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Celebratio Mathematica

Michael F. Atiyah

Some later works

Commentary by M. Atiyah

In the de­tailed com­ment­ar­ies to the earli­er volumes, there is an ex­tens­ive ac­count of my main math­em­at­ic­al col­lab­or­a­tions, More re­cently, I was asked to provide an in­teg­rated ana­lys­is of these col­lab­or­a­tions, both from the math­em­at­ic­al and per­son­al points of view. This ap­peared as [72] and is ap­pro­pri­ately the first art­icle in this volume [6 of the Col­lec­ted works].

My mono­graph [8] writ­ten jointly with Nigel Hitchin, and based on the Port­er lec­tures I gave at Rice Uni­versity in 1987, is now out of print and so is in­cluded in Volume 6 [of the Col­lec­ted works]. It provides a full ac­count of the res­ults an­nounced briefly in pa­pers [1] and [2] which ap­peared in Volume 5. The main res­ult is the ex­pli­cit de­term­in­a­tion of the nat­ur­al met­ric on the mod­uli space of centered \( \mathit{SU}(2) \)-mono­poles of charge 2. The key prop­erty which is ex­ploited is that the met­ric is hy­per­kähler, a prop­erty arising from su­per-sym­metry, and of in­creas­ing in­terest to geo­met­ers and phys­i­cists. An ex­pos­it­ory ac­count of hy­per­kähler man­i­folds is con­tained in [17]. The next pa­per [4] is a lengthy one cen­ter­ing round co­homo­lo­gic­al and arith­met­ic­al as­pects of the Dede­kind \( \eta \)-func­tion, in­ter­preted in terms of the in­dex the­or­em in vari­ous forms. It was the sub­ject of the Rademach­er Lec­tures that I gave at the Uni­versity of Pennsylvania in 1987.

The spec­tac­u­lar res­ults of Don­ald­son on 4-di­men­sion­al man­i­folds which led to his Fields Medal are briefly de­scribed in the cita­tion [3] I presen­ted at the ICM in Berke­ley in 1986. Four years later, there was a sig­ni­fic­ant fol­low up when I wrote the cita­tion [20] for Wit­ten’s Fields Medal at the Kyoto ICM (1990). Wit­ten is the only phys­i­cist so far to have re­ceived the Fields Medal and, al­though some eye­brows were raised at the time, there are few who would now dis­pute the enorm­ous in­flu­ence of his work on math­em­at­ics.

The new vis­tas opened up in geo­metry by Don­ald­son and Wit­ten trans­formed the sub­ject in the last part of the 20th cen­tury, and this is re­flec­ted in my own pa­pers, many of which were ex­pos­it­ory or spec­u­lat­ive. Don­ald­son the­ory and the new poly­no­mi­al knot in­vari­ants of Vaughan Jones (also a Fields Medal­ist) were dis­cussed by me in [6], at the Her­mann Weyl Sym­posi­um, where I spec­u­lated on the pos­sible phys­ic­al sig­ni­fic­ance of the geo­met­ric res­ults. This chal­lenge was suc­cess­fully taken up by Wit­ten in the next few years and led to the emer­gence of to­po­lo­gic­al quantum field the­or­ies (TQFT). I sum­mar­ized this story in an ax­io­mat­ic form that I thought would be pal­at­able to math­em­aticians in [7], and I elab­or­ated on the 2-di­men­sion­al case later in [38]. I am glad to say that math­em­aticians did in fact take on board the concept of a TQFT and built sub­stan­tially on it, at least in the 3-di­men­sion­al con­text.

The Jones the­ory, trans­formed by Wit­ten in­to a TQFT, was the sub­ject of my Lincei Lec­tures in [15], de­livered in Florence in 1988, and was giv­en a pop­u­lar ex­pos­i­tion at the Roy­al In­sti­tu­tion [67] shortly af­ter­wards. A slightly dif­fer­ent present­a­tion was the sub­ject of my Mil­ne Iec­ture in Ox­ford [9]. A to­po­lo­gic­al sub­tlety in the the­ory, the “grav­it­a­tion­al an­om­aly”, led to the note [14].

Wit­ten’s in­ter­pret­a­tion of Don­ald­son the­ory as a TQFT in­volved an elab­or­ate su­per-sym­met­ric ver­sion of Yang–Mills the­ory, whose math­em­at­ic­al sig­ni­fic­ance was far from clear. My pa­per [16] with my last stu­dent, Lisa Jef­frey, gave a form­al math­em­at­ic­al ex­plan­a­tion of it in terms of an equivari­ant (in­fin­ite-di­men­sion­al) Euler class. A very dif­fer­ent ob­ject, with a sim­il­ar nature, was in­tro­duced in my joint pa­per [10] with Graeme Segal as a \( K \)-the­ory in­ter­pret­a­tion of what phys­i­cists called the “string the­ory Euler num­ber of an or­bi­fold”. This was per­haps the first in­dic­a­tion that \( K \)-the­ory might have rel­ev­ance for string the­ory (bey­ond its role in in­dex the­ory).

