a s a y o u n g m a n i n o r d e r t o b e

l e d i n t o p r o d u c t i v e a r e a s o f re s e a r c h .

T h e i n s t i tu t e a t P r i n c e t o n , w h i c h i s w h e r e I w e n t

a f te r I t o o k m y P h . D . , s e r v e s t h a t p u r p o s e v e r y w e ll .

I h a d d o n e m y P h . D . , w r i t t e n m y t h e s is , b u t I w a s s ti ll

l o o k i n g a r o u n d f o r a n ic h e i n l if e m a t h e m a t i c a l l y . I

d i d n ' t k n o w w h e r e I w a s g o i ng , w h a t I w a s g o i n g to

b e d o i n g . I w e n t t o th i s l ar g e c e n t re w h e r e t h e r e w e r e

l o ts o f v e r y a b le y o u n g m e n , a n d o l d e r m e n , f r o m

d i f f e re n t p a r t s o f th e w o r l d w i t h l o t s o f d i f f e r e n t i d e a s.

A f t e r a y e a r o r s o t h e r e , I c a m e a w a y f u l l o f n e w i d e a s

a n d n e w d i re c t io n s . T h a t h a d a n e n o r m o u s b e a r i n g o n

m y s u b s e q u e n t m a t h em a t ic a l d e v e l o p m e n t .

M I N I O : W h o w a s it a t P r i n c e t o n w h o i n f lu e n c e d y o u

m o s t ?

AT IYAH : W e l l , I t h i n k i t w a s n ' t s o m u c h t h e p e r -

m a n e n t s t af f. I w e n t i n 1 95 5 a n d m a n y o f t h e p e o p l e

t h e r e w e r e p r o b a b l y s l i g h t l y o l d e r t h a n t h e c o m p a r a b l ep e o p l e w h o g o n o w .

I m e t H i r z e b r u c h , S e r re , B o tt , S i n g e r . . . I m e t t h e m

a ll w h e n I w a s a t t h e i n s ti t u t e . K o d a i r a a n d S p e n c e r

w e r e t h e r e t o o . T h a t w h o l e g r o u p o f p e o p l e - - I g o t t o

k n o w t h e m , a n d I w a s i n f l u e n c e d b y t he i r m a t h e -

m a t i c s . I t ' s n o t a n a c c i d e n t t h a t I ' v e c o l l a b o r a t e d s u b -

s e q u e n f l y w i t h t h e s e s a m e p e o p l e I m e t a t t h e i n st i -

t u t e .

I t h i n k t h e r e i s a n o t h e r a s p e c t . N o t o n l y d o y o u a l t er

y o u r p o i n t o f v i e w a n d y o u r w o r k , b u t i t p u t s y o u i n

c o n t a c t w i t h o t h e r a c t iv e p e o p l e a n d y o u k e e p u p t h e s e

p e r s o n a l c o n t a c t s ; t h e y a r e v e r y i m p o r t a n t i n m a i n -

t a i n in g y o u r m a t h e m a t i c a l d e v e l o p m e n t a f t e r w a r d s .

M e e t i n g p e o p l e f r o m d i f fe r e n t c o u n t r i e s is i m p o r -

t a n t - m a t h e m a t i c s is V e ry i n t e rn a t i o n al a n d t h e s e

c e n t r e s p r o v i d e a n o p p o r t u n i t y w h i c h i s v e r y h a r d t o

g e t o t h e r w i s e .

MIN IO : C o n f e r e n c e s a l s o p r o v i d e a n o p p o r t u n i t y f o r

m e e t i n g p e o p l e , b u t m a y b e l e s s o f a n o p p o r t u n i t y f o r

w o r k i n g t o g e t h e r a n d r e a l l y l e a r n i n g t h i n g s ?

AT IYAH : Y e s , c o n f e r e n c e s a r e v e r y u s e f u l , b u t p r o b -

a b l y n o t s o u s e f u l f o r t h e y o u n g p e r s o n s t a r t i n g o f f.

T h e y a r e u s e f u l fo r t h e p e r s o n w h o i s e s t a b l i s h e d . I fy o u k n o w o t h e r p e o p l e w e l l a l r e a dy , a n d y o u a r e a c -

f i v e , t h e n i n a v e r y s h o r t t i m e y o u c a n b e n e f i t f r o m a

r a p i d e x c h a n g e o f i d e a s . I f y o u a r e a y o u n g s t u d e n t ,

o r p o s t - d o c t o r a l , y o u c a n ' t r e a l l y t a l k t o a lo t o f p e o p l e

b e c a u s e y o u d o n ' t r e a l ly k n o w t h e m , y o u a r e i n h ib -

i te d , a n d a l s o y o u d o n ' t u n d e r s t a n d e n o u g h t o f o l lo w

w h a t t h e y a r e s a y in g . T h e n y o u n e e d m u c h l o n g e r

e x p o s u r e , I t h i nk . Y o u n e e d a y e a r o r s o t o a b s o r b

t h i n g s s l o w l y , t o g e t t o k n o w p e o p l e w e l l . S o I t h i n k

co n f e r en ces f u l f i l l d i f f e r en t f u n c t i o n s .

