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Home , Andrew Casson, Barry Mazur, David Gabai, Morton Brown, Property P conjecture, Tomasz Mrowka

2007 Veblen Prize

The 2007 Oswald Veblen Prize in Geometry was awarded at the 113th Annual Meeting of the AMS in NewOrleansinJanuary2007. TheVeblenPrizeisawardedeverythreeyearsfor a notable researchmemoirin geometryor that has appeared during the previous five years in a recognized North American journal (until 2001 the prize was usually awarded every five years). Es- tablished in 1964, the prize honors the memory of Oswald Veblen (1880–1960), who served as presi- dent of the AMS during 1923–1924. It carries a cash award of US$5,000. The Veblen Prize is awarded by the AMS Coun- cil acting on the recommendation of a selection Peter Kronheimer committee. For the 2007 prize, the members of the selection committee were: Cameron M. Gordon, MichaelJ.Hopkins(chair),andRonaldJ.Stern. “Embedded surfaces and the structure of Previous recipients of the Veblen Prize are: Donaldson’s polynomial invariants”, J. Differential Christos D. Papakyriakopoulos (1964), Raoul H. Geom.41(1995),no.3,573–734. Bott (1964), (1966), Since 1982, most of the progress in four-dimen- and (1966), Robion C. Kirby (1971), sional differential topology has arisen from the Dennis P. Sullivan (1971), William P. Thurston applications of pioneered by S. K. (1976), James Simons (1976), Mikhael Gromov Donaldson. In particular, Donaldson’s polynomial (1981), Shing-Tung Yau (1981), Michael H. Freed- invariants have been used to prove a variety of man (1986), (1991), Clifford H. results about the topology and geometry of four- Taubes (1991), Richard Hamilton (1996), Gang . This paper is one of the pinnacles of Tian (1996), Jeff Cheeger (2001), this development. It gives a conceptual framework (2001), Michael J. Hopkins (2001), and and an organizing principle for some of the dis- (2004). parate observations and calculations of Donaldson The 2007 Veblen Prize was awarded to two invariants that had been made earlier, it reveals pairs of collaborators: to Peter Kronheimer and a deep structure encoded in the Donaldson in- Tomasz Mrowka, and to Peter Ozsváth and variants which is related to embedded surfaces in four-manifolds, and it has been the point of Zoltán Szabó. The text that follows presents the departure and the motivating example for impor- selection committee’s citations, brief biograph- tant further developments, most spectacularly ical sketches, and the awardees’ responses upon for Witten’s introduction of the so-called Seiberg- receivingtheprize. Witteninvariants. “The genus of embedded surfaces in the projec- Citation: Peter Kronheimer and Tomasz tiveplane”,Math. Res. Lett.1(1994),no.6,797–808. Mrowka This paper proves the Thom conjecture, which The2007VeblenPrizeinGeometryisawardedtoPe- claims that if C is a smooth holomorphic curve in ter Kronheimer and Tomasz Mrowka for their joint CP2, and C′ is a smoothly embedded oriented two- contributions to both three- and four-dimensional representing the same homology class as topology through the development of deep analy- C,thenthegenusofC′ satisfiesg(C′) ≥ g(C). tical techniques and applications. In particular the “Witten’s conjecture and property P”, Geometry prizeisawardedfortheirseminalpapers: and Topology8(2004),295–310.

April 2007 Notices of the AMS 527 Here the authors use their earlier development Heworksmainlyontheanalyticaspectsofgauge of Seiberg-Witten monopole to theories and applications of gauge theory to prob- prove the for knots. In other lemsinlow-dimensionaltopology. 3 words, if K ⊂ S is a nontrivial , and Kp/q is the three-manifoldobtainedbyp/q Dehnsurgeryalong Response: Peter Kronheimer and Tomasz