Mag­net­ic mono­poles are, like in­stan­tons, solitons and their the­ory is in some sense “in­teg­rable”. In par­tic­u­lar, they are de­scribed by Hitchin’s “spec­tral curve”. My pa­per [18], based on work with my former stu­dent Mi­chael Mur­ray, es­tab­lishes a con­nec­tion between these spec­tral curves and those arising from the Yang–Bax­ter equa­tions. The res­ult is in­triguing but still mys­ter­i­ous and it has not yet been taken any fur­ther.

My work on mono­poles in [8] had been much in­flu­enced by Nick Man­ton whom I met as a gradu­ate stu­dent in Cam­bridge. In fact it was my old friend and con­tem­por­ary John Polk­ing­horne, just on the verge of for­sak­ing phys­ics for the Church, who sug­ges­ted to me that I might find Nick in­ter­est­ing to talk to. It was Nick who told me that the met­ric on the mono­pole mod­uli space should de­term­ine the low-en­ergy dy­nam­ics, and this even­tu­ally led to [8]. Sub­sequently he got me in­ter­ested in skyrmi­ons, a clas­sic­al non-lin­ear soliton mod­el of the nuc­le­us that was in­tro­duced much earli­er by Tony Skyrme and which has now reac­quired some pop­ular­ity. Nick was per­suaded that skyrmi­ons had something in com­mon with mono­poles. After much struggle we fi­nally found a link (ac­tu­ally via in­stan­tons) and this led to our two pa­pers [11] and [22].

The next set of four short art­icles are con­cerned with the gen­er­al re­la­tion between geo­metry and phys­ics. In [26] I re­spon­ded to a de­lib­er­ately pro­voc­at­ive art­icle by Jaffe and Quinn, and the lec­ture [31] fol­lows up the same theme. [5] was my lec­ture at the first ICIAM con­fer­ence and aimed at a broad audi­ence of ap­plied math­em­aticians, while [42] was de­livered on the oc­ca­sion of my hon­or­ary de­gree at Brown Uni­versity.

Between 1990 and 1995, much of my time was taken up by my du­ties as Pres­id­ent of the Roy­al So­ci­ety and, from the large num­ber of speeches I had to de­liv­er in that ca­pa­city. I have se­lec­ted a small sample of three. The first two [21], [43] and [23] were de­livered in Phil­adelphia at a joint meet­ing with the Amer­ic­an Philo­soph­ic­al So­ci­ety. [24] was only one of my five Pres­id­en­tial Ad­dresses to the Roy­al So­ci­ety where I con­cen­trated on math­em­at­ics. I tried to de­scribe its fun­da­ment­al nature, par­tic­u­larly in re­la­tion to sci­ence. I elab­or­ated the view put for­ward more briefly in [21] of math­em­at­ics as a lan­guage. A slightly dif­fer­ent per­spect­ive on math­em­at­ics, re­lated more to philo­soph­ic­al ques­tions, is con­tained in my book re­view [29].

The next eight pa­pers re­late to in­di­vidu­al math­em­aticians and their work. [39] was a lec­ture de­livered at the Roy­al So­ci­ety in con­nec­tion with the lay­ing of a com­mem­or­ative stone to Dir­ac in West­min­ster Ab­bey. [34] and [53] were lec­tures in hon­our of my friend and col­lab­or­at­or Fritz Hirzebruch, and [27] was my con­tri­bu­tion to the col­lec­ted works of an­oth­er close col­league, Raoul Bott. [40] was giv­en at a con­fer­ence hon­our­ing Ro­ger Pen­rose and it gave me an op­por­tun­ity of ac­know­ledging the in­flu­ence of my Ox­ford pro­fess­or­i­al col­league and erstwhile fel­low stu­dent. [41] and [44] were ob­it­u­ary art­icles for my first teach­er J. A. Todd and the great Ja­pan­ese math­em­atician K. Kodaira, at whose feet I had stud­ied in Prin­ceton in 1955.

The ob­it­u­ary art­icle [58] on Her­mann Weyl was a very un­usu­al one and calls for some com­ment. Weyl died in 1955, but for some reas­on the Na­tion­al Academy of Sci­ences had not pub­lished an ob­it­u­ary in the years after his death. I was ex­tremely sur­prised to be asked to un­der­take this so many years later. Giv­en the wide range and im­port­ance of Weyl’s work, it would have been a her­culean task to have pro­duced a really com­pre­hens­ive ob­it­u­ary, and per­haps this is why the pro­ject lapsed. However, Weyl has al­ways been one of my her­oes, and the present in­ter­ac­tion between geo­metry and phys­ics is so much in the spir­it of Weyl’s own work that I felt I could not turn it down. I used the oc­ca­sion to as­sess Weyl’s con­tri­bu­tions in the ret­ro­spect­ive light of the pro­gress of the last fifty years, with em­phas­is on the re­la­tions to phys­ics.