MIN IO : W h a t a b o u t t h e I n t e rn a t i o n a l C o n g r e s s ?

AT IYAH : W e l l , I t h i n k t h e I n t e r n a t i o n a l C o n g r e s s i s

t o t a ll y d i ff e r e n t . I ' v e b e e n t o e v e r y I n t e r n a t io n a l C o n -

g r e s s s i n ce 1 9 54 , I t h i n k ; t h e b en e f i t I h av e d e r i v e d

f r o m t h o s e i s v e r y m i x e d .

T h e f i rs t o n e , w h i c h I a t t e n d e d a s a y o u n g s t u d e n t ,

w a s g r e a t . I h a d a c h a n c e t o h e a r H e r m a n n W e y l g i v e

a t a l k a n d i t w a s a t r e m e n d o u s p s y c h o l o g i c a l b o o s t . I

f el t I w a s o n e o f a l a rg e c o m m u n i t y o f se v e r a l t h o u s a n d

m a t h e m a t i c i a n s . I d i d n ' t u n d e r s t a n d m o s t o f t h e t al k s.

I ' d g o to t h e m a n d b e l os t . I d o n ' t t h i n k I g a i n e d a n y -

14 THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 1, 1984



M. Atiyah (right) talks to F. Hirzebruch

t h i n g i n te r m s o f c o n c r e t e m a t h e m a t i c a l u n d e r -

s t a nd ing , bu t t he psy c ho log ic a l boo s t wa s subs t a n t ia l .

Now a s I ge t o lde r t he I n t e r na t iona l Congr e ss i s o f

l it tl e va lue . I go ou t o f a se nse o f d u t y - - I ha ve f unc -

t i ons t o pe r f o r m - - to t a lk t o pe op le , t o g ive l e c tu r e s . Idon ' t r e a l l y be ne f i t be c a use t he r e a r e so m a ny pe op le .

Some lec tures I qui te enjoy; I th ink the Inte rna t iona l

C o n g r e s s h a s s o m e b e n e f it , b u t n o t v e r y m u c h .

B e s i d e s th e b e n e f i t t h e y g i v e t o y o u n g p e o p l e ,

g iv ing the m som e se nse o f in t e r na tiona l i de n t i t y , t he ir

m a in f unc t ion i s p r oba b ly t o he lp pe op le f r om c oun-

t r i e s ou t s id e t he sm a l l c i r c u it o f ve r y a c t i ve m a the -

m a t ic a l c oun t r ie s . I f yo u c om e f rom W e s te r n E ur ope

or t he Uni t e d S t a t e s t he y a r e p r oba b ly no t ve r y e s se n-

t ia l. But i f yo u co me f rom Afr ica or Asia or Easte rn

E u r o p e , w h e r e t h e o p p o r t u n i t y f o r t r a v e ll i n g a n d

m e e t ing pe op le i s m u c h l e s s, t he n I t h ink i t is the onec ha nc e you ha ve o f se e ing wha t i s go ing on . I suspe c t

tha t i s i t s main jus t i f ica t ion.

M I N I O: Do yo u th ink tha t t he F i e lds M e da l s se r ve a

use f u l f unc t ion?

A T I Y A H : W e l l, I s u p p o s e in s o m e m i n o r w a y . I

think i t ' s a good thing tha t Fie lds Medals a re not l ike

the Nob e l p r iz e s . T he Nob e l p r i z e s d i s to r t s c i e nc e ve r y

ba d ly , e spe c i a l l y phys i c s . T he p r e s t i ge t ha t goe s w i th

t h e N o b e l p r iz e s , a n d t h e h o o p l a h t h a t g o e s w i t h

(center). Michael Artin is pictured on left.

t h e m , a n d t h e w a y u n i v er s it i e s b u y u p N o b e l

p r i z e m e n- - th a t i s t e r ri b ly d i sc on t inu ous . T he d i ff e r-

e n c e b e t w e e n s o m e o n e g e t t in g a p r i ze a n d n o t g e t ti n g

one i s a toss-up- - i t i s a very a r t i f ic ia l dis t inc t ion. Yet ,

i f yo u ge t t he Nob e l P r iz e a nd I don ' t , t he n y ou ge ttw ic e t he sa l a r y a nd your un ive r s i t y bu i lds you a b ig

lab; I th ink tha t i s very unfor tuna te .

But in m a the m a t i c s t he F i e lds M e da l s don ' t ha ve a ny

ef fec t a t al l, so the y do n ' t have a nega t ive e f fec t. They

a r e g iv e n t o y o u n g p e o p l e a n d a r e m e a n t t o b e a fo r m

o f e n c o u r a g e m e n t t o t h e m a n d t o th e m a t h e m a ti c a l

wor ld a s a whole .

I w a s e n c o u r a g e d b y g e t t i n g a F ie l d s M e d a l . I t

he lpe d m y se l f - c onf ide nc e , m y m or a l e . I don ' t know

w h e t h e r i f I h a d n ' t g o t a M e d a l it w o u l d h a v e b e e n

any di f fe rent , but cer ta inly ge t t ing i t a t tha t s tage gave

m e e n c o u r a g e m e n t a n d m a d e m e e n t hu s ia s ti c . S o It h ink in t ha t s e nse i t c a n he lp .

