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Celebratio Mathematica

Michael F. Atiyah

General papers

Commentary by M. Atiyah

On nu­mer­ous oc­ca­sions I have giv­en “gen­er­al” talks and some of these have been pub­lished. The de­gree of gen­er­al­ity has var­ied, de­pend­ing on the nature of the audi­ence. In some cases I was talk­ing to pro­fes­sion­al math­em­aticians but in oth­ers I was al­most the only math­em­atician in the room. Giv­ing such gen­er­al talks (and, even more, writ­ing them up!) is hard work. It re­quires much great­er thought on the ma­ter­i­al and present­a­tion than for a nor­mal sem­in­ar, but it is a worth­while and im­port­ant activ­ity.

Pa­per [1] was a lec­ture giv­en at the cen­ten­ary meet­ing of the Lon­don Math­em­at­ic­al So­ci­ety, and it gave me an op­por­tun­ity of ex­plain­ing my views on al­geb­ra­ic to­po­logy. Not hav­ing had an or­tho­dox train­ing as a to­po­lo­gist I ten­ded to see to­po­logy as a power­ful tool that should be more widely ap­pre­ci­ated and used. My lec­ture was a plea for this point of view.

In 1968 I re­ceived an hon­or­ary de­gree from the Uni­versity of Bonn on the oc­ca­sion of its 150th an­niversary. While I was of course de­lighted by the hon­our, par­tic­u­larly be­cause of my close re­la­tion with Hirzebruch and his Arbeit­sta­gung, I was less than pleased when I dis­covered that I had to de­liv­er a lec­ture to the en­tire Sci­ence Fac­ulty (plus spouses). I did my best to ex­plain in very broad terms why gen­er­al ideas like sym­metry, con­tinu­ity, and prob­ab­il­ity were now so prom­in­ent in math­em­at­ics. The lec­ture (de­livered in Eng­lish) was in sure course trans­lated in­to Ger­man, suit­ably em­bel­lished with il­lus­tra­tions and, at Hirzebruch’s sug­ges­tion, pub­lished in a glossy pop­u­lar sci­ence magazine [2].

My lec­ture [4] at the IMA meet­ing provided an oc­ca­sion for an ex­pres­sion of my philo­soph­ic­al views on the styles of math­em­at­ic­al re­search. The text is es­sen­tially the tran­script of a tape re­cord­ing, which ex­plains its rather loose and verb­ose form. It con­tains noth­ing par­tic­u­larly pro­found or pro­voc­at­ive.

In 1975 I was asked to de­liv­er the Bakeri­an lec­ture [8] of the Roy­al So­ci­ety. Look­ing back through the re­cords I failed to find any pure math­em­atician among my pre­de­cessors, and this only con­firmed my view of the dif­fi­cult task a math­em­atician faces when ad­dress­ing a really gen­er­al sci­entif­ic audi­ence. Clearly one could not talk down pat­ron­iz­ingly to such dis­tin­guished sci­ent­ists, and the talk should also have some in­tel­lec­tu­al weight. On the oth­er hand most sci­ent­ists, par­tic­u­larly on the bio­lo­gic­al side, would know only rather ba­sic clas­sic­al math­em­at­ics. How was I to cope with this di­lemma? After much thought I de­cided to fo­cus on al­geb­ra­ic equa­tions, as be­ing fa­mil­i­ar, and to talk about gen­er­al ideas of struc­ture, cul­min­at­ing in the Weil con­jec­tures. It was such a dif­fi­cult task that I felt com­pelled to write the talk ver­batim in ad­vance and then to read it — the only oc­ca­sion when I have gone to this length of pre­par­a­tion.

An­oth­er Roy­al So­ci­ety pub­lic­a­tion, but of quite a dif­fer­ent kind, was my ob­it­u­ary [6] for Hodge. As his stu­dent and dis­ciple, this was a task I read­ily un­der­took. I had no prob­lem deal­ing with the math­em­at­ic­al side and for­tu­nately Hodge had left ex­tens­ive notes on the per­son­al side which I re­lied on heav­ily. A few years be­fore, on the oc­ca­sion of Hodge’s sev­en­ti­eth birth­day con­fer­ence, I had giv­en a lec­ture try­ing to sum­mar­ize Hodge’s math­em­at­ic­al con­tri­bu­tion, put in­to its his­tor­ic­al con­text. Hodge was in the audi­ence and was kind enough to say af­ter­wards that I had giv­en a very ac­cur­ate de­scrip­tion of how his math­em­at­ic­al ideas had evolved. I hope the same is true of the ob­it­u­ary.