The year 1900 saw the great lec­ture by Hil­bert at the Par­is ICM, when he pro­duced his fam­ous list of prob­lems. It was widely re­cog­nised that this feat could not be suc­cess­fully re­peated but vari­ous at­tempts were made to re­cog­nise the end of the cen­tury (and of the mil­len­ni­um). The three next art­icles re­flect this. [45] was giv­en at the open­ing of the new math­em­at­ic­al centre at the Amer­ic­an Uni­versity of Beirut in Le­ban­on, my fath­er’s home coun­try. [55] was an ex­tremely broad-brush sur­vey lec­ture I gave, look­ing back at the math­em­at­ics of the 20th Cen­tury. This lec­ture was giv­en in many places and was re­pro­duced in many journ­als and in dif­fer­ent lan­guages. The IMU com­mis­sioned a spe­cial volume for 2000 for which I was one of the four ed­it­ors. I failed to con­trib­ute an art­icle but my co-ed­it­ors kindly al­lowed me to write a lengthy pre­face which is re­pro­duced as [46]. My most re­cent sur­vey on geo­metry and phys­ics is [70], giv­en on the oc­ca­sion of the Gel­fand 90th birth­day con­fer­ence in Har­vard. It con­cludes with some spec­u­la­tion about the fu­ture sig­ni­fic­ance of all these de­vel­op­ments for both math­em­at­ics and phys­ics.

The next six pa­pers all had their ori­gin in a simple ques­tion about geo­metry that was put to me by Mi­chael Berry. It emerged from a study (jointly with Jonath­an Rob­bins) of the spin stat­ist­ics the­or­em of quantum mech­an­ics and it con­cerned con­fig­ur­a­tions of \( n \) dis­tinct points in \( \mathbb{R}^3 \). It turned out to be re­mark­ably fruit­ful and it en­abled me to re-enter math­em­at­ic­al re­search after my long peri­od of in­volve­ment with Trin­ity and the Roy­al So­ci­ety. In [32] I de­scribed the prob­lem and gave a first solu­tion, but I also made a con­jec­ture which would lead to a much more el­eg­ant solu­tion. In [52] I de­duced some co­homo­lo­gic­al con­sequences, while in [51] I elab­or­ated on my con­jec­ture, de­fin­ing a cer­tain \( n\times n \) de­term­in­ant as­so­ci­ated to \( n \) points. The con­jec­ture as­serts that this de­term­in­ant nev­er van­ishes. In [56] I col­lab­or­ated with Paul Sutcliffe whose com­pu­ta­tion­al skills en­abled us to provide strong nu­mer­ic­al evid­ence in fa­vour of the con­jec­ture. We also re­versed the prob­lem by study­ing those con­fig­ur­a­tion of points which max­im­ized the norm of the de­term­in­ant. We found these were re­mark­ably sym­met­ric con­fig­ur­a­tions, sim­il­ar to those arising in a vari­ety of phys­ic­al prob­lems. In my “Le­onardo” lec­ture at Mil­an I gave a sur­vey of such prob­lems and this was then writ­ten up by Paul as our second joint pa­per [63]. Fi­nally, in a slightly dif­fer­ent dir­ec­tion, my pa­per [60] with Ro­ger Bielawski gave yet an­oth­er solu­tion to the Berry–Rob­bins prob­lem. This came from a study of Nahm’s equa­tion and was re­lated to oth­er ques­tions in phys­ics.

It also gen­er­al­ized the prob­lem for Lie groups — the ori­gin­al prob­lem be­ing the case of \( U(n) \).

The fi­nal five pa­pers are on phys­ics or on closely re­lated geo­metry. My joint pa­per [50] with Mal­da­cena and Vafa ori­gin­ated from an earli­er dis­cus­sion I had had with Vafa about de­riv­ing an open string — closed string du­al­ity in 6 di­men­sions by go­ing up to 7 di­men­sions. This fit­ted in with cur­rent ideas on M-the­ory and was my in­tro­duc­tion to the sub­ject. The much lar­ger joint pa­per with Wit­ten [59] star­ted off as a fol­low-up try­ing to ex­ploit a hid­den tri­al­ity in [50]. However, as a res­ult of a two-month stay at Cal Tech it grew in­to a much more am­bi­tious pro­ject. I learnt a great deal more phys­ics and con­trib­uted a mod­est amount of to­po­logy. I re­por­ted on some of this at a con­fer­ence in Durham at which Jür­gen Berndt was present. He sug­ges­ted that my res­ults on the qua­ternion pro­ject­ive plane should ex­tend to the Cay­ley plane and even to a cer­tain ho­mo­gen­eous space of the ex­cep­tion­al group \( E_6 \). This led in due course to our joint pa­per [62] and to my edu­ca­tion on the ex­cep­tion­al Lie groups.

The last two pa­pers are both con­cerned with gen­er­al­iz­a­tions of \( K \)-the­ory mo­tiv­ated by phys­ics. My joint pa­per with Mi­chael Hop­kins [66] arose from my try­ing to un­der­stand “ori­enti­fold string the­ory” while I was at Cal Tech, and fol­low­ing up a sug­ges­tion of Hop­kins. The pa­per [69] with Graeme Segal deals with a more sub­stan­tial ex­ten­sion or “twist­ing” of \( K \)-the­ory which is needed for M-the­ory and has re­cently be­come pop­u­lar. [69] is a full ac­count of this the­ory and will be fol­lowed later by a second pa­per deal­ing with its re­la­tions with co­homo­logy. These two pa­pers amp­li­fy (and cor­rect) the sketch out­lined in [53].