I f ound ou t t ha t i n a f e w c oun t r i e s t he M e da l s ha ve

a lo t o f p r e s t i g e - - f o r e xa m ple , J a pa n . Ge t t i ng a F i e lds

M e da l i n Ja pa n i s l ike ge t ti ng a No be l P r i ze . So wh e n

I go to Ja pa n a nd a m in t r oduc e d , I f e e l l i ke a Nobe l

P r i ze w inne r . Bu t in t h is c oun t r y , n ob od y no t i c es a t

all.

MIN IO : Do you f i nd t ha t m a the m a t i c i a ns ge ne r a l l y

are t rea ted s igni f icant ly di f fe rent ly in di f fe rent coun-

tr ies?

THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 1, 19 84 15



AT IYAH : W e l l m a t h e m a t i c s c a n m e a n s l ig h t l y d i f-

f e r e n t t h i n g s i n d i f f e r e n t c o u n t r i e s , o f c o u r s e . P a r t ic -

u l a rl y th e d i v i s io n b e t w e e n m a t h e m a t i c s , a p p l i e d

m a t h e m a t i c s , a n d p h y s i c s i s q u i t e d i f f e r e n t in th i s

c o u n t r y ; in m o s t o t h e r c o u n t r i e s , p u r e m a t h e m a t i c s i s

m u c h m o r e s e p a r a t e d . T h i s p r o b a b l y h a s a g e n e r al e f-

f e ct o n w h a t p e o p l e t h i n k a b o u t m a t h e m a t ic i a n s ; t h e y

d o n ' t i d e n ti f y t h e m q u i t e s o n a r r o w l y h e r e w i t h p u r e

m a t h e m a t i c s a s t h e y d o i n A m e r i c a , w h e r e m a t h e m a -

t i c i a n m e a n s p u r e m a t h e m a t i c i a n .

O t h e r w i s e , I s u p p o s e i t i s t r u e i n F r a n c e t h a t m a t h -

e m a t i c i a n s h a v e a h i g h e r s t a t u s , t r a d i ti o n a l l y . T h a t is

b e c a u s e F r a n c e h a s a tr a d i ti o n o f a s s i g n i n g h i g h e r

s t a t u s to p h i l o s o p h y , l i te r a t u r e , a n d t h e a rt s, a n d

m a t h e m a t i c s b e l o n g s t o th a t g r o u p . W h e r e a s i n t hi s

c o u n t r y , t h e y n e v e r a t t a c h m u c h w e i g h t t o t h e s e

t h i n g s .

I n G e r m a n y a l so p r o f e s s o r s t r a d i ti o n a l ly h a d a

h i g h e r s t a tu s , t h o u g h t h a t is n o w r a p i d l y c h a n g i n g .

I t h i n k t h e r e a r e o b v i o u s n a t i o n a l d i f f e r e n c e s , a b o u t

h o w p e o p l e v i e w m a t h e m a t i c s o r u n i v e r s it i es . B u t t h a t

i s c h a n g i n g - - t h e d i s ti n c t io n s b e t w e e n d i f f e r e n t c u l-

t u r e s a r e b l u r r i n g .

M I N I O : I h a v e a f e w q u e s t io n s h er e a b o u t h o w y o u

w o r k . F o r e x a m p l e , w h a t s o r t o f m e n t a l i m a g e s d o y o u

u s e ?

AT IYAH : I ' m n o t s u r e I k n o w t h e a n s w e r . I t h i n k I

d o h a v e a v is u a l p i c t u r e in m y m i n d s o m e t i m e s , s o m e

s c h e m a t i c d i a g r a m . B u t w h e t h e r t h a t i s r e a l l y a p i c t u r e

o r w h e t h e r i t i s p u r e l y s y m b o l i c I d o n ' t k n o w . I t h i n k

i t i s a v e r y d i f f ic u l t q u e s t i o n , m o r e t o d o w i t h t h e g e n -

e r al n a t u r e o f p s y c h o l o g y t h a n m a t h e m a t i c s .

MIN IO : I s u p p o s e t h e q u e s t io n w a s m e a n t t o d r a w

a d i s t i n c t i o n b e t w e e n g e o m e t r i c a l i n t u i t i o n a n d a l g e -

b r a i c m a n i p u l a t i o n .

AT IYAH : Y e s , t h e r e a r e d i f f e r e n c e s t h e r e . I s u s p e c t

t h a t d i c h o t o m y i s q u i t e r e a l i n t h e b r a i n . I w o r k w i t h

t h i n g s t h a t a r e m o r e g e o m e t r i c a l b u t I ' m n o t t h e s o r t

o f p e r s o n l ik e T h u r s t o n w h o s e e s c o m p l i c a t e d , m u l t i-

d i m e n s io n a l g e o m e t r y i n t ha t w a y . M y g e o m e t r y i s

r a t h e r m o r e f o r m a l. B u t I' m n o t a n a l g e b r a is t e i t h e r - -

I d o n ' t j u s t e n j o y m a n i p u l a t i o n s . P e r h a p s I a m n o ts u f fi c ie n t ly e x t r e m a l f o r t h e p s y c h o l o g y t o b e e v i d e n t ;

I ' m s o r t o f t h e o r d i n a r y m a n i n th e m i d d l e .

