by András Stipsicz and Zoltán Szabó
We met John in the early 90s, as students eager to understand more about the smooth invariants introduced by Simon Donaldson a couple of years earlier. These invariants had shown amazing success in discovering the exotic behavior of four-dimensional manifolds, and for us this seemed the most exciting development in contemporary topology. Before meeting John, we visited Columbia University, and got a version of the then recent book by Robert (Bob) Friedman and Morgan on complex surfaces from Friedman himself (on floppy disks, which then led us to various adventures of printing a several-hundred-page book in installments of the ten-page daily limit imposed on graduate students). Bob also advised us to contact John. We finally met the latter in Princeton in the fall of 1992, where he announced a course on the “Mayer–Vietoris principle for Donaldson invariants” (or something very similar). The course was beautifully designed, in John’s signature style — clear, to the point, seemingly easy, yet containing a lot of information. John stayed in Princeton one day a week, and that day was packed: a lecture, a seminar, meetings with students, a full day of concentration. He was very generous with his time. We were a little bit lost at that time, as we were learning the theory mostly from books and papers. To our great relief, even though we were not affiliated with either Princeton or Columbia, John agreed to help and suggested some problems. After the completion of the course in Princeton, he continued advising us — we now visited him regularly in New York, usually on Wednesday afternoons at Columbia. The regular weekly “check-in” on the work was a tremendous help: it gave a certain rhythm, and the advice we received on those afternoons helped us to steer our research further.
The two main problems back then in smooth four-dimensional topology revolved around
- the blow-up formula (how do the invariants change when the manifold is blown up, that is, given as the connected sum with the complex projective plane with its orientation reversed), and
- the invariants of elliptic surfaces (what are the Donaldson invariants of the infinite family of simply connected elliptic surfaces fibered over the complex projective line).
Peter Ozsváth, Tom Leness, Jim Bryan (and probably other students) were in the “blow-up” camp, while we, together with Paolo Lisca, formed the “elliptic” people. At the Parks City Mathematics Institute (PCMI) Summer program in 1994 (which took place just months before the discovery of Seiberg–Witten theory) we even organized a soccer game between the “blow-ups” and the “elliptics”, inspired by the ongoing soccer world championship in the US. The European dominance in the elliptic camp determined the outcome, which probably would have been very different if we had opted for baseball, say, instead of soccer as our sport of choice.
Friedman and Morgan’s calculation of (part of) the Donaldson invariants
of Dolgachev surfaces
[1]
already verified a novel
four-dimensional phenomenon: topological four-manifolds can admit
infinitely many distinct smooth structures. The full calculation, and
indeed the verification of the fact that these complex surfaces are
diffeomorphic if and only if they are deformation equivalent, was
however open. In a remarkable work with
\Kieran O’Grady
[2],
and
relying on the complex algebraic geometric
reformulation
of the theory,
John carried out the calculation of
further parts of Donaldson’s instanton invariants for complex surfaces
homeomorphic to the famous \( K3 \) surface (and its non-spin analogue).
(A parallel effort was carried out independently by
Stefan Bauer
[e3]
and
Christian Okonek
and
Antonius Van de Ven
[e1].)
The calculations showed that, indeed, in these cases
diffeomorphism and deformation equivalence coincides. Furthermore,
they provided the stepping stone to extend this
principle to all (simply connected) elliptic surfaces; a version of
which then formed the core of the first paper on the subject
by the present authors
[e4].
There were other, similar calculations, e.g., those of Paolo
Lisca
[e5],
which then culminated in work of
Tom Mrowka
and John
[4];
and those of
Ron Fintushel
and
Ron Stern
[e9],
who determined the
entire Donaldson series, and could rephrase the results in the
language of basic classes just discovered by
Peter Kronheimer
and Tom
Mrowka
[e2].
In this direction, another major piece of
mathematics played a crucial role:
The \( L^2 \)-moduli space
and a vanishing theorem for Donaldson polynomial invariants, the book John wrote
in collaboration
with Tom Mrowka and
Danny Ruberman
[3].
This text provided the first theoretically solid
underpinning of the calculation of the invariants for four-manifolds
we constructed by “cut-and-paste” methods, still a dominant way to
find four-manifolds with various interesting properties.
The introduction of the Seiberg–Witten equations into the study of four-manifolds brought revolutionary changes [e8], [e7], [e6], and John was at the forefront of all these actions. In fact, we learned some of these new developments directly from him, while he was visiting Irvine at the time and giving lectures at Princeton. These discussions then led to joint works — for example, to the joint paper with Cliff Taubes on the generalized Thom conjecture [6]. John’s book on the theory [5] was probably the first detailed account, and had a profound impact on the field. He devoted several papers to the structure and properties of Seiberg–Witten invariants of four-manifolds (see, e.g., [7]).
In the mid 1990s John entered the field of mathematical physics, and then undertook work on the Ricci flow and the expansion and completion of Perelman’s work on Thurston’s geometrization conjecture. Yet he has always remained interested in developments in low dimensional topology, and has closely followed the advances, for example, in Heegaard Floer homology.
John has had a strong impact on his students. We have learned many aspects of a mathematician’s life from him: he has influenced how we advise our students, he has shown us how to serve the community by other means (like refereeing or journal editing), and how to deliver lectures (prepare!). We may try to imitate him in all these things, but we will probably never reach his standards. His style, his calm and friendly approach to problems (both within and outside of mathematics) and to his colleagues continue to have a profound impact on our mathematical lives. The decade between the mid 1980s and the mid 1990s brought turbulent and very interesting times in four-manifold topology. It has been a pleasure and a privilege to be part of it with the guidance we have received from John. We are very grateful for the opportunity to work with him, and to be part of a community driven by researchers and personalities like John.
András Stipsicz received his PhD in 1994 under the guidance of John Morgan and Ted Petrie from Rutgers University, and after postdoctoral years at UC Irvine and semester visits in the Max Planck Institute and at MSRI Berkeley, he assumed a position at the Rényi Institute of Mathematics in Budapest, Hungary, where he is a professor and the director of the Institute.
Zoltán Szabó got his PhD at Rutgers University; his advisors were John Morgan and Ted Petrie. After postdoctoral years in Princeton and a year at the University of Michigan in Ann Arbor, he became a professor at the Department of Mathematics at Princeton University.