Works

[1]M. F. Atiyah and N. J. Hitchin: “Low en­ergy scat­ter­ing of nona­beli­an mono­poles,” Phys. Lett. A 107 : 1 (1985), pp. 21–​25. MR 778313 Zbl 1177.​53069 article

[2]M. Atiyah, N. J. Hitchin, J. T. Stu­art, and M. Tabor: “Low-en­ergy scat­ter­ing of nona­beli­an mag­net­ic mono­poles [and dis­cus­sion],” Philos. Trans. Roy. Soc. Lon­don Ser. A 315 : 1533 (1985), pp. 459–​469. MR 836746 article

[3]M. Atiyah: “On the work of Si­mon Don­ald­son,” pp. 3–​6 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Berke­ley, CA, 3–11 Au­gust 1986), vol. 2. Edi­ted by A. Gleason. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1987. MR 934209 Zbl 0666.​01010 incollection

[4]M. Atiyah: “The log­ar­ithm of the Dede­kind \( \eta \)-func­tion,” Math. Ann. 278 : 1–​4 (1987), pp. 335–​380. MR 909232 Zbl 0648.​58035 article

[5]M. Atiyah: “To­po­logy and dif­fer­en­tial equa­tions,” pp. 45–​52 in ICIAM ’87: Pro­ceed­ings of the first In­ter­na­tion­al Con­fer­ence on In­dus­tri­al and Ap­plied Math­em­at­ics (Par­is, 29 June–3 Ju­ly 1987). Edi­ted by J. McK­enna and R. Tem­am. SIAM (Phil­adelphia, PA), 1988. MR 976850 Zbl 0687.​35001 incollection

[6]M. Atiyah: “New in­vari­ants of 3- and 4-di­men­sion­al man­i­folds,” pp. 285–​299 in The math­em­at­ic­al her­it­age of Her­mann Weyl (Durham, NC, 12–16 May 1987). Edi­ted by R. O. Wells, Jr. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 48. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1988. Rus­si­an trans­la­tion pub­lished in Us­pekhi Mat. Nauk 45:4(274) (1990). MR 974342 Zbl 0667.​57018 incollection

[7]M. Atiyah: “To­po­lo­gic­al quantum field the­or­ies,” Inst. Hautes Études Sci. Publ. Math. 68 : 1 (January 1988), pp. 175–​186. MR 1001453 Zbl 0692.​53053 article

[8]M. Atiyah and N. Hitchin: The geo­metry and dy­nam­ics of mag­net­ic mono­poles. M. B. Port­er Lec­tures. Prin­ceton Uni­versity Press, 1988. Rus­si­an trans­la­tion pub­lished as Geo­met­riya i di­n­ami­ka mag­nit­nykh mono­polei (1991). MR 934202 Zbl 0671.​53001 book

[9]M. F. Atiyah: “Geo­metry, to­po­logy and phys­ics,” Quart. Journ. Roy­al As­tro­phys­ics Soc. 29 : 3 (September 1988), pp. 287–​299. De­livered as the 11th Ar­thur Mil­ne Lec­ture at Ox­ford Uni­versity. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. article

[10]M. Atiyah and G. Segal: “On equivari­ant Euler char­ac­ter­ist­ics,” J. Geom. Phys. 6 : 4 (1989), pp. 671–​677. MR 1076708 Zbl 0708.​19004 article

[11]M. F. Atiyah and N. S. Man­ton: “Skyrmi­ons from in­stan­tons,” Phys. Lett. B 222 : 3–​4 (1989), pp. 438–​442. MR 1001325 article

[12]M. F. Atiyah: “The geo­metry and phys­ics of knots,” pp. 1–​17 in Minicon­fer­ence on geo­metry and phys­ics (Can­berra, 20–23 Feb­ru­ary 1989). Edi­ted by M. N. Barber and M. K. Mur­ray. Pro­ceed­ings of the Centre for Math­em­at­ic­al Ana­lys­is 22. Aus­trali­an Na­tion­al Uni­versity (Can­berra), 1989. See also 1990 book of the same title. MR 1027859 Zbl 0696.​57002 incollection

[13]M. Atiyah: “New in­vari­ants of 3- and 4-di­men­sion­al man­i­folds,” Us­pekhi Mat. Nauk 45 : 4(274) (1990), pp. 3–​16, 192. Rus­si­an trans­la­tion of art­icle from The math­em­at­ic­al her­it­age of Her­mann Weyl (1988). MR 1075385 Zbl 0709.​57018 article

[14]M. Atiyah: “On fram­ings of 3-man­i­folds,” To­po­logy 29 : 1 (1990), pp. 1–​7. MR 1046621 Zbl 0716.​57011 article

[15]M. Atiyah: The geo­metry and phys­ics of knots. Lezioni Lincee. Cam­bridge Uni­versity Press, 1990. These notes arise from lec­tures presen­ted in Florence un­der the aus­pices of the Ac­ca­demia dei Lincei. Rus­si­an trans­la­tion pub­lished as Geo­met­riya i fiz­ika uzlov (1995). See also Minicon­fer­ence on geo­metry and phys­ics (1989). MR 1078014 Zbl 0729.​57002 book

[16]M. F. Atiyah and L. Jef­frey: “To­po­lo­gic­al Lag­rangi­ans and co­homo­logy,” J. Geom. Phys. 7 : 1 (1990), pp. 119–​136. MR 1094734 Zbl 0721.​58056 article