I f y o u a s k T h u r s t o n , p e r h a p s h e s a y s t h a t h e d o e s

s e e c o m p l i c a t e d p i c t u r e s i n h i s m i n d a n d a l l h e h a s t o

d o i s d r a w i t o n p a p e r t o g e t t h e p ro o f . A s k T h o m p s o n

h o w h e s e e s a g r o u p ; I d o n ' t k n o w w h a t h i s a n s w e r

i s. T h e r e a r e d i f f e r e n c e s . I t i s a c o m p l i c a t e d q u e s t i o n ,

b u t i t is t h r e e - q u a r t e r s p s y c h o l o g y a n d o n l y o n e -

q u a r t e r m a t h e m a t i c s .

MIN IO : H o w i m p o r t a n t i s m e m o r y f o r y o u r w o r k ?

AT IYAH : I m e n t i o n e d t h a t w h e n I w a s f if t ee n I w a s

v e r y k e e n o n d o i n g c h e m i s t r y . I d i d a w h o l e y e a r o f

c h e m i s t r y , t h e n I g a v e i t u p f o r t h e s i m p l e r e a s o n t h a t

i n c h e m i s tr y y o u h a v e t o m e m o r i z e v a s t a m o u n t s o f

f a ct s. T h e r e w e r e b i g b o o k s I h a d i n in o r g a n ic c h e m -

i st ry a n d I s i m p l y h a d t o r e m e m b e r t h a t y o u p r o d u c e d

d i f f e r e n t c h e m i c a ls o u t o f d i f f e r e n t s u b s t a n c e s b y v a r -

i o u s p r o c e s s e s . T h e a m o u n t o f st r u c t u r e th a t h e l p e d

y o u t o r e m e m b e r t h e s e t h i n g s w a s i n fi n it e si m a l. O r -

g a n i c c h e m i s t r y w a s a b i t b e t t e r. C o m p a r e d w i t h t h i s,

i n m a t h e m a ti c s y o u n e e d e d p r ac ti ca ll y n o m e m o r y a t

a ll. Y o u d i d n ' t n e e d t o m e m o r i z e f ac ts ; a ll y o u n e e d e d

t o d o w a s t o h a v e a n u n d e r s t a n d i n g o f t h e w a y t h e

w h o l e t h i n g f i t t e d t o g e t h e r . S o I t h i n k , i n t h a t s e n s e ,

m a t h e m a t i c i a n s d o n ' t r e a ll y n e e d t h e s o r t o f m e m o r y

t h a t s c i e n t i st s o r m e d i c a l s t u d e n t s d o .

M e m o r y i s i m p o r t a n t i n m a t h e m a t i c s i n a d i f f e re n t

w a y . I w i ll b e th i n k i n g a b o u t s o m e t h i n g a n d s u d d e n l y

i t w i l l d a w n o n m e t h a t t h i s i s r e l a t e d t o s o m e t h i n g

e l s e I h e a r d a b o u t la s t w e e k , l a s t m o n t h , t a l k i n g t o

s o m e b o d y . M u c h o f m y w o r k h a s c o m e t h a t w a y . I g o

a r o u n d s h o p p i n g , t a lk i n g t o p e o p l e , I g e t th e i r i d e a s,

h a lf u n d e r s t o o d , p i g e o n - h o l e d i n t h e b a c k o f m y m i n d ,

I h a v e t h i s v a s t c a r d - i n d e x o f b i t s o f m a t h e m a t i c s f r o m

a ll o f t h o s e a r e a s . S o I t h i n k m e m o r y p l a y s a r o l e in

m a t h e m a t i c s , b u t i t is a d i f f e r e n t s o r t o f m e m o r y t h a n

y o u w o u l d u s e i n o t h e r ar e as .

M I N I O : W h e n y o u ' r e w o r k i n g d o y o u k n o w if a r e-

s u l t i s t r u e e v e n i f y o u d o n ' t h a v e a p r o o f ?

A T I Y AH : T o a n s w e r t h a t q u e s t i o n I s h o u l d f ir s t

p o i n t o u t t h a t I d o n ' t w o r k b y t r y i n g t o s o l v e p r o b l e m s .

I f I ' m i n t e r e s t e d i n s o m e t o p i c t h e n I j u s t t r y to u n -d e r s t a n d it; I j u s t g o o n t h i n k i n g a b o u t i t a n d t r y i n g

t o d i g d o w n d e e p e r a n d d e e p e r . I f I u n d e r s t a n d i t,

t h e n I k n o w w h a t i s r ig h t a n d w h a t i s n o t ri g ht .

O f c o u r s e i t i s a l so p o s s i b l e t h a t y o u r u n d e r s t a n d i n g

h a s b e e n f a u lt y , a n d y o u t h o u g h t y o u u n d e r s t o o d i t

b u t i t t u r n s o u t e v e n t u a l l y t h a t y o u w e r e w r o n g .