In 1975 the IMA or­gan­ized a meet­ing in Cam­bridge to cel­eb­rate Lit­tle­wood’s 90th birth­day, and [7] is again the tran­script of a tape re­cord­ing. The lack of pol­ish will be im­me­di­ately ap­par­ent to the read­er. I was sup­posed to talk about something “ex­cit­ing”, so I de­cided to dis­cuss the Bern­stein poly­no­mi­al and re­lated ques­tions. I was by no means an ex­pert on this sub­ject, though it was re­lated to my earli­er pa­per [e1], but I found it a very fas­cin­at­ing area and had tried to fol­low de­vel­op­ments. I had first met the Bern­stein poly­no­mi­al in Gel­fand’s house in Mo­scow. Gel­fand in­tro­duced Bern­stein to me and told him to ex­plain his res­ults. However, Bern­stein was young and bash­ful, so his slow present­a­tion ex­as­per­ated Gel­fand, who then in­ter­rup­ted and car­ried on the ex­plan­a­tion him­self with great verve. No won­der I as­so­ci­ated the top­ic with “ex­cite­ment”!

At the ICME Con­gress in Karls­ruhe in 1976 I was in­vited to give a plen­ary ad­dress [9]. The audi­ence con­sisted largely of school teach­ers and oth­ers in­ter­ested in math­em­at­ic­al edu­ca­tion. This was clearly an oc­ca­sion for a very broad-brush present­a­tion, and I tried to put the de­vel­op­ment of mod­ern math­em­at­ics in­to some sort of his­tor­ic­al per­spect­ive. There had been a great deal of non­sense talked (and im­ple­men­ted!) about mod­ern math­em­at­ics in schools, and I thought I should at­tempt to clear the air.

On my re­turn from Prin­ceton in 1973 I joined the Coun­cil of Lon­don Math­em­at­ic­al So­ci­ety and from 1974 to 1976 I was Pres­id­ent. The fi­nal duty of the Pres­id­ent is the pres­id­en­tial ad­dress, and this is clearly a rather spe­cial and chal­len­ging oc­ca­sion. In my case the chal­lenge was en­hanced by the pres­ence in the audi­ence of my wife and two sons, all math­em­at­ic­ally trained! I de­cided to il­lus­trate some im­port­ant and deep math­em­at­ic­al de­vel­op­ments by us­ing very simple ele­ment­ary ex­amples. The pub­lished form [11] of my pres­id­en­tial ad­dress is es­sen­tially the same as the spoken ver­sion. This is a rule I gen­er­ally fol­low. I res­ist the tempta­tion to en­large or im­prove the pub­lished ver­sion of a talk, in the be­lief that the in­form­al style of a lec­ture has much to com­mend it. Too of­ten math­em­at­ics in print is heavy, form­al, and pedant­ic, mak­ing im­possible de­mands on the read­er.

The Math­em­at­ic­al As­so­ci­ation has a long tra­di­tion of choos­ing its pres­id­ents al­tern­ately from the uni­versit­ies and the schools. In 1981–82 I be­came pres­id­ent, and for some time af­ter­wards my edu­ca­tion­al in­volve­ment in­creased, cul­min­at­ing in mem­ber­ship of the Cock­croft Com­mit­tee. This was a good ex­ample of the snow­ball route to be­com­ing an “ex­pert” — each com­mit­tee adding a lay­er to one’s cre­den­tials and hence lead­ing to the next com­mit­tee. Pres­id­ents, in their pres­id­en­tial ad­dresses, fre­quently speak on edu­ca­tion­al policy and usu­ally cas­tig­ate the Gov­ern­ment of the time. Not feel­ing an au­thor­ity on these mat­ters I chose in­stead, in my ad­dress [13], to make a plea for geo­metry. All my re­search has had a pro­nounced geo­met­ric­al fla­vor, and I tried to ex­plain my view of geo­metry in terms that would be in­tel­li­gible to school teach­ers, and might be rel­ev­ant in the classroom.