[17]M. Atiyah: “Hy­per-Kähler man­i­folds,” pp. 1–​13 in Com­plex geo­metry and ana­lys­is (Pisa, 23–27 May 1988). Edi­ted by V. Vil­lani. Lec­ture Notes in Math­em­at­ics 1422. Spring­er (Ber­lin), 1990. MR 1055838 incollection

[18]M. Atiyah: “Mag­net­ic mono­poles and the Yang–Bax­ter equa­tions,” pp. 2761–​2774 in To­po­lo­gic­al meth­ods in quantum field the­ory (Trieste, 11–15 June 1990), published as In­ter­nat. J. Mod­ern Phys. A 6 : 16 (1991). MR 1117746 Zbl 0757.​53038 incollection

[19]M. Atiyah and N. Khitchin: Geo­met­riya i di­n­ami­ka mag­nit­nykh mono­polei [The geo­metry and dy­nam­ics of mag­net­ic mono­poles]. Mir (Mo­scow), 1991. Rus­si­an trans­la­tion of The geo­metry and dy­nam­ics of mag­net­ic mono­poles (1988). MR 1137269 Zbl 0754.​53003 book

[20]M. Atiyah: “On the work of Ed­ward Wit­ten,” pp. 31–​35 in Pro­ceed­ings of the In­ter­na­tion­al Con­gress of Math­em­aticians (Kyoto, 21–29 Au­gust 1990), vol. 1. Edi­ted by I. Satake. Math­em­at­ic­al So­ci­ety of Ja­pan (Tokyo), 1991. MR 1159202 Zbl 0742.​01013 incollection

[21]M. F. Atiyah: “Math­em­at­ics: Queen and ser­vant of the sci­ences,” Proc. Am. Phil. Soc. 137 : 4 (December 1993), pp. 527–​531. 250th an­niversary is­sue. Re­pub­lished in Asi­an J. Math. 3:1 (1999) and The founders of in­dex the­ory (2009). article

[22]M. F. Atiyah and N. S. Man­ton: “Geo­metry and kin­emat­ics of two skyrmi­ons,” Comm. Math. Phys. 153 : 2 (1993), pp. 391–​422. MR 1218307 Zbl 0778.​53051 article

[23]M. F. Atiyah: “Amer­ic­an Philo­soph­ic­al So­ci­ety din­ner ad­dress, 30 April 1993,” Proc. Am. Phil. Soc. 137 : 4 (1993), pp. 704–​707. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. article

[24]M. F. Atiyah: “An­niversary ad­dress by the Pres­id­ent,” Roy­al So­ci­ety News supplement (December 1994), pp. i–​iv, 535–​540. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. See also ver­sion in Notes and Re­cords Roy. Soc. Lon­don 49:1 (1995). article

[25]A. Jaffe and F. Quinn: “Re­sponse to com­ments on ‘The­or­et­ic­al math­em­at­ics’,” Bull. Amer. Math. Soc. (N.S.) 30 : 2 (1994), pp. 208–​211. Re­sponse to an art­icle in Bull. Amer. Math. Soc. 30:2 (1994). MR 1254077 article

[26]M. Atiyah, A. Borel, G. J. Chaitin, D. Friedan, J. Glimm, J. J. Gray, M. W. Hirsch, S. Mac Lane, B. B. Man­del­brot, D. Ruelle, A. Schwarz, K. Uh­len­beck, R. Thom, E. Wit­ten, and C. Zee­man: “Re­sponses to ‘The­or­et­ic­al math­em­at­ics: To­ward a cul­tur­al syn­thes­is of math­em­at­ics and the­or­et­ic­al phys­ics’, by A. Jaffe and F. Quinn,” Bull. Am. Math. Soc., N.S. 30 : 2 (1994), pp. 178–​207. MR 1254073 Zbl 0803.​01014 ArXiv math.​HO/​9404229 article

[27]M. Atiyah: “Con­tri­bu­tion to the col­lec­ted works of Raoul Bott,” pp. xxix–​xxx in Raoul Bott: Col­lec­ted pa­pers, vol. 2: Dif­fer­en­tial op­er­at­ors. Edi­ted by R. D. MacPh­er­son. Con­tem­por­ary Math­em­aticians. Birkhäuser (Bo­ston), 1994. MR 1290365 incollection

[28]M. Atiyah: Geo­met­riya i fiz­ika uzlov [The geo­metry and phys­ics of knots]. Mir (Mo­scow), 1995. Rus­si­an trans­la­tion of The geo­metry and phys­ics of knots (1990). See also Minicon­fer­ence on geo­metry and phys­ics (1989). MR 1397266 book

[29]M. F. Atiyah: “Book re­view: ‘Con­ver­sa­tions on mind, mat­ter and math­em­at­ics’,” Times High. Educ. Sup­pl. (29 September 1995). Book by Jean-Pierre Changeux and Alain Connes (Prin­ceton Uni­versity Press, 1995). Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. article

[30]M. Atiyah: “Ad­dress of the Pres­id­ent, Sir Mi­chael Atiyah, O.M., giv­en at the an­niversary meet­ing on 30 Novem­ber 1994,” Notes and Re­cords Roy. Soc. Lon­don 49 : 1 (1995), pp. 141–​151. See also ver­sion in Roy­al So­ci­ety News (Decem­ber 1994). MR 1325201 Zbl 0978.​01520 article