B r o a d l y s p e a k i n g , o n c e y o u r e a l ly f e el t h a t y o u u n -

d e r s t a n d s o m e t h i n g a n d y o u h a v e e n o u g h e x p e ri e n ce

w i t h t h a t t y p e o f q u e s t i o n t h r o u g h l o ts o f e x a m p l e s

a n d t h r o u g h c o n n e c t i o n s w i t h o t h e r th i n g s, y o u g e t a

f e e l i n g f o r w h a t i s g o i n g o n a n d w h a t o u g h t t o b e

r ig h t . A n d t h e n t h e q u e s t i o n i s: H o w d o y o u a c t u a ll yp r o v e i t ? T h a t m a y t a k e a l o n g t im e .

T h e I n d e x T h e o r e m , f o r e x a m p l e , w a s f o r m u l a t e d

a n d w e k n e w i t s h o u l d b e t r u e . B u t it t o o k u s a c o u p l e

o f y e a r s t o g e t a p r o o f . T h a t w a s p a r t y b e c a u s e d i f-

f e r e n t te c h n i q u e s w e r e i n v o l v e d , a n d I h a d t o l e a r n

s o m e n e w t h i n g s t o g e t th e p r o o f , i n t h a t c a se s e v e ra l

p r o o f s . I d o n ' t p a y v e r y m u c h a t t e n t io n t o t h e i m p o r -

t a n c e o f p r o o f s . I th i n k i t i s m o r e i m p o r t a n t t o u n d e r -

s t a n d s o m e t h i n g .

M I N I O : T h e n w h a t i s t h e i m p o r t a n c e o f a p r o o f ?




A T I Y A H : W e l l, a p r o o f i s i m p o r t a n t a s a c h e c k o n

y o u r u n d e r s t a n d i n g . I m a y th i n k I u n d e r s t a n d , b u t t h e

p r o o f is t h e c h e c k t h a t I h a v e u n d e r s t o o d , t h a t ' s a ll. I t

i s t h e la s t s t a g e i n t h e o p e r a t i o n - - a n u l t im a t e c h e c k - -

b u t i t i s n ' t t h e p r i m a r y t h i n g a t a l l .

I r e m e m b e r o n e t h e o r e m t h at I p r o v e d a n d y e t I

r e al ly c o u l d n ' t s e e w h y i t w a s t ru e . I t w o r r i e d m e f o r

y e a r s a n d y e a r s . I t h a d t o d o w i t h t h e r e l a t i o n s h i p

b e t w e e n K - T h e o r y a n d r e p r e s e n t a t i o n s o f f in i te

g r o u p s . T o p r o v e t h e t h e o r e m I h a d t o b r e a k u p t h e

g r o u p i n to s o l v a b l e g r o u p s a n d c y cl ic g r o u p s ; t h e r e

w e r e l o ts a n d l o t s o f i n d u c t i o n s a n d t h e r e w e r e v a r i o u s

b i t s o n t h e w a y . I n o r d e r f o r t h e p r o o f t o w o r k , e v e r y

s i n gl e th i n g h a d t o g o j u s t r i g h t - - y o u h a d t o b e r e-

m a r k a b l y l u c k y , s o t o s p e a k . I w a s s t a g g e r e d t h a t i t a ll

w o r k e d a n d I k e p t t h i n k i n g t h a t i f a n y o n e l i n k o f t h is

c h a i n w e r e t o s n a p , i f t h e r e w a s s o m e f l a w in t h e a r-

g u m e n t , t h e w h o l e t h i n g w o u l d c o l la p s e . B e c a u se I

d i d n ' t u n d e r s t a n d i t, it m i g h t n o t b e t r u e a t a ll . I k e p t

w o r r y i n g a b o u t it , a n d f i v e o r s ix y e a r s l a t e r I u n d e r -

s t o o d w h y i t h a d t o b e t r u e . T h e n I g o t a n e n t i r e l y

d i f f e re n t p r o o f b y g o i n g f r o m f in i te g r o u p s t o c o m p a c t

g r o u p s . U s i n g q u i t e d i f f e r e n t te c h n i q u e s , i t w a s q u i t e

c l ea r w h y i t h a d t o b e t r u e .

M I N I O : D o y o u s e e s o m e w a y of c o m m u n i c a t in g

t hi s u n d e r s t a n d i n g t o s o m e o n e w i t h o u t t h e p r o o f s ?

A T I Y A H : W e l l , I t h i n k i d e a l l y a s y o u a r e t r y i n g t o

c o m m u n i c a t e m a t h e m a t i c s , y o u o u g h t to b e t ry i n g to

c o m m u n i c a t e u n d e r s t a n d i n g . I t i s r e l at i v el y e a s y to d o

t h is i n c o n v e r s a t io n . W h e n I c o l la b o r a te w i t h p e o p l e ,

w e e x c h a n g e i d e a s a t th i s le v el of u n d e r s t a n d i n g - - w e

u n d e r s t a n d t o p i c s a n d w e c l in g to o u r i n tu i t io n .I f I g i v e ta l k s , I t r y a l w a y s t o c o n v e y t h e e s s e n t i a l

i n g r e d i e n t s o f a to p i c. W h e n i t c o m e s t o w r i t i n g p a p e r s

o r b o o k s , h o w e v e r , t h e n i t i s m u c h m o r e d i ff ic u lt . I

d o n ' t t e n d t o w r i t e b o o k s . I n p a p e r s I t r y to d o a s m u c h

a s I c a n i n w r i t i n g a n a c c o u n t a n d a n i n t r o d u c t i o n

w h i c h g i v e s t h e id e a s . B u t y o u a re c o m m i t t e d t o

w r i t i n g a p r o o f i n a p a p e r , s o y o u h a v e t o d o t h a t.