K with q ≠ 0, then π1(Kp/q ) must be nontrivial. The Mrowka proof is a beautiful work of synthesis which draws We are honored, surprised, and delighted to be upon advances made in the fields of gauge theory, selected, together with Peter Ozsváth and Zoltán symplectic and , and Szabó, as recipients of the Oswald Veblen Prize in overthepasttwentyyears. Geometry. The Thom conjecture, and other related ques- Biographical Sketch: Peter Kronheimer tions concerning the genus of embedded 2-mani- Born in London, Peter Kronheimer was educated folds in 4-manifolds, are natural and central at the City of London School and Merton College, questions in 4-dimensional differential topolo- Oxford. He obtainedhis B.A. in 1984 and his D.Phil. gy. After ’s work in gauge theory openedupthisfield,theseproblemsbecametempt- in 1987 under the supervision of Michael Atiyah. ing targets for the newly available techniques. In After a year as a Junior Research Fellow at Bal- the summer of 1989, we were both at MSRI and liol and two years at the Institute for Advanced discussed the idea of using “singular instantons” Study, he returned to Merton as Fellow and Tutor to prove such conjectures. But it was not until in . In 1995 he moved to Harvard Uni- two years later, when we spent a month together versity, where he is now William Caspar Graustein at Oberwolfach, that a proof began to emerge of ProfessorofMathematics.ArecipientoftheFörder- a version of the Thom conjecture for embedded preis from the Mathematisches Forschungsinstitut 2-manifolds in K3 surfaces. This theorem and its Oberwolfach, and a Whitehead Prize from the proof filled our first two joint papers,and provided London Mathematical Society, he was elected a thefirsttrulysharpresultsforthegenusproblem. FellowoftheRoyalSocietyin1997. In March 1993, we met at Columbia, at the in- Next to mathematics, his main pastime has vitation of John Morgan. We understood that the often been music—the horn in particular, which he singular instanton techniques that we had used studied as a pupil of Ifor James. Peter Kronheimer for the genus problem should lead to universal lives in Newton, Massachusetts, with his wife Jenny relations among the values of Donaldson’s poly- andtwosons,MatthewandJonathan. nomial invariants for 4-manifolds. At the time, calculating Donaldson’s invariants in special cases Biographical Sketch: Tomasz Mrowka was a challenging occupation. Although we could Tom Mrowka is a professor at the Massachusetts seehow to proverelations, no coherent picture was Institute of Technology. He received his under- emerging; and at the end of this visit, Peter headed graduate degree from MIT in 1983 and attended to LaGuardia with the forest still not visible for the graduate school at the University of California at trees. New York’s “Blizzard of the Century” closed Berkeley, receiving his Ph.D. under the direction of the airport, and we worked together for another Clifford H. Taubes in 1989. After graduate school day, during which we noticed that our relations im- he held postdoctoral positions at the Mathematical plied a simple linear recurrence relation for certain values of Donaldson’s invariants. This soon led to Sciences Research Institute in Berkeley (1988–89), a beautiful structure theorem for the polynomial Stanford (1989–91) and Caltech (1991–92). He held invariants in terms of “basic classes”, intricately a professorship at Caltech from 1992 until 1996 entwined with the genus question through an “ad- and was a visiting professor at Harvard (spring of junctioninequality”.Thesedevelopmentsprovided 1995) and at MIT (fall of 1995) before returning to a proof of the Thom conjecture for a large class of MITpermanentlyinthefallof1996. algebraic surfaces, though the original version He received the National Young Investigator for the complex projective plane had to wait until Grant of the NSF in 1993 and was a Sloan Founda- 1994 and the introduction of the Seiberg-Witten tion Fellow from 1993 to 1995. He gave an invited equations. lecture in the topology section of the 1994 Inter- While techniques from gauge theory revolution- national Congress of Mathematicians in Zurich, ized the field of 4-manifolds, providing answers the lectures at the Institute for to many important questions, these ideas had no Advanced Study in Princeton in 1999, the Stanford comparable impact in 3-dimensional topology, Distinguished Visiting Lecture Series in 2000, the wherethequestionsandtoolsremainedverydiffer- Joseph Fels Ritt Lectures at Columbia University in ent. Around 1986, Andreas Floer used Yang-Mills 2004, and the 23rd Friends of Mathematics Lecture gauge theory to define his “instanton homology” atKansasStateUniversityin2005. groups for 3-manifolds. The Euler number of these