[31]M. Atiyah: “Re­flec­tions on geo­metry and phys­ics,” pp. 1–​6 in Sur­veys in dif­fer­en­tial geo­metry: Pro­ceed­ings of the con­fer­ence on geo­metry and to­po­logy (Cam­bridge, MA, 23–25 April 1993). Edi­ted by C.-C. Hsiung and S.-T. Yau. Sur­veys in dif­fer­en­tial geo­metry (Journ­al of Dif­fer­en­tial Geo­metry sup­ple­ments) 2. In­ter­na­tion­al Press (Cam­bridge, MA), 1995. MR 1375254 Zbl 0867.​57031 incollection

[32]M. Atiyah: “The geo­metry of clas­sic­al particles,” pp. 1–​15 in Pro­ceed­ings of the con­fer­ence on geo­metry and to­po­logy held at Har­vard Uni­versity (Cam­bridge, MA, 23–25 April 1993). Edi­ted by C.-C. Hsiung and S.-T. Yau. Sur­veys in Dif­fer­en­tial Geo­metry 2. In­ter­na­tion­al Press (Somerville, MA), 1995. MR 1919420 Zbl 1050.​55502 incollection

[33]M. Atiyah: “Re­flec­tions on geo­metry and phys­ics,” pp. 423–​428 in Geo­metry, to­po­logy, & phys­ics: Lec­tures of a con­fer­ence in hon­or of Raoul Bott’s 70th birth­day (Har­vard Uni­versity, April 1993). Edi­ted by S.-T. Yau. Con­fer­ence pro­ceed­ings and lec­ture notes in geo­metry and to­po­logy 4. In­ter­na­tion­al Press (Cam­bridge, MA), 1995. MR 1358626 incollection

[34]M. Atiyah: “Friedrich Hirzebruch: An ap­pre­ci­ation,” pp. 1–​5 in Pro­ceed­ings of the Hirzebruch 65 con­fer­ence on al­geb­ra­ic geo­metry (Bar-Il­an Uni­versity, Ramat Gan, Is­rael, 2–7 May 1993). Edi­ted by M. Teich­er. Is­rael Math­em­at­ic­al Con­fer­ence Pro­ceed­ings 9. Bar-Il­an Uni­versity (Ramat Gan, Is­rael), 1996. MR 1360493 Zbl 0834.​01012 incollection

[35]P. Ex­n­er: “Proofs and phys­ics, their in­ter­re­la­tion­ship,” Pok­roky Mat. Fyz. As­tro­nom. 41 : 2 (1996), pp. 73–​81. Ad­apt­a­tion and Czech trans­la­tion of art­icle from Bull. Amer. Math. Soc. 30:2 (1994). MR 1454823 article

[36]M. F. Atiyah: “The work of Ed­ward Wit­ten,” pp. 514–​518 in Fields Medal­lists’ lec­tures. Edi­ted by M. F. Atiyah and D. Iag­ol­nitzer. World Sci­entif­ic Series in 20th Cen­tury Math­em­at­ics 5. World Sci­entif­ic (River Edge, NJ), 1997. MR 1622921 incollection

[37]M. F. Atiyah: “The work of Si­mon Don­ald­son,” pp. 377–​380 in Fields Medal­lists’ lec­tures. Edi­ted by M. F. Atiyah and D. Iag­ol­nitzer. World Sci­entif­ic Series in 20th Cen­tury Math­em­at­ics 5. World Sci­entif­ic (River Edge, NJ), 1997. MR 1622912 incollection

[38]M. F. Atiyah: “An in­tro­duc­tion to to­po­lo­gic­al quantum field the­or­ies,” pp. 1–​7 in Pro­ceed­ings of Gökova geo­metry-to­po­logy con­fer­ence 1996 (Gökova, Tur­key, 27–31 May 1996), published as Turk­ish J. Math. 21 : 1. Issue edi­ted by S. Ak­bu­lut, T. Önder, and R. J. Stern. Sci­entif­ic and Tech­no­lo­gic­al Re­search Coun­cil of Tur­key (Ank­ara), 1997. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. MR 1456155 Zbl 0890.​57019 incollection

[39]M. F. Atiyah: “The Dir­ac equa­tion and geo­metry,” pp. 108–​124 in A. Pais, M. Jac­ob, D. I. Olive, and M. F. Atiyah: Paul Dir­ac: The man and his work. Edi­ted by P. God­dard. Cam­bridge Uni­versity Press, 1998. MR 1606723 incollection

[40]M. Atiyah: “Ro­ger Pen­rose: A per­son­al ap­pre­ci­ation,” pp. 3–​7 in The geo­met­ric uni­verse (Ox­ford, June 1996). Edi­ted by S. A. Hug­gett, L. J. Ma­son, K. P. Tod, S. Tsou, and N. M. J. Wood­house. Ox­ford Uni­versity Press (New York), 1998. MR 1634501 Zbl 0904.​01010 incollection