The re­cord [16] of my in­ter­view with Minio is hardly one of my “works”, but it ex­pressed my views on many math­em­at­ic­al top­ics in a spon­tan­eous and un­in­hib­ited way. To have omit­ted it would have looked like cen­sor­ship! Of course, some of my off-the-cuff re­marks are a bit out­rageous, and out­rage has been duly re­gistered par­tic­u­larly for my re­marks about the clas­si­fic­a­tion of fi­nite simple groups. While I do not wish to re­tract any­thing I said, I could have been more dip­lo­mat­ic and a writ­ten art­icle would have been more bal­anced. Per­haps I should add here my be­lief that the most im­port­ant out­come of the search for clas­si­fic­a­tion was the dis­cov­ery of the Mon­ster group. This is clearly a mys­ter­i­ous ob­ject hav­ing deep con­nec­tions with im­port­ant top­ics such as mod­u­lar forms, and re­mains to be prop­erly un­der­stood.

In my in­ter­view with Minio I had com­men­ted with ap­prov­al on the ab­sence of No­bel Prizes in math­em­at­ics. Iron­ic­ally a short while later I heard that I had been awar­ded the Fel­trinelli Prize by the Itali­an Academy (the “Lincei”). While lack­ing the fame or no­tori­ety of the No­bel Prizes it was com­par­able fin­an­cially and my math­em­at­ic­al pre­de­cessors were Hadam­ard, Lef­schetz, and Leray: I was in ex­cel­lent com­pany. The present­a­tions were to be made in Rome by the Pres­id­ent of the Re­pub­lic, and I was asked to pre­pare a speech of about 25 minutes on my work. This was clearly a ser­i­ous task and I gave the mat­ter much thought. Pre­sum­ably I was sup­posed to sur­vey my math­em­at­ic­al work in a way that would not be en­tirely in­com­pre­hens­ible to the dis­tin­guished audi­ence at the Itali­an Academy. Even­tu­ally, with some mis­giv­ing, I pro­duced [26], and set off with my wife for the ce­re­mony in Rome. For­tu­nately, Rome is not Stock­holm and the Itali­ans have a dif­fer­ent way of do­ing things. There was no lack of ce­re­mony with im­pos­ing guards­men in uni­form and a beau­ti­fully fur­nished salon. Moreover, Pres­id­ent Per­tini duly turned up. However, as the pro­ceed­ings pro­gressed, time was run­ning short, the Pres­id­ent had oth­er ob­lig­a­tions to at­tend to and I was asked to com­press my care­fully pre­pared speech in­to 10 minutes! I hast­ily de­cided which pas­sages to se­lect. I can­not now re­mem­ber what I omit­ted but no doubt I left out all the more math­em­at­ic­al bits. In a way it was a great re­lief. So, al­though my speech was es­sen­tially not giv­en, the writ­ten ver­sion now serves a dif­fer­ent pur­pose by provid­ing a brief over­view of the pa­pers col­lec­ted here.

In Oc­to­ber 1983 the In­sti­tut des Hautes Études Sci­en­ti­fiques cel­eb­rated its 25th an­niversary. As someone who had been closely in­volved with its de­vel­op­ment I at­ten­ded the cel­eb­ra­tions and gave one of three sci­entif­ic lec­tures (I nar­rowly es­caped hav­ing to give one of the form­al min­is­teri­al-type ad­dresses!). I chose to sur­vey the cur­rent in­ter­ac­tion between geo­metry and ana­lys­is [25]. I jus­ti­fied my choice partly by the fact that the three Fields Medal­lists at the Warsaw Con­gress (Connes, Thur­ston, Yau) all worked in this area. I also re­ferred to the work of Don­ald­son and Freed­man: it did not take much foresight to guess that they would fig­ure amongst Fields Medal­lists at the next Con­gress.