528 Notices of the AMS Volume 54, Number 4 homology groups recaptured an integer invariant introduced by Andrew Casson which had already been seen to imply results about surgery on 3- manifolds, including partial results in the direction of the “Property P” conjecture. It seems likely that Floer himself foresaw the possibility of using his homology theory to prove stronger results in the same direction; in particular, he established an “exact triangle” relating the Floer homology groups of the 3-manifolds obtained by three different surg- eries on a knot. But a missing ingredient at this time was any very general result stating that these homology groups were not trivial. In 1995 Yasha EliashbergvisitedHarvardandlecturedonhiswork Peter Ozsváth Zoltán Szabó with Bill Thurston on foliations and contact struc- tures. It was apparent that these results could be combinedwithworkofCliffTaubesandDaveGabai Citation: Peter Ozsváth and Zoltán Szabó to give a non-vanishing theorem for a version of The 2007 Veblen Prize is awarded to Peter Ozsváth Floer’s homology groups defined using the Seiberg- and Zoltán Szabó in recognition of the contribu- tions they have made to 3- and 4-dimensional Witten equations, and to show, for example, that topology through their Heegaard Floer homology these versions of Floer homology encode sharp theory. informationaboutthe genus of embedded surfaces Ozsváth and Szabó have developed this theory in 3-manifolds. This was the first strong indication in a highly influential series of over twenty papers that, by combining non-vanishing theorems with producedin the lastfive years,andindoingsohave surgery exact triangles, one would be able to use generateda remarkable amountofactivityin3-and Floergroupstoobtainsignificantnewresultsabout 4-dimensional topology. Specifically, they are cited Dehn surgery. This hope was eventually realized fortheirpapers in our joint paper with the other co-recipients of “Holomorphic disks and topological invariants this prize, Peter Ozsváth and Zoltán Szabó, on for closed three-manifolds”, Ann. of Math. (2) 159 lens space surgeries, and in the eventual resolu- (2004), 1027-1158; tion of the Property P conjecture using instanton “Holomorphic disks and three-manifold invari- homology. In the meantime, Ozsváth and Szabó’s ants:propertiesandapplications”,Ann. of Math.(2) 159 “Heegaard Floer theory” has transformed the field (2004), 1159-1245; “Holomorphicdisks andgenus bounds”, Geome- once again: it has led to a wealth of new results, try and Topology8(2004),311-334. and open problems to attract a new generation of The Heegaard Floer homology of a 3-manifold researchers. plays a role, in the context of the Seiberg-Witten Peter would like to thank his wife Jenny and all invariants of 4-manifolds, analogous to that played his family for their love and support; and his mathe- by LagrangianFloer homology in the context of the maticalmentors,MichaelAtiyah,SimonDonaldson, Donaldson invariants. There is also a version for andDominicWelsh,fortheirguidance. knots, whose Euler characteristic is the Alexander Tom thanks Gigliola, Mario, and Sofia for the polynomial. It detects the genus of a knot and also joy they bring. Tom also thanks his teachers Victor whether or not a knot is fibered. The combinatorial Guillemin, Richard Melrose, , Stephen nature of these invariants has led to many deep Smale, John Stallings, and Rob Kirby for lighting up applications in 3-dimensional topology. Among very different directions and ways of thinking at theseareresultsaboutDehnsurgeryonknots,such the beginning of his mathematical journey. Special as the Dehn surgery characterization of the unknot, thanks to his advisor Cliff Taubes and to John Mor- strong restrictions on lens space and other Seifert fiber space surgeries, and dramatic new results gan whose confidence and interest at the beginning on unknotting numbers. Ozsváth and Szabó have ofhiscareerwerecrucial. used Heegaard Floer homology to define a contact Finally we would both like to thank the Ameri- structure invariant, which has led to new results can Mathematical Society for recognizing the field in 3-dimensional contact topology. They have al- ofgaugetheoryandlow-dimensionaltopologywith so defined a new concordance invariant of knots, thisyear’sVeblenPrize.Wefeelprivilegedtobecho- which gives a lower bound on the 4-ball genus. The sen, together with our co-recipients, as representa- 4-dimensional version of Heegaard Floer homol- tives for an area of research that has seen so many ogy has enabled them to give gauge-theory-free excitingdevelopments. proofs of many of the results in 4-dimensional

