Celebratio Mathematica

I first met John Mor­gan in the 1980s. I was a Ritt as­sist­ant pro­fess­or1 at Columbia Uni­versity, and I was sit­ting in the com­mon lounge. John, who was chair­man of the de­part­ment at the time, ap­proached me and took a seat next to me. He star­ted talk­ing about an ar­gu­ment re­gard­ing teach­ing du­ties that had oc­curred between me and a seni­or col­league in the de­part­ment. John was at the same time kind, firm and con­vin­cing. I ap­pre­ci­ated the way he handled the “in­cid­ent”.

Some time later, we star­ted work­ing on a com­mon pro­ject. At the be­gin­ning of the 1980s, Si­mon Don­ald­son had defined poly­no­mi­al func­tions on the second co­homo­logy of smooth com­pact 4-man­i­folds by eval­u­at­ing products of co­homo­logy classes on mod­uli spaces of ASD con­nec­tions, or rather its Uh­len­beck com­pac­ti­fic­a­tion. Don­ald­son’s poly­no­mi­als, which were very hard to com­pute, could be used to prove that cer­tain couples of homeo­morph­ic smooth 4-man­i­folds are not dif­feo­morph­ic. If the 4-man­i­fold un­der­lies a com­plex pro­ject­ive sur­face \( S \) then the Uh­len­beck com­pac­ti­fic­a­tion is iden­ti­fied with an al­gebro-geo­met­ric com­pac­ti­fic­a­tion of the mod­uli space of stable al­geb­ra­ic vec­tor bundles on \( S \), and hence one could try com­put­ing the poly­no­mi­als by al­gebro-geo­met­ric meth­ods. John and I did ex­actly that for a series of el­lipt­ic sur­faces (which in­clude el­lipt­ic K3 sur­faces) that were known to be homeo­morph­ic but were sus­pec­ted to be pair­wise nondif­feo­morph­ic. We did prove that the el­lipt­ic sur­faces are pair­wise nondif­feo­moroph­ic; the out­put of our work is the Spring­er LNM 1545 Dif­fer­en­tial to­po­logy of com­plex sur­faces [1]. It took us a long time to com­plete the pro­ject, and we spent many hours in John’s apart­ment, writ­ing to­geth­er on John’s black Next com­puter, in­ter­rup­ted every once in a while by John’s daugh­ter Bri­anna or his son Jake. Note: the in­tro­duc­tion of Seiberg–Wit­ten in­vari­ants a few years later provided a much short­er proof of this res­ult (and it paved the way for oth­er spec­tac­u­lar res­ults on the clas­si­fic­a­tion of smooth 4-man­i­folds).

In the aca­dem­ic year 2002–03 we had a spe­cial year in geo­metry in the Di­par­ti­mento di Matem­at­ica of Uni­versità di Roma La Sapi­enza (I had moved back to Italy in 1994, and I had been in La Sapi­enza since 1997), and we asked John wheth­er he could give a few talks on Perel­man’s re­cently pos­ted pa­pers on the Ricci flow and the Poin­caré con­jec­ture. John agreed and gave a series of lec­tures on the sub­ject. It was ex­cit­ing to at­tend a series of beau­ti­ful lec­tures on such a hot top­ic.

John is cap­able of speak­ing math in a way that is already suit­able to be writ­ten down. He once told me, “I enter the classroom and I start giv­ing the lec­ture; I do not need to pre­pare.” Work­ing with him, I star­ted ap­pre­ci­at­ing the im­port­ance of writ­ing math­em­at­ics clearly and I had some glimpses of the beauty of to­po­logy. I wish I had taken more ad­vant­age of John’s deep know­ledge of the sub­ject. In fact, I could have taken ad­vant­age of John’s know­ledge of many dif­fer­ent sub­jects. He star­ted out as a to­po­lo­gist, but he of­ten col­lab­or­ated with al­geb­ra­ic geo­met­ers, and be­came an ex­pert in gauge the­ory and the Ricci flow. He once told me that he in­ten­ded to give a course on the Weil con­jec­tures (“speak of an over­arch­ing ana­logy” is what he said apro­pos of the Weil Con­jec­tures) — hardly something to ex­pect from a math­em­atician with John’s back­ground!

Born in Mel­bourne (Aus­tralia) and raised in Roma (Italy), Kier­an O’Grady re­ceived his PhD from Brown Uni­versity in 1986 un­der the dir­ec­tion of Joe Har­ris. After spend­ing a few more years in the US he re­turned to Italy in 1994. He has been a Pro­fess­or of Geo­metry at Sapi­enza Uni­versità di Roma since 1997.