George Or­well’s 1984 made such an im­pact on our time that the even­tu­al ar­rival of that year was greeted with some fore­bod­ing. I was per­suaded (against my bet­ter judge­ment) to take part in a pub­lic meet­ing in Lo­c­arno, where a num­ber of speak­ers from dif­fer­ent fields were asked to de­scribe the chal­lenge of 1984. I found my­self in un­usu­al and dis­tin­guished com­pany. It was in­deed a pleas­ure to make the ac­quaint­ance of Sir Karl Pop­per, the well-known philo­soph­er, and to meet Sir John Ec­cles, who had shared the No­bel Prize for Physiology with Hodgkin and Hux­ley. This was however severe com­pet­i­tion, from the point of view of the gen­er­al pub­lic. Al­though I had pre­pared a care­ful ad­dress [22], [24] on the com­puter re­volu­tion, as seen by a math­em­atician, I doubt if this is what the audi­ence wanted. The pro­ceed­ings were sponsored by, and even­tu­ally pub­lished in, an Itali­an magazine. They reached a per­haps more re­cept­ive and ap­pro­pri­ate audi­ence when Ka­hane, Pres­id­ent of ICMI, sug­ges­ted that the art­icle be presen­ted to a con­fer­ence on com­puters and math­em­at­ic­al equa­tion.

An­oth­er but more ser­i­ous oc­ca­sion when I had to ad­dress a non-spe­cial­ist audi­ence took place at Col­mar in 1983, at a small meet­ing or­gan­ized by the European Sci­ence Found­a­tion. This con­sisted of rep­res­ent­at­ives of dif­fer­ent dis­cip­lines, from bio­logy to art his­tory, try­ing to de­scribe the nature of pro­gress in their field: what are the cri­ter­ia used to as­sess “pro­gress”? It was a ser­i­ous and in­ter­est­ing meet­ing with high-level par­ti­cip­a­tion. Pa­pers were writ­ten and cir­cu­lated in ad­vance, with sub­sequent com­ment­ary and dis­cus­sion at the meet­ing, so I did not ac­tu­ally de­liv­er my own con­tri­bu­tion [21]. My ana­lys­is of the situ­ation in math­em­at­ics re­peats many of the points I had made in pre­vi­ous art­icles, but it rep­res­ents per­haps the most care­fully thought-out ver­sion of my views on math­em­at­ics and its place in so­ci­ety.

Works

[1]M. F. Atiyah: “The role of al­geb­ra­ic to­po­logy in math­em­at­ics,” J. Lon­don Math. Soc. 41 : 1 (1966), pp. 63–​69. See also Geo­met­rie (1972). MR 0187231 Zbl 0137.​17301 article

[2]M. F. Atiyah: “Wan­del und Forts­ch­ritt in der Math­em­atik,” Bild der Wis­senschaft 4 (1969), pp. 315–​323. Re­pub­lished in Math­em­atiker über die Math­em­atik (1974) and Atiyah’s Col­lec­ted works, vol. 1. article

[3]M. F. Atiyah: “The role of al­geb­ra­ic to­po­logy in math­em­at­ics,” pp. 74–​83 in Geo­met­rie. Edi­ted by K. Strubeck­er. Wege der Forschung 177. Wis­senschaft­liche Buchgesell­schaft (Darm­stad), 1972. See also J. Lon­don Math. Soc. 41:1 (1966). incollection

[4]M. F. Atiyah: “How re­search is car­ried out,” Bull. IMA 10 (1974), pp. 232–​234. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 1. article

[5]M. F. Atiyah: “Wan­del und Forts­ch­ritt in der Math­em­atik,” pp. 202–​218 in Math­em­atiker über die Math­em­atik. Edi­ted by M. Otte. Wis­senschaft und Öf­fent­lich­keit. Spring­er (Ber­lin), 1974. Re­pub­lished from Bild der Wis­senschaft 4 (1969). incollection

[6]M. F. Atiyah: “Wil­li­am Val­lance Douglas Hodge 1903–1975,” Biog. Mem. Fel­lows Roy. Soc. Lond. 22 (1976), pp. 169–​192. Re­pub­lished in Col­lec­ted works, vol. 1 and with modi­fic­a­tions in Bull. Lon­don Math. Soc. 9:1 (1977). article