[41]M. F. Atiyah: “Ob­it­u­ary: John Ar­thur Todd,” Bull. Lon­don Math. Soc. 30 : 3 (1998), pp. 305–​316. MR 1608134 Zbl 0927.​01042 article

[42]M. Atiyah: “Math­em­at­ics and the real world,” pp. 807–​812 in Cur­rent and fu­ture chal­lenges in the ap­plic­a­tions of math­em­at­ics, published as Quart. Ap­pl. Math. 56 : 4 (1998). MR 1668728 Zbl 1159.​00312 incollection

[43]M. Atiyah: “Math­em­at­ics: Queen and ser­vant of the sci­ences,” pp. xxiii–​xxvi in Sir Mi­chael Atiyah: A great math­em­atician of the twen­ti­eth cen­tury, published as Asi­an J. Math. 3 : 1 (1999). Re­pub­lished from Proc. Am. Phil. Soc. 137:4 (1993). See also The founders of in­dex the­ory (2009). Zbl 0957.​01038 incollection

[44]M. F. Atiyah: “Ob­it­u­ary: Kuni­hiko Kodaira,” Bull. Lon­don Math. Soc. 31 : 4 (1999), pp. 489–​493. MR 1687532 Zbl 0928.​01020 article

[45]M. Atiyah: “Geo­metry and phys­ics in the 20th cen­tury,” pp. 1–​9 in The math­em­at­ic­al sci­ences after the year 2000 (Beirut, 11–15 Janu­ary 1999). Edi­ted by K. Bit­ar, A. Chamsed­dine, and W. Sabra. World Sci­entif­ic (River Edge, NJ), 2000. MR 1799435 Zbl 0994.​01011 incollection

[46]M. F. Atiyah: “Pre­face” in Math­em­at­ics: Fron­ti­ers and per­spect­ives. Edi­ted by V. Arnold, M. Atiyah, P. Lax, and B. Mazur. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 2000. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. incollection

[47]M. Atiyah: Equivari­ant co­homo­logy and rep­res­ent­a­tions of the sym­met­ric group. Pre­print, 2000. ArXiv math/​0012215 techreport

[48]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” Amer. Math. Mon. 108 : 7 (2001), pp. 654–​666. See also ver­sions in Math. Today 37:2 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wia­dom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1862105 Zbl 1081.​01502 article

[49]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” Math. Today 37 : 2 (2001), pp. 46–​53. See also ver­sions in Amer. Math. Mon. 108:7 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wia­dom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1840757 article

[50]M. Atiyah, J. Mal­da­cena, and C. Vafa: “An M-the­ory flop as a large \( N \) du­al­ity,” J. Math. Phys. 42 : 7 (2001), pp. 3209–​3220. MR 1840340 Zbl 1061.​81056 article

[51]M. Atiyah: “Con­fig­ur­a­tions of points,” pp. 1375–​1387 in To­po­lo­gic­al meth­ods in the phys­ic­al sci­ences (Lon­don, 2000), published as R. Soc. Lond. Philos. Trans. Ser. A 359 : 1784 (July 2001). MR 1853626 Zbl 1037.​58007 incollection

[52]M. Atiyah: “Equivari­ant co­homo­logy and rep­res­ent­a­tions of the sym­met­ric group,” Chinese Ann. Math. Ser. B 22 : 1 (2001), pp. 23–​30. MR 1823127 Zbl 1057.​20007 article

[53]M. Atiyah: “\( K \)-the­ory past and present,” pp. 411–​417 in Sitzungs­berichte der Ber­liner Math­em­at­ischen Gesell­schaft, 1997–2000. Ber­liner Math­em­at­ischen Gesell­schaft, 2001. MR 2091892 Zbl 1061.​19500 ArXiv math/​0012213 incollection

[54]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” pp. 1–​21 in Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (Tianjin, China, 9–13 Oc­to­ber 2000). Edi­ted by S.-S. Chern, L. Fu, and R. M. Hain. Nankai Tracts in Math­em­at­ics 5. World Sci­entif­ic (River Edge, NJ), 2002. See also ver­sions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wia­dom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1945353 Zbl 1038.​01014 incollection

[55]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” Bull. Lond. Math. Soc. 34 : 1 (2002), pp. 1–​15. See also ver­sions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), N.T.M. 10:1 (2002), Wia­dom. Mat. 39 (2003), Adv. Math. (China) 33:1 (2004). MR 1866422 Zbl 1022.​01007 article

[56]M. Atiyah and P. Sutcliffe: “The geo­metry of point particles,” R. Soc. Lond. Proc. Ser. A 458 : 2021 (2002), pp. 1089–​1115. MR 1902577 Zbl 1010.​58015 article

[57]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” N.T.M. (N.S.) 10 : 1 (2002), pp. 25–​39. See also ver­sions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), Bull. Lond. Math. Soc. 34:1 (2002), Wia­dom. Mat. 39 (2003) and Adv. Math. (China) 33:1 (2004). MR 1894494 Zbl 0991.​01015 article

[58]M. F. Atiyah: “Her­mann Weyl: 1885–1955,” Biog. Mem. Nat. Acad. Sci. 82 (2002), pp. 3–​17. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. article

[59]M. Atiyah and E. Wit­ten: “\( M \)-the­ory dy­nam­ics on a man­i­fold of \( G_2 \) holonomy,” Adv. The­or. Math. Phys. 6 : 1 (2002), pp. 1–​106. MR 1992874 Zbl 1033.​81065 article

