Celebratio Mathematica

I came to Prin­ceton as a gradu­ate stu­dent in 1989, at an ex­cit­ing time in low-di­men­sion­al to­po­logy. Four-di­men­sion­al dif­fer­en­tial to­po­logy was un­der­go­ing a re­volu­tion, pi­on­eered by the work of Si­mon Don­ald­son. Don­ald­son had re­cently dis­covered a way to use the mod­uli space of solu­tions to the (met­ric-de­pend­ent) Yang–Mills equa­tions to ac­cess in­form­a­tion about the smooth struc­tures on four-man­i­folds which had evaded un­der­stand­ing for dec­ades.

Al­though Prin­ceton’s math de­part­ment at the time lacked ex­pert­ise in these ex­cit­ing de­vel­op­ments, my ad­visor Bill Browder en­thu­si­ast­ic­ally en­cour­aged me to try to learn the newly emer­ging the­ory. Dur­ing my third year of gradu­ate school, I taught a year-long course out of the re­cently pub­lished book by Don­ald­son and Peter Kron­heimer, The Geo­metry of Four-Man­i­folds [e1], which helped give me some ground­ing in the sub­ject.

But learn­ing the ba­sics is very dif­fer­ent from find­ing a cur­rent prob­lem to think about, so after learn­ing what I could by my­self, I began ask­ing out­side ex­perts for guid­ance. I was for­tu­nate that John Mor­gan was will­ing to help me out. Moreover, dur­ing my fourth year of gradu­ate school, Browder in­vited Mor­gan to come to Prin­ceton to teach a course on gauge the­ory and four-man­i­folds.

Once a week, John came to Prin­ceton to lec­ture to a packed room, with an audi­ence con­sist­ing of a mix of stu­dents, postdocs, and oth­er fac­ulty. Dur­ing lunch breaks, John and I would meet over greasy ho­agies, dis­cuss­ing math­em­at­ics. Prin­ceton, which un­til then felt some­what isol­ated from the to­po­logy scene, seemed to spring to life. Zoltán Szabó and An­drás Stip­sicz, who were PhD stu­dents at Rut­gers, also at­ten­ded John’s course. They, too, ended up work­ing with him; and all of us were able to talk to each oth­er about gauge the­ory.

I will say a few words about the math that John and I dis­cussed, and refer the read­er to a more de­tailed ac­count in Don­ald­son’s art­icle in this volume. Re­call that a closed, ori­ented four-man­i­fold is en­dowed with a nonde­gen­er­ate bi­lin­ear form on its two-di­men­sion­al co­homo­logy, its in­ter­sec­tion form. Cor­res­pond­ingly, the second Betti num­ber can be de­com­posed as \( b_2=b_2^++b_2^- \), where \( b_2^+ \) is the di­men­sion of a max­im­al sub­space of \( H^2(X) \) on which the in­ter­sec­tion form is pos­it­ive def­in­ite. As a part of Don­ald­son’s re­volu­tion in smooth four-man­i­fold to­po­logy, he had defined in­ter­est­ing to­po­lo­gic­al in­vari­ants for closed, smooth four-man­i­folds with \( b_2^+ > 1 \). These in­vari­ants are defined by count­ing the num­ber of anti-self-dual con­nec­tions on the prin­cip­al \( SU(2) \)-bundle over the four-man­i­fold. To ob­tain a sens­ible point count, one must first cut down the mod­uli space of solu­tion by two-di­men­sion­al con­straints cor­res­pond­ing to two-di­men­sion­al ho­mo­logy classes on \( X \). Hence, Don­ald­son’s in­vari­ants can be thought of form­ally as mul­ti­lin­ear func­tions on \( H_2(X) \).

John sug­ges­ted I think about the “blowup for­mula” for Don­ald­son in­vari­ants: how are the Don­ald­son in­vari­ants of \( M_1 \) re­lated to the Don­ald­son in­vari­ants for \( M_1\#M_2 \), when \( M_2 \) has \( b_2^+(M_2)=0 \) [e2]? In par­tic­u­lar, when \( M_1 \) is an al­geb­ra­ic sur­face and \( M_2 \) is \( -{\mathbb CP}^2 \) (i.e., the com­plex pro­ject­ive plane en­dowed with the ori­ent­a­tion op­pos­ite to the one it in­her­its nat­ur­ally as an al­geb­ra­ic vari­ety), this ques­tion asks how the Don­ald­son poly­no­mi­als of an al­geb­ra­ic sur­face \( M_1 \) are re­lated to the Don­ald­son poly­no­mi­als of the al­geb­ra­ic sur­face ob­tained by blow­ing up \( M_1 \) at a point.