[7]M. F. Atiyah: “Sin­gu­lar­it­ies of func­tions,” Bull. Inst. Math. Ap­pl. 12 : 7 (1976), pp. 203–​206. Pa­per presen­ted at the Sym­posi­um on Ex­cite­ment in Math­em­at­ics. MR 0590054 article

[8]M. F. Atiyah: “Bakeri­an Lec­ture, 1975: Glob­al geo­metry,” Proc. Roy. Soc. Lon­don Ser. A 347 : 1650 (1976), pp. 291–​299. Re­pub­lished in Amer. Math. Mon. 111:8 (2004). MR 0462821 article

[9]M. F. Atiyah: “Trends in pure math­em­at­ics,” pp. 71–​74 in Pro­ceed­ings of the Third In­ter­na­tion­al Con­gress on Math­em­at­ic­al Edu­ca­tion (Karls­ruhe, 16–21 Au­gust 1976). Edi­ted by H. Athen and H. Kunle. Uni­versität Karls­ruhe, 1977. Re­prin­ted in Atiyah’s Col­lec­ted works, vol. 1. incollection

[10]M. F. Atiyah: “Wil­li­am Val­lance Douglas Hodge,” Bull. Lon­don Math. Soc. 9 : 1 (1977), pp. 99–​118. Re­pub­lished with modi­fic­a­tions from Biog. Mem. Fel­lows Roy. Soc. Lond. 22 (1976). MR 0427007 Zbl 0343.​01010 article

[11]M. F. Atiyah: “The unity of math­em­at­ics,” Bull. Lon­don Math. Soc. 10 : 1 (1978), pp. 69–​76. Bul­gari­an trans­la­tion pub­lished in Fiz.-Mat. Spis. 22(55):1 (1979). MR 0476223 Zbl 0376.​00001 article

[12]M. F. Atiyah: “The unity of math­em­at­ics,” Fiz.-Mat. Spis. 22(55) : 1 (1979), pp. 11–​18. Bul­gari­an trans­la­tion of art­icle in Bull. Lon­don Math. Soc 10:1 (1979). MR 543366 Zbl 0464.​00034 article

[13]M. Atiyah: “What is geo­metry?,” Math. Gaz. 66 : 437 (1982), pp. 179–​184. Nor­we­gi­an trans­la­tion pub­lished in Nor­mat 31:2 (1983). A Czech trans­la­tion pub­lished in Pok­roky Mat. Fyz. As­tro­nom. 29:4 (1984). MR 677460 Zbl 0549.​51001 article

[14]M. Atiyah: “Hva er geo­metri?” [What is geo­metry?], Nor­mat 31 : 2 (1983), pp. 70–​74. Nor­we­gi­an trans­la­tion of art­icle in Math. Gaz. 66:437 (1982). See also Pok­roky Mat. Fyz. As­tro­nom. 29:4 (1984). MR 702261 article

[15]M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Eta in­vari­ants, sig­na­ture de­fects of cusps, and val­ues of \( L \)-func­tions,” Ann. Math. (2) 118 : 1 (1983), pp. 131–​177. An ad­dendum was pub­lished in Ann. Math. 119:3 (1984). MR 707164 Zbl 0531.​58048 article

[16]R. Minio: “An in­ter­view with Mi­chael Atiyah,” Math. In­tel­li­gen­cer 6 : 1 (1984), pp. 9–​19. Slov­e­ni­an trans­la­tion pub­lished in Obzornik Mat. Fiz. 31:5–6 (1984). Czech trans­la­tion pub­lished in Pok­roky Mat. Fyz. As­tro­nom. 31:3 (1986). MR 735359 article

[17]R. Minio: “An in­ter­view with Mi­chael Atiyah,” Obzornik Mat. Fiz. 31 : 5–​6 (1984), pp. 129–​142. Slov­e­ni­an trans­la­tion of art­icle in Math. In­tell. 6:1 (1984). MR 782244 article

