Celebratio Mathematica

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].