Celebratio Mathematica

The to­po­logy of smooth 4-man­i­folds was re­volu­tion­ised in the early 1980s by the work of Si­mon Don­ald­son, and then turned on its head by Ed Wit­ten in the 1990s [e1], [e2]. John Mor­gan was a cent­ral fig­ure in the de­vel­op­ment and use of both Don­ald­son poly­no­mi­al in­vari­ants and Seiberg–Wit­ten the­ory. His con­tri­bu­tions in­cluded deep and found­a­tion­al the­or­ems, a vast amount of ex­pos­i­tion open­ing up the field to oth­ers, and the train­ing of stu­dents who went on to be­come key mem­bers of the next gen­er­a­tion of lead­ers in the field.

The aim of this note is to give a brief ac­count of some of these con­tri­bu­tions, and some per­son­al re­min­is­cences based on my time as a gradu­ate stu­dent in Columbia Uni­versity in the 1990s.

Ac­cord­ing to Math­s­cinet, John has 30 pub­lic­a­tions on the sub­ject of gauge the­ory, with pub­lic­a­tion dates from 1987 to 2003. He was the first (with Fried­man) to ex­hib­it a 4-man­i­fold with in­fin­itely many smooth struc­tures, and to show the vast dif­fer­ence between smooth and to­po­lo­gic­al auto­morph­ism groups of 4-man­i­folds; these res­ults were an­nounced in 1985 [1]. Much of his work was fo­cussed on the smooth to­po­logy of al­geb­ra­ic sur­faces. Key res­ults in­cluded the clas­si­fic­a­tion of el­lipt­ic sur­faces, with vari­ous coau­thors in­clud­ing Fried­man, Mrowka, and O’Grady, and the fact that com­plex al­geb­ra­ic sur­faces (up to de­form­a­tion) ad­mit fi­nitely many smooth struc­tures; also the de­tailed study with Mrowka and Ruber­man of mod­uli spaces of ASD con­nec­tions on cyl­indric­al end man­i­folds [5], [4], [2], [3]. These res­ults were im­port­ant pre­curs­ors to Kron­heimer–Mrowka’s struc­ture the­or­em for Don­ald­son poly­no­mi­al in­vari­ants [e3].

Mor­gan was one of the key early ad­op­ters of Seiberg–Wit­ten the­ory. His found­a­tion­al work in­cluded a proof with Szabó and Taubes of the gen­er­al­ised Thom con­jec­ture for em­bed­ded sur­faces of non­neg­at­ive self-in­ter­sec­tion in sym­plect­ic 4-man­i­folds, prov­ing that sym­plect­ic­ally em­bed­ded sur­faces min­im­ise genus in their ho­mo­logy classes [7]. This was based on a glu­ing the­or­em for Seiberg–Wit­ten mod­uli spaces when a 4-man­i­fold is a uni­on of two sub­man­i­folds glued along a sur­face times a circle, res­ult­ing in a product for­mula for the Seiberg–Wit­ten in­vari­ants of such a man­i­fold. An­oth­er found­a­tion­al pa­per, this time with Mrowka and Szabó, gives glu­ing for­mu­las along 3-tori, en­abling the com­pu­ta­tion of Seiberg–Wit­ten in­vari­ants of 4-man­i­folds ob­tained by gen­er­al­ised log trans­form, which is one of the ba­sic con­struc­tion tools in 4-di­men­sion­al to­po­logy [8]. These glu­ing the­or­ems of Mor­gan (with Szabó–Taubes and with Mrowka–Szabó) have been es­sen­tial tools in sub­sequent cal­cu­la­tions of Seiberg–Wit­ten in­vari­ants in oth­er key con­struc­tions of 4-man­i­folds, in­clud­ing ra­tion­al blow­downs ( Fin­tushel–Stern and Park), and knot sur­ger­ies (Fin­tushel–Stern) [e4], [e6], [e5]. The ba­sic toolkit used by the mod­ern to­po­lo­gist to ex­hib­it exot­ic smooth struc­tures on 4-man­i­folds is built on the work of Mor­gan and his col­lab­or­at­ors.

Mor­gan is a pro­lif­ic ex­pos­it­or whose work on the sub­ject of gauge the­ory in­cludes four books, at least one set of lec­ture notes, and many sur­vey art­icles. Of par­tic­u­lar note was his in­tro­duct­ory book on Seiberg–Wit­ten the­ory which was avail­able in pre­print form with­in a year of the ap­pear­ance of the new in­vari­ants, and thus en­abled many young to­po­lo­gists to get in­to the sub­ject [6]. He also taught vari­ous gradu­ate courses on this top­ic; I re­mem­ber in par­tic­u­lar one course that gave an ex­pos­i­tion of Taubes’ work on Seiberg–Wit­ten in­vari­ants of sym­plect­ic 4-man­i­folds, quite soon after Taubes’ work ap­peared [e8].

Ac­cord­ing to the Math­em­at­ics Gene­a­logy Pro­ject, John Mor­gan su­per­vised 27 PhD stu­dents and has 158 des­cend­ents. Of these, 13 of his stu­dents wrote theses in or re­lated to gauge the­ory, and these ac­count for 126 of his math­em­at­ic­al des­cend­ents. His stu­dents in­clude cur­rent lead­ing fig­ures in low-di­men­sion­al to­po­logy such as Oz­sváth and Szabó, the in­vent­ors of Hee­gaard Flo­er ho­mo­logy, and Stip­sicz, among whose many achieve­ments is his text­book with Gom­pf which has been a gate­way to low-di­men­sion­al to­po­logy for count­less stu­dents [e7], [e9].

As a PhD ad­visor, John was al­ways avail­able and sup­port­ive to his stu­dents. He was much in de­mand as an ad­visor, and had quite a large group of mu­tu­ally sup­port­ing stu­dents dur­ing my time in Columbia. This was a great learn­ing en­vir­on­ment, and John was al­ways ready to en­cour­age and par­ti­cip­ate in read­ing sem­inars. I re­call stim­u­lat­ing read­ing sem­inars based on Kron­heimer and Mrowka’s pa­per on mono­poles and con­tact struc­tures; on \( J \)-holo­morph­ic curves and Gro­mov–Wit­ten in­vari­ants; and on the proof of Flo­er’s ex­act tri­angle the­or­em in in­stan­ton Flo­er ho­mo­logy. I found John in­spir­a­tion­al as a teach­er and as a math­em­atician, and con­sider my­self very for­tu­nate to have had the be­ne­fit of his train­ing and ex­ample.

Brendan Owens re­ceived his PhD in 2000 from Columbia Uni­versity, as a stu­dent of John Mor­gan. His re­search is in smooth low-di­men­sion­al to­po­logy, and he is a Pro­fess­or of Math­em­at­ics at the Uni­versity of Glas­gow in Scot­land.