Proceedings of Symposia in

PURE MATHEMATICSVolume 99

Modern Geometry:A Celebration of the Workof Simon Donaldson

Vicente MunozIvan SmithRichard P. ThomasEditors

Volume 99

Modern Geometry:A Celebration of the Workof Simon Donaldson

Vicente MunozIvan SmithRichard P. ThomasEditors

NathalieWahl,2017

PURE MATHEMATICSProceedings of Symposia in

Volume 99

Modern Geometry:A Celebration of the Workof Simon Donaldson

Vicente MunozIvan SmithRichard P. ThomasEditors

2010 Mathematics Subject Classification. Primary 32J25, 32L05, 53C07, 53C44, 53D35,53D40, 53D50, 57R55, 57R57, 57R58.

Library of Congress Cataloging-in-Publication Data

Names: Munoz, V. (Vicente), 1971– editor. | Smith, Ivan, 1973– editor. | Thomas, Richard P.,1972– editor.

Title: Modern geometry : a celebration of the work of Simon Donaldson / Vicente Munoz, IvanSmith, Richard P. Thomas, editors.

Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Pro-ceedings of symposia in pure mathematics ; volume 99 | Includes bibliographical references.

Identifiers: LCCN 2017052437 | ISBN 9781470440947 (alk. paper)

Subjects: LCSH: Donaldson, S. K. | Manifolds (Mathematics) | Four-manifolds (Topology) | Ge-ometry. | Topology. | AMS: Several complex variables and analytic spaces – Compact analyticspaces – Transcendental methods of algebraic geometry. msc | Several complex variables andanalytic spaces – Holomorphic fiber spaces – Holomorphic bundles and generalizations. msc |Differential geometry – Global differential geometry – Special connections and metrics on vectorbundles (Hermite-Einstein-Yang-Mills). msc | Differential geometry – Global differential geometry– Geometric evolution equations (mean curvature flow, Ricci flow, etc.). msc | Differential geome-try – Symplectic geometry, contact geometry – Global theory of symplectic and contact manifolds.msc | Differential geometry – Symplectic geometry, contact geometry – Floer homology and coho-mology, symplectic aspects. msc | Differential geometry – Symplectic geometry, contact geometry– Geometric quantization. msc | Manifolds and cell complexes – Differential topology – Differ-entiable structures. msc | Manifolds and cell complexes – Differential topology – Applicationsof global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants. msc |Manifolds and cell complexes – Differential topology – Floer homology. msc

Classification: LCC QA613 .M6345 2018 | DDC 516/.07–dc23

LC record available at https://lccn.loc.gov/2017052437

DOI: http://dx.doi.org/10.1090/pspum/099

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Contents

Preface vii

Graded linearisationsGergely Berczi, Brent Doran, and Frances Kirwan 1

Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizationsAliakbar Daemi and Kenji Fukaya 23

Weinstein manifolds revisitedYakov Eliashberg 59

Remarks on Nahm’s equationsNigel Hitchin 83

Conjectures on counting associative 3-folds in G2-manifoldsDominic Joyce 97

Toward an algebraic Donaldson-Floer theoryJun Li 161

Introduction to a provisional mathematical definition of Coulomb branches of3-dimensional N = 4 gauge theories

Hiraku Nakajima 193

An overview of knot Floer homologyPeter Ozsvath and Zoltan Szabo 213

Descendants for stable pairs on 3-foldsRahul Pandharipande 251

The Dirichlet problem for the complex homogeneous Monge-Ampere equationJulius Ross and David Witt Nystrom 289

Kahler-Einstein metricsGabor Szekelyhidi 331

Donaldson theory in non-Kahlerian geometryAndrei Teleman 363

Two lectures on gauge theory and Khovanov homologyEdward Witten 393

v

Preface

Simon Donaldson has been one of the central figures in modern geometry forthirty-five years, and remains as active today as ever. His work has revolutionisednumerous fields; the breadth of the essays in this volume are testament to hisprofound influence across different areas of differential and algebraic geometry, andits connections to topology, to analysis and to theoretical physics.

Simon Kirwan Donaldson was born on August 20th, 1957, in Cambridge, U.K.He attended secondary school at Sevenoaks in Kent, and was a mathematics under-graduate at Pembroke College, Cambridge, before going on to doctoral work underthe joint supervision of Michael Atiyah and Nigel Hitchin at Oxford. After hisDPhil degree, Donaldson became a Research Fellow at All Souls College, Oxford,and then (with a year at the Institute for Advanced Study in Princeton as inter-mission) the Wallis Professor at Oxford. He remained in Oxford until 1997, thenspent one year at Stanford, California, before returning to the U.K. with a Chairat Imperial College, London. In 2014 he joined the Simons Center for Geometryand Physics at Stony Brook, and now divides his time between there and Imperial.