This prob­lem had vary­ing levels of dif­fi­culty, ac­cord­ing to how many times one used ho­mo­logy classes from the \( M_2 \) side: the more such ho­mo­logy classes one uses, the deep­er one finds one­self in the Uh­len­beck com­pac­ti­fic­a­tion of the mod­uli space of anti-self-dual con­nec­tions. For ex­ample, for four ho­mo­logy classes on the \( M_2 \) side, the an­swer had been worked out by Fried­man and Mor­gan [2], who ex­pressed this in terms of a fa­mil­i­ar (top-strat­um) sin­gu­lar­ity in a mod­uli space, de­scribed by a neigh­bor­hood of a re­du­cible con­nec­tion. For my thes­is [e3], I worked out the cases which re­quired un­der­stand­ing a four-di­men­sion­al strat­um in the Uh­len­beck com­pac­ti­fic­a­tion. Oth­er re­lated res­ults were ob­tained by Bry­an [e7] and Le­ness [e8].

This was a prob­lem I found deeply com­pel­ling; it fit very well with my math­em­at­ic­al tem­pera­ment. John was also an ideal ad­visor. In ad­di­tion to find­ing an ex­cel­lent prob­lem, he dis­pelled many math­em­at­ic­al mis­con­cep­tions I had, see­ing things with his hall­mark clar­ity, and con­vey­ing that clar­ity with great pa­tience.

Any­one who has made some com­pletely mis­in­formed re­mark while work­ing with John is no doubt fa­mil­i­ar with that dis­tant look he gets in his eyes, while in­ton­ing quietly, in his gentle Texas drawl “I think we’re talk­ing cross-pur­poses here.” I very quickly learned to try not to talk cross-pur­poses with him!

After John’s course ended, we con­tin­ued our reg­u­lar meet­ings. I took the train once a week from Prin­ceton to New York City, where John would wel­come me in his beau­ti­ful apart­ment. We chat­ted over hot tea in his liv­ing room. Our ses­sions were some­times in­ter­rup­ted by a stroll to a loc­al res­taur­ant, where the math­em­at­ic­al con­ver­sa­tions would con­tin­ue over sand­wiches.

Soon John and I real­ized that an in­ter­play between the form­al struc­ture of Don­ald­son in­vari­ants, com­bined with some plaus­ible form­al prop­er­ties of mod­uli spaces [1], would un­lock the blowup for­mula in gen­er­al. We set about writ­ing this up, but this pro­ject was in­ter­rup­ted by three events, two math­em­at­ic­al and one per­son­al.

First, in the back­drop of Kron­heimer and Mrowka’s amaz­ing “struc­ture the­or­em” for Don­ald­son poly­no­mi­als [e5], Fin­tushel and Stern com­puted the blowup for­mu­las, in the case where is \( M_2=-{\mathbb CP}^2 \), by it­er­at­ing the blowup pro­ced­ure, and us­ing a re­la­tion as­so­ci­ated to em­bed­ded spheres with self-in­ter­sec­tion num­ber \( -2 \) [e6].

Second, I gradu­ated and moved to Cali­for­nia for a postdoc­tor­al po­s­i­tion at Cal­tech, un­der the su­per­vi­sion of Tom Mrowka.

Third, and most im­port­antly, dur­ing my first Fall as a postdoc, Nath­an Seiberg and Ed­ward Wit­ten [e4] re­vo­lu­tion­ized gauge the­ory by the in­tro­duc­tion of a new equa­tion which ap­peared to give us ac­cess to the same four-di­men­sion­al in­form­a­tion with re­fresh­ingly few­er tech­nic­al dif­fi­culties. In a mat­ter of months, the en­tire com­munity of smooth four-man­i­fold to­po­lo­gists had turned from anti-self-dual Yang–Mills con­nec­tions to solu­tions to the Seiberg–Wit­ten equa­tions.