M o s t b o o k s n o w a d a y s t e n d t o b e t o o f o r m a l m o s t o f

t h e t i m e , t h e y g i v e t o o m u c h i n th e w a y o f f o rm a l

p r o o f s , a n d n o t n e a r l y e n o u g h i n t h e w a y o f m o ti v a -

t i o n a n d i d e a s . O f c o u r s e i t i s d i f fi c u l t t o d o t h a t - - t o

g i v e m o t i v a t i o n a n d i d e a s .T h e r e a r e s o m e e x c e p t i o n s . I t h i n k t h e R u s s i a n s a r e

a n e x c e p t i o n . I t h i n k t h e R u s s i a n t r a d i t i o n i n m a t h e -

m a t i cs h a s b e e n l e s s fo r m a l i z e d a n d s t r u c t u r e d t h a n

t h e W e s t e r n t r a d it i o n , w h i c h i s u n d e r t h e i n f lu e n c e o f

F r e n c h m a t h e m a t i c s . F r e n c h m a t h e m a t i c s h a s b e e n

d o m i n a n t a n d h a s l e d t o a v e r y fo r m a l s c h o o l. I t h i n k

i t i s v e r y u n f o r t u n a t e t h a t m o s t b o o k s t e n d t o b e

w r i t t e n i n t h is o v e r l y a b s t ra c t w a y a n d d o n ' t t r y to

c o m m u n i c a t e u n d e r s t a n d i n g .

B u t i t i s h a r d t o c o m m u n i c a t e u n d e r s t a n d i n g b e -

c a u s e t h a t is s o m e t h i n g y o u g e t b y l iv i n g w i t h a

p r o b l e m f o r a lo n g t i m e . Y o u s t u d y i t, p e r h a p s f o r

y e a r s , y o u g e t t h e f e e l o f i t a n d i t i s i n y o u r b o n e s .

Y o u c a n ' t c o n v e y th a t t o a n y b o d y e ls e . H a v i n g s t u d i e d

t h e p r o b l e m f o r f i ve y e a r s y o u m a y b e a b l e t o p r e s e n t

i t i n s u c h a w a y t h a t i t w o u l d t a k e s o m e b o d y e ls e le s s

t i m e t o g e t t o t h a t p o i n t t h a n i t t o o k y o u , b u t i f t h e y

h a v e n ' t s t r u g g l e d w i t h t h e p r o b l e m a n d s e e n al l t h e

p i tf a ll s, t h e n t h e y h a v e n ' t r e a ll y u n d e r s t o o d it.

M I N I O : W h e r e d o y o u g e t y o u r i d ea s f o r w h a t y o u

a r e d o i n g ? D o y o u j u s t s it d o w n a n d s a y , " A ll r ig h t ,

I ' m g o i n g t o d o m a t h e m a t i c s f o r t w o h o u r s " ?

A T I Y A H : I t h i n k t h a t if y o u a r e a ct iv e l y w o r k i n g i n

m a t h e m a t i c a l r e s e a r c h , t h e n t h e m a t h e m a t i c s i s a l w a y s

w i t h y o u . W h e n y o u a r e t h in k i n g a b o u t p r o b l em s ,

t h e y a r e a l w a y s t h e re . W h e n I g e t u p i n th e m o r n i n g

a n d s h a v e , I ' m th i n k i n g a b o u t m a t h e m a t i c s . W h e n I

h a v e m y b r e a k f a s t, I a m s til l t h i n k i n g a b o u t m y p r o b -

l e m s . W h e n I a m d r i v i n g m y c a r , I a m s ti ll t h i n k i n g

a b o u t m y p r o b l e m s . A l l i n v a r io u s d e g r e e s o f c o n c e n -

t r a t i on .

S o m e t i m e s y o u w o n d e r w h e t h e r i t i s w o r t h w h i l e

t h i n k i n g a b o u t it w h i l e y o u a r e d o i n g t h e s e t h in g s ,

w h e t h e r i t r e a l l y h e l p s . Y o u a r e j u s t t u r n i n g i t o v e r

i d l y i n y o u r m i n d .

T h e r e a r e o c c a s io n s w h e n y o u s it d o w n i n th e

m o r n i n g a n d s t a rt to c o n c e n t r a t e v e r y h a r d o n s o m e -

t h i n g . T h a t k i n d o f a c u t e c o n c e n t r a t i o n i s v e r y d i f fi c u lt

f o r a lo n g p e r i o d o f t i m e a n d n o t a l w a y s v e r y s u c -

c e s sf u l . S o m e t i m e s y o u w i l l g e t p a s t y o u r p r o b l e m

w i t h c a r e f u l t h o u g h t . B u t t h e r e a l l y i n t e r e s t i n g i d e a s

o c c u r a t ti m e s w h e n y o u h a v e a f la s h o f in s p i ra t io n .