April 2007 Notices of the AMS 529 topology obtained in the last decade using Donald- to whom I have turned many times for insight son and Seiberg-Witten theory, such as the Thom and advice. I would also like to thank R. Fintushel, Conjecture on the minimal genus of a smooth R.Kirby,andR.Sternforhelpingtomakethefieldso representative of the homology class of a curve of pleasant socially and so rich mathematically. I am degree d in CP 2, and the Milnor Conjecture on the deeply grateful to my undergraduate teachers P. J. unknottingnumberofatorusknot. Cohen,R.L.Cohen,R.J.Milgram,andfellowstudent D. B. Karagueuzian, for introducing me to mathe- Biographical Sketch: Peter Ozsváth matics. I would also like to thank my more recent Peter Ozsváth was born on October 20, 1967, in collaborators C. Manolescu, S. Sarkar, A. Stipsicz, Dallas, Texas. He received his B.S. from Stanford D. P. Thurston, and my former students E. Grigsby, University (1989) and his Ph.D. from Princeton M. Hedden, P. Sepanski, and the many additional University (1994) under the direction of John W. members of the Columbia mathematics depart- Morgan. He held postdoctoral positions at Caltech, ment who are constantly bringing new insights to the Max-Planck-Institut in Bonn, the Mathematical anexcitingandever-developingfield. Sciences Research Institute in Berkeley, and the Institute for Advanced Study in Princeton. He held Biographical Sketch: Zoltán Szabó faculty positions at , Michigan Zoltán Szabó was born in Budapest, Hungary, in State University, and the University of California, 1965. He did his undergraduate studies at Eötvös Berkeley. He has been on the faculty at Columbia Loránd University in Budapest, and then his gradu- University since 2002. ate studies at Rutgers University with John Morgan Ozsváth received a National Science Foundation and Ted Petrie. He has worked at Princeton Uni- Postdoctoral Fellowship and an Alfred P. Sloan versity since graduating in 1994. He also spent a Research Fellowship. His invited lectures include year at the University of Michigan in 1999–2000. an Abraham Robinson Lecture at He has been a professor at Princeton University (2003), a William H. Roever Lecture at Washington since 2002. He received a Sloan Research Fellow- University St. Louis (2004), a Kuwait Foundation ship and a Packard Fellowship. He was an invited Lecture at Cambridge University (2006), and a lecturer at the 2006 International Congress of lecture in the topology section of the International Mathematicians in Madrid, and a plenary speaker CongressofMathematicians(2006). at the 2004 European Congress of Mathematics in Stockholm. Szabó’s main research interests are Response: Peter Ozsváth smooth 4-manifolds, 3-manifolds, knots, Heegaard Floer homology, , and gauge I am greatly honored to be a co-recipient of of the theory. Oswald Veblen Prize, along with my long-time col- laborator Zoltán Szabó, and also Peter Kronheimer Response: Zoltán Szabó and Tomasz Mrowka, whose work has always been I am greatly honored to be named, along with Peter aprofoundsourceofinspirationforme. Kronheimer, Tom Mrowka, and Peter Ozsváth, as a Heegaard Floer homology grew out of our ef- recipientoftheOswaldVeblenPrize.Thejointwork forts at understanding gauge theory from a more with Peter Ozsváth which is noted here grew out of combinatorial point of view. The mathematical our attempts to understand Seiberg-Witten moduli starting point was Yang-Mills theory and then spaces over three-manifolds where the metric de- Seiberg-Witten theory, which started with the work generates along a surface. This led to the construc- of S. K. Donaldson, A. Floer, and C. H. Taubes. But tionofHeegaardFloerhomologythatinvolvedboth we have also been fortunate to be able to draw on topological tools, such as Heegaard diagrams, and the work of many interlocking neighboring fields, tools from symplectic geometry, such as holomor- including contact and symplectic geometry, where phic disks with Lagrangian boundary constraints. at various times the work of Y. Eliashberg and E. The time spent on investigating Heegaard Floer Giroux provided answers to fundamental ques- homology and its relationship with problems in tions, and also three-manifold topology, where the low-dimensional topology was rather interesting. questions raised by C. Gordon serve as a guiding I am very glad that this effort was rewarded by the light and the work of D. Gabai provides powerful prizecommittee. tools which fit very neatly into the context of Floer First of allI would like to thank my wife, Piroska, homology. forhersupportovertheyears.Ialsoowe alottomy I would like to thank my family and friends co-author Peter Ozsváth whose boundless energy for their support throughout the years, and I also made this work possible, and to my thesis advisor, owe a great debt of gratitude to my teachers, co- John Morgan, who introduced me to the world of authors, and students. In particular, I thank Zoltán gauge theory. for those many caffeinated mathematical sessions. I also thank my thesis advisor J. W. Morgan for introducing me to gauge theory and T. S. Mrowka,

530 Notices of the AMS Volume 54, Number 4