I owe a great deal to the ment­or­ing John gave me as a gradu­ate stu­dent; but I also had the good for­tune to be­ne­fit from his ment­or­ing later in my ca­reer. In 2002, I was offered a pro­fess­or­ship at Columbia Uni­versity, where I be­came John’s col­league. Shortly af­ter­wards, John be­came Chair of the math de­part­ment. Dur­ing those years, Columbia’s math­em­at­ics de­part­ment blos­somed. In to­po­logy and geo­metry alone, Mikhail Khovan­ov, Robert Lip­shitz, Dusa Mc­Duff, Cipri­an Man­oles­cu, and Dylan Thur­ston soon joined the de­part­ment, all this thanks in large part to John’s lead­er­ship.

Moreover, John en­cour­aged me early to step out of my fairly isol­ated com­fort zone and run sem­inars for gradu­ate stu­dents on Hee­gaard Flo­er ho­mo­logy, which Zoltán Szabó and I were just start­ing to de­vel­op. I ac­quired many very tal­en­ted stu­dents then, in an ex­cit­ing at­mo­sphere, sur­roun­ded by many postdocs and oth­er young­er fac­ulty mem­bers. Fri­days, for ex­ample, fea­tured two or three out­side speak­ers in low-di­men­sion­al to­po­logy.

Dur­ing this time, I was able to see up close John’s ad­min­is­trat­ive skills. His un­re­lent­ing vis­ion for how to foster math­em­at­ic­al activ­ity com­bined well with his knack for avoid­ing con­flicts with col­leagues, draw­ing on his pa­tience and in­ner peace.

These were skills he also ap­plied with suc­cess when he be­came the first dir­ect­or of the Si­mons Cen­ter for Geo­metry and Phys­ics. His first chal­lenge there, he ex­plained to me, was to find the best avail­able chef for the din­ing hall. A din­ing hall that can at­tract re­search­ers with qual­ity food would help to keep a co­hes­ive com­munity of schol­ars, and John took this task ser­i­ously. He meth­od­ic­ally ate meals at fine res­taur­ants and ho­tels in Long Is­land un­til he was sat­is­fied he had found the right chef. After re­cruit­ing that chef, he turned to build­ing a top-rate math­em­at­ics de­part­ment, re­cruit­ing both Si­mon Don­ald­son and Kenji Fukaya as per­man­ent mem­bers. To this day, the SCGP con­tin­ues on the mo­mentum ini­ti­ated by John.

I would like to say a few more words here about John’s math­em­at­ic­al style. He has an abil­ity to learn new sub­jects with what ap­pears to be un­canny ease, to see through the com­plex­it­ies of those sub­jects, and then to con­trib­ute sig­ni­fic­antly to their fur­ther de­vel­op­ment. Be­fore my stu­dent days, John had done this already with ra­tion­al ho­mo­topy the­ory and then with hy­per­bol­ic geo­metry. I was a stu­dent in his hey­day as a gauge the­or­ist. Soon af­ter­wards, he turned his at­ten­tion to un­der­stand­ing the phys­ics con­nec­ted to the mod­ern de­vel­op­ments. When Grigori Perel­man an­nounced his proof of the Poin­caré con­jec­ture, which at the time was in­scrut­able even to many spe­cial­ists, John turned his for­mid­able math­em­at­ic­al skills to thor­oughly di­gest­ing that proof, and present­ing it to the wider math­em­at­ic­al audi­ence, in a col­lab­or­a­tion with Gang Tian [4], [5]. All of these con­tri­bu­tions were made with John’s dis­tinct­ive en­thu­si­asm and clar­ity of vis­ion.

I am grate­ful to John’s ment­or­ship. The math­em­at­ic­al com­munity has be­nefited greatly from his many con­tri­bu­tions to to­po­logy and geo­metry.

Peter Oz­sváth is a pro­fess­or of math­em­at­ics at Prin­ceton Uni­versity. He wrote his thes­is on gauge the­ory at Prin­ceton in 1994 un­der the su­per­vi­sion of John Mor­gan. Oz­sváth’s re­search in­terests are at the in­ter­face between low-di­men­sion­al to­po­logy and sym­plect­ic geo­metry.