[18]M. F. Atiyah, H. Don­nelly, and I. M. Sing­er: “Sig­na­ture de­fects of cusps and val­ues of \( L \)-func­tions: The non­split case. Ad­dendum to: ‘Eta in­vari­ants, sig­na­ture de­fects of cusps, and val­ues of \( L \)-func­tions’,” Ann. Math. (2) 119 : 3 (1984), pp. 635–​637. Ad­dendum to an art­icle in Ann. Math. 118:1 (1983). MR 744866 article

[19]M. F. Atiyah: “Co je geo­met­rie?” [What is geo­metry?], Pok­roky Mat. Fyz. As­tro­nom. 29 : 4 (1984), pp. 213–​217. Czech trans­la­tion of art­icle in Math. Gaz. 66:437 (1982). See also Nor­mat 31:2 (1983). MR 762935 article

[20]M. F. Atiyah: “Rivoluzione in­form­at­ica e matem­at­ica” [Math­em­at­ics and the com­puter re­volu­tion], Nuova Civiltà della Mac­chine 2 : 3 (1984), pp. 27–​32. Itali­an trans­la­tion of a lec­ture giv­en at the Lo­c­arno Con­fer­ence “1984: Comin­cia il fu­turo” (May 1984), pub­lished in The in­flu­ence of com­puters and in­form­at­ics on math­em­at­ics and its teach­ing (1986). article

[21]M. F. Atiyah: “Identi­fy­ing pro­gress in math­em­at­ics,” pp. 24–​41 in The iden­ti­fic­a­tion of pro­gress in learn­ing (Col­mar, France, 26–28 March 1983). Edi­ted by T. Häger­strand. Cam­bridge Uni­versity Press, 1985. MR 807796 incollection

[22]M. Atiyah: “Math­em­at­ics and the com­puter re­volu­tion,” pp. 43–​51 in The in­flu­ence of com­puters and in­form­at­ics on math­em­at­ics and its teach­ing (Stras­bourg, March, 1985). Edi­ted by A. G. Howson and J.-P. Ka­hane. Cam­bridge Uni­versity Press, 1986. Re­prin­ted in Atiyah’s Col­lec­ted works, vol. 1, pp. 327–348.

[23]R. Minio: “An in­ter­view with Mi­chael Atiyah,” Pok­roky Mat. Fyz. As­tro­nom. 31 : 3 (1986), pp. 154–​168. Czech trans­la­tion of art­icle from Math. In­tell. 6:1 (1984). MR 857260 article

[24]M. F. Atiyah: “Math­em­at­ics and the com­puter re­volu­tion,” pp. 43–​51 in The in­flu­ence of com­puters and in­form­at­ics on math­em­at­ics and its teach­ing (Stras­bourg, March 1985). Edi­ted by A. G. Howson and J.-P. Ka­hane. Cam­bridge Uni­versity Press, 1986. Lec­ture giv­en at the Lo­c­arno Con­fer­ence “1984: Comin­cia il fu­turo” (May 1984). Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 1. Itali­an trans­la­tion pub­lished in Nuova Civiltà della Mac­chine 2:3 (1984). incollection

[25]M. F. Atiyah: “Geo­metry and ana­lys­is in the nine­teen eighties,” pp. 317–​326 in Col­lec­ted works, vol. 2: \( K \)-the­ory. Ox­ford Sci­ence Pub­lic­a­tions. Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 1988. incollection

[26]M. F. Atiyah: “Speech on con­fer­ment of Fel­trinelli Prize,” pp. 309–​316 in Col­lec­ted works, vol. 2: \( K \)-the­ory. Ox­ford Sci­ence Pub­lic­a­tions. Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 1988. Atiyah re­ceived the Ant­o­nio Fel­trinelli Prize from the Ac­ca­demia Nazionale dei Lincei in 1981. incollection

[27]M. Atiyah: Col­lec­ted works, vol. 1: Early pa­pers; gen­er­al pa­pers. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford and New York), 1988. MR 951892 Zbl 0935.​01034 book

[28]M. F. Atiyah: “Bakeri­an Lec­ture, 1975: Glob­al geo­metry,” Amer. Math. Mon. 111 : 8 (2004), pp. 716–​723. Re­pub­lished from Proc. Roy. Soc. Lon­don Ser. A 347:1650 (1976). MR 2091548 Zbl 1187.​58001 article