[60]M. Atiyah and R. Bielawski: “Nahm’s equa­tions, con­fig­ur­a­tion spaces and flag man­i­folds,” Bull. Braz. Math. Soc. (N.S.) 33 : 2 (2002), pp. 157–​176. MR 1940347 Zbl 1022.​22007 ArXiv math/​0110112 article

[61]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” Wia­dom. Mat. 39 (2003), pp. 47–​63. Pol­ish trans­la­tion of Bull. Lond. Math. Soc. 34:1 (2002). See also ver­sions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), N.T.M. 10:1 (2002) and Adv. Math. (China) 33:1 (2004). MR 2043772 article

[62]M. Atiyah and J. Berndt: “Pro­ject­ive planes, Severi vari­et­ies and spheres,” pp. 1–​27 in Sur­veys in dif­fer­en­tial geo­metry: Lec­tures on geo­metry and to­po­logy held in hon­or of Calabi, Lawson, Siu, and Uh­len­beck (Har­vard Uni­versity, 3–5 May 2002). Edi­ted by S.-T. Yau. Sur­veys in Dif­fer­en­tial Geo­metry (Journ­al of Dif­fer­en­tial Geo­metry sup­ple­ments) 8. In­ter­na­tion­al Press (Somerville, MA), 2003. MR 2039984 Zbl 1057.​53040 ArXiv math/​0206135 incollection

[63]M. Atiyah and P. Sutcliffe: “Poly­hedra in phys­ics, chem­istry and geo­metry,” Mil­an J. Math. 71 : 1 (2003), pp. 33–​58. MR 2120915 Zbl 1050.​52002 article

[64]The founders of in­dex the­ory: Re­min­is­cences of Atiyah, Bott, Hirzebruch, and Sing­er. Edi­ted by S.-T. Yau. In­ter­na­tion­al Press (Somerville, MA), 2003. 2nd edi­tion was pub­lished in 2009. MR 2136846 Zbl 1072.​01021 book

[65]M. Atiyah: Col­lec­ted works, vol. 6. Ox­ford Sci­ence Pub­lic­a­tions. The Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 2004. MR 2160826 Zbl 1099.​01024 book

[66]M. Atiyah and M. Hop­kins: “A vari­ant of \( K \)-the­ory: \( K_\pm \),” pp. 5–​17 in To­po­logy, geo­metry and quantum field the­ory (Ox­ford, 24–29 June 2002). Edi­ted by U. Till­mann. Cam­bridge Uni­versity Press, 2004. MR 2079369 Zbl 1090.​19004 incollection

[67]M. F. Atiyah: “The geo­metry and phys­ics of knots,” pp. 289–​304 in Col­lec­ted works, vol. 6. Ox­ford Sci­ence Pub­lic­a­tions. Clar­en­don Press and Ox­ford Uni­versity Press (Ox­ford, New York), 2004. See also Minicon­fer­ence on geo­metry and phys­ics (1989), the 1990 book of the same title and the Rus­si­an trans­la­tion Geo­met­riya i fiz­ika uzlov (1995). incollection

[68]M. Atiyah: “Math­em­at­ics in the 20th cen­tury,” Adv. Math. (China) 33 : 1 (2004), pp. 26–​40. See also ver­sions in Amer. Math. Mon. 108:7 (2001), Math. Today 37:2 (2001), Con­tem­por­ary trends in al­geb­ra­ic geo­metry and al­geb­ra­ic to­po­logy (2002), N.T.M. 10:1 (2002), Bull. Lond. Math. Soc. 34:1 (2002) and Wia­dom. Mat. 39 (2003). MR 2058440 article

[69]M. Atiyah and G. Segal: “Twis­ted \( K \)-the­ory,” Ukr. Mat. Visn. 1 : 3 (2004), pp. 287–​330. MR 2172633 Zbl 1151.​55301 article

[70]M. Atiyah: “The in­ter­ac­tion between geo­metry and phys­ics,” pp. 1–​15 in The unity of math­em­at­ics (Cam­bridge, MA, 31 Au­gust–4 Septem­ber 2003). Edi­ted by P. I. Ètin­gof, V. Re­takh, and I. M. Sing­er. Pro­gress in Math­em­at­ics 244. Birkhäuser (Bo­ston, MA), 2006. MR 2181802 Zbl 1118.​58002 incollection

[71]The founders of in­dex the­ory: Re­min­is­cences of Atiyah, Bott, Hirzebruch, and Sing­er, 2nd edition. Edi­ted by S.-T. Yau. In­ter­na­tion­al Press (Somerville, MA), 2009. Ori­gin­al edi­tion was pub­lished in 2003. MR 2547480 Zbl 1195.​01090 book

[72]M. F. Atiyah: “A per­son­al his­tory,” pp. 5–​15 in The founders of in­dex the­ory: Re­min­is­cences of and about Sir Mi­chael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Sing­er, 2nd edition. Edi­ted by S.-T. Yau. In­ter­na­tion­al Press (Somerville, MA), 2009. Re­pub­lished in Atiyah’s Col­lec­ted works, vol. 6. incollection