T h o s e a r e m o r e h a p h a z a r d b y t h e ir n a t u re ; t h e y m a yo c c u r j u s t i n c a s u a l c o n v e r s a t i o n . Y o u w i l l b e t a l k i n g

w i t h s o m e b o d y a n d h e m e n t i o n s s o m e t h i n g a n d y o u

t h in k , " G o o d G o d , y e s, t h a t i s j u s t w h a t I n e e d . . , i t

e x p l a in s w h a t I w a s t h i n k i n g a b o u t la s t w e e k . " A n d

y o u p u t t w o t h in g s t o g e th e r , y o u f u s e t h e m a n d s o m e -

t h i n g c o m e s o u t o f i t. P u t t i n g t w o t h i n g s t o g e t h e r , l ik e

a j i g s a w p u z z l e , i s i n s o m e s e n s e r a n d o m . B u t y o u

h a v e t o h a v e t h e s e t h i n g s c o n s t a n t l y t u r n i n g o v e r in

y o u r m i n d s o t h a t y o u c a n m a x i m i z e t h e o p p o r t u n i t i e s

f o r r a n d o m i n t e r a c ti o n . I t h i n k P o i n c a r 6 s a i d s o m e -

t h i n g l i ke t ha t . I t i s a k i nd o f p r oba b i l i s f i c e f f e c t: i de a s

s p i n a r o u n d i n y o u r m i n d a n d t h e f r u it f u l i n te r a c ti o n sa r is e o u t o f s o m e r a n d o m , f o r t u n a t e m u t a t i o n . T h e

s k il l i s t o m a x i m i z e t h i s d e g r e e o f r a n d o m n e s s s o t h a t

y o u i n c r e a s e t h e c h a n c e s o f a f r u i t fu l i n te r a c t io n .

F r o m m y p o i n t o f v ie w , t h e m o r e I t al k w i t h d i f f e r e n t

t y p e s o f p e o p l e , t h e m o r e I th i n k a b o u t d i f f er e n t b i t s

o f m a t h e m a t i c s , t h e g r e a t e r t h e c h a n c e t h a t I a m g o i n g

t o g e t a f r e s h i d e a f r o m s o m e o n e e l s e t h a t is g o in g t o

c o n n e c t u p w i t h s o m e t h i n g I k n o w .

F o r e x a m p l e , t h e I n d e x T h e o r e m w a s p a r t l y a m a t t e r

o f c h a n c e . S i n g e r a n d I h a p p e n e d t o b e w o r k i n g a t

O x f o r d o n t h i n g s r e l a te d t o t h e R i e m a n n - R o c h T h e -

THE MATHEMATICAL 1NTELLIGENCER VOL. 6, NO. 1, 19 84 1 7



Michael Atiyah (left) and Laurent Schwartz (right) in Japan.

orem, coming out of Hirzebruch's work. We were

playing around and we had the idea of looking for a

formula with the Dirac operator. And then Smale

passed thro ugh and we talked to him. He told us that

he had read a paper just the other day by Gel'fand,

which had been about the general question of the

index of operators, and he suggested that it might

have some relation to what we were doing. I found

this paper very difficult to understa nd, but it was the

general formulation of the problem and we were

looking at an impor tan t special case. The n we realized

that we had to generalize what we were doing and

that led to the whole thing. But it was Smale passingthrough that put us on the right track.

Another example is the work I did on instantons.

That was also a bit of a chance. I knew that Roger

Penrose and his group were working on geometrical

aspects of physics and one of them named Richard

Ward was doing some nice things. He was giving a

seminar and I asked myself, "Should I go to it or not,

would it be a bit of a bore? Well, all right." So I went

to the seminar. It was a very clear seminar; I under-

stood what he was doing, and I came away saying,

"Gee, that's really good. " I went back and spent three

days thinking very hard about i t , and suddenly i tdawned on me what was going on, how it related to

algebraic geometry. From then on, the thing took off.

I could easily have not gone to that seminar and the

subject would still be where it was. The gap between

the mathematicians and physicists was very big; I

doubt whether the ideas would have been taken up

nearly so rapidly. But of course, a lot of time you go

to seminars and don't get ideas.

MINIO: Do you have a favourite theorem or

problem?

A T I Y A H : That's not such a serious question because

I don't really believe in theorems, per se. I believe in

mathematics as sort of an entity; a theorem is just a

staging post. I know lots of nice nuggets, nice facts,

nice things, but I don't attach much importance to

the m individually. I guess the same goes for problems.

I don't want to give the impression that I think of

mathematics as simply abstract theory with no body

to it. A theory is interesting because it solves lots of

special problems and puts them in proper context; it

enables you to understand them all. Quite often a

theory evolves because somebody solved some very

hard problem first and then you try to understandwhat is going on- -y ou build a superstructure around

it. Soft theory which has no hard problems in it is of

no us e .

MINIO : What is your feeling about the classification

of finite simple groups?

A T I Y A H : Well, I have s lightly mixed feelings about

that. First of all, it takes so many pages to prove; it

seems to me the degree of understanding must be

pretty limited if that is the only way it can be done.One would hope that a much better understanding of

all this would emerge. Maybe I am wrong in this, but

I believe that if it is going to emerge, it is going to

come from people who look at groups from an extro-

verted rather than introverted way.

As groups arise in nature, they are things that move

things around, they are transformations or permuta-

tions. In the abstract version, you think of the group

as an internal structure with its own multiplication--

that's a very introverted point of view. If you only

allow yourself to use the introverted point of view, you




have a very limited array of techniques. But if you

think of the group throu gh its manifestations, from the

outside world, t hen you have the whole of the outside

world to help you. And you have got, or ought to

have, a much more powerful unders tanding. My idea,

my dream, is that one should prove these deep theo-

rems about groups by using the fact that the group

occurs in some natural context as a group of transfor-

mations, and t hen out of this the structure should be-

come transparent.

Also, I am not entirely sure how important this

whole result is. Some people will say that in mathe-

matics the most important thing is to set up an axiom

system with axioms one, two, three. There are ob-

jects-groups, spaces. The problem is then to classify

all such objects. I don't think that is the right point of

view. The goal is to understand the nature of these

things a nd to us e them; classification simply gives you

an indication of the scope of the theory.

For example, the classification of Lie groups is a bit

peculiar. You have this list of groups, both classical

and exceptional. But for most practical purposes, you

just use the classical groups. The exceptional Lie

groups are just there to show you that the theor y is a

bit bigger; it is pretty rare that they ever turn up. And

the theory of Lie groups would not be very different

if the classification had been vastly more complicated,

if there had been infinitely many more of these excep-

tional groups.

So I don't think it makes much difference to math-

ematics to know that there are different kinds

of simple groups or not. It is a nice intellectual end-

point , but I don' t think i t has any fundamental

importance.

MINIO: But if there were another way of doing it,

some extroverted point of view, would that have more

of an impact?

A T I Y A H : That would have an impact in the sense

that it would show people that one can do things in

some other way. But I don't think the result has fun-

damental importance; it doesn't compare with, say,

the theory of group representations.

The classification point of view can be greatly ex-

aggerated. It is a focus for a while, it points out niceproblems and challenges. But if it takes a lot of effort,

one suspects that there ought to be better ways to do

it. Getting a better way to do it may in itself be inter-

esting and show new ideas and new techniques at

work. You see this result, it looks nice, you'v e got this

long complicated proof, it is a challenge to find better

ways to prove it. The search ma y be beneficial, but the

benefits come more from the new ideas than from the

fact that you've got the new proof.

George Mackey once said to me something that I

think is very true. In a given area of mathematics the

things that are important are quite often not the most

technically difficult pa rt s- -t he hardest things to

prove. They are quite often the more elementary

things because those are the parts that have the widest

interaction with other fields and other areas--they

have the widest impact.

There are many things about group theory which

are tremendously important and occur in all sorts of

mathematics everywhere. Those tend to be the moreelementary things: basic ideas about groups, about

homomorphisms, about representations. General fea-

tures, general techniques--those are the things that

are really important.

The same is true in analysis. There are some very

fine points in proving exactly under which conditions

a Fourier series may converge; these are technically

demanding, very interesting. But for the rest of math-

ematicians who use Fourier analysis they are not really

importa nt. The specialists in a field, of course, get en-

amoured of the hard technical problems. But from the

point of view of mathema ticia ns as a whole, while th ey

admire them, they don't use them.

MINIO: Who is your most admired mathematician?

A T I Y A H : Well, I think that is rather easy. The

person I admire most is Her mann Weyl. I have found

that in almost e verything I have ever done in mathe-

matics, Hermann Weyl was there first. Most of the

areas I have worked in were areas where he worked

and did pioneering, very deep work himself--except

topology, of course, wh ich c ame after his time. But he

had interests in group theory, representation theory,

differential equations, spectral properties of differen-tial equations, differential geometry, theoretical

physics; nearly everything I have done is very much

in the spirit of the sort of things he worked in. And I

entirely agree with his conceptions about mathematics

and his view about what are the interesting things in

mathematics.

I heard him at the International Congress in Am-

sterdam. He gave the Fields medals ther e, to Serre and

Kodaira. Then I went to the Institute at Princeton, but

he was in Zurich at the time and he died there. I never

saw him at Princeton, I only saw him that one time.

So it wasn't personal contact that makes me admirehim.

For many years whe never I got into a different topic

I found out who was behind the scene, and sure

enough, it was Hermann Weyl. I feel my centre of

gravity is in the same place as his. Hflbert was more

algebraical; I don't think he had quite the same geo-

metrical insights. Von Neumann was more analytical

and worked more in applied areas. I think Hermann

Weyl is clearly the person I identify with most in terms

of mathematical philosophy and mathematical inter-

ests.

THE MATHEMATICAL INTELLIGENCER VOL. 6, NO. 1, 19 84 1 9

An Interview With Michael Atiyah

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