Donaldson was an invited speaker at the 1982 ICM inWarsaw, and was awardedthe Fields Medal at the 1986 ICM in Berkeley. Amongst his many other awards arethe King Faisal International Prize (2006), the Nemmers Prize (2008), the ShawPrize (2009, joint with Cliff Taubes), and the Breakthrough Prize (2015). He wasknighted in the 2012 New Year Honours list for services to mathematics.

Whilst still a graduate student, in 1982, Donaldson overturned the world oflow-dimensional topology, bringing to bear methods from classical gauge theoryand the Yang-Mills equations – ideas later recast by Witten in terms of quantumfield theory – to prove new constraints on the topology of smooth four-dimensionalmanifolds, the nature of which have no analogue in either lower or higher di-mensions. Celebrated results in this period include: the diagonalisability theo-rem1 for the intersection forms of definite four-manifolds; the disproof of the four-dimensional s-cobordism conjecture and introduction of his polynomial invariantsof four-manifolds; the Donaldson-Uhlenbeck-Yau (DUY) theorem describing thesolutions of the Hermitian-Yang-Mills equations on Kahler manifolds; and his workon Nahm’s equations and monopoles.

Whilst his work in low-dimensional topology dominated four-manifold theoryfrom 1982–1994, Donaldson later made profound contributions to three quite differ-ent areas. In 1996 he introduced Lefschetz pencils into symplectic topology, provingthe first general existence theorem for symplectic hypersurfaces. At the core of this

1The frontispiece to this volume, painted by Nathalie Wahl, merges Simon’s childhood passionfor sailing with an abstracted version of the renowned image of the cobordism underlying thediagonalisability theorem. Readers might look for hints of other theorems hidden in the painting!

vii

viii PREFACE

work is an estimated transversality or quantitative Sard theorem, established via anovel h-principle based on analytical methods of approximately holomorphic geom-etry. In an attempt to fit Floer’s symplectic-geometric invariants into the formalismof topological quantum field theory, in analogy with the expected and known struc-tures for gauge-theoretic Floer homology, he introduced the triangle product inLagrangian Floer cohomology, and the quantum category of a symplectic manifold– the cohomological version of which became the Fukaya category, central in mirrorsymmetry. At around the same time, Donaldson laid out a program in higher-dimensional gauge theory suggesting generalisations of both instanton theory andLagrangian Floer theory to G2 and Spin(7)-manifolds, a program in rapid currentdevelopment.

In the mid 1990s, Donaldson began studying the existence question for con-stant scalar curvature Kahler metrics – the higher-dimensional analogue of theconstant curvature metrics on Riemann surfaces provided by the uniformisationtheorem. Over the following two decades, he introduced a huge array of new ideasinto this part of complex differential geometry, partly based on intuitions derivedfrom infinite-dimensional moment maps and ideas around geometric quantisation.He eventually successfully resolved (in 2013, with Xiuxiong Chen and Song Sun)the existence question for Kahler-Einstein metrics on Fano manifolds, as conjec-tured by Yau and Tian – a landmark achievement, once again binding togetherideas from algebraic geometry and from infinite-dimensional analysis. Whilst theDUY theorem relied essentially on the link between stability of bundles and theexistence of special-curvature connections, the results in complex geometry estab-lish a “more non-linear” analogue, reformulating the existence of Kahler-Einsteinmetrics in terms of the stability of the varieties themselves. Donaldson will givethe opening lecture at the ICM in Rio in 2018, the 4th ICM which he will address.

Donaldson’s influence on mathematics reaches very much further than his bodyof published results. He has had a huge number of graduate students (44 studentsand 132 descendents so far, according to the Mathematics Genealogy database).Our own extraordinarily priviliged experiences of being his students were that onewas not just given a thesis problem, one was given a whole raft of problems, earlyentry to an intellectual landscape which other people had scarcely begun to think ofpopulating. Donaldson suggested key examples which paved routes through theseuncharted territories and made them familiar, generously leaving the impression onehad surveyed and discovered the contours of the theory for oneself. Many peoplehave worked on his suggestions without formally being his students or postdocs:he has always been incredibly generous with his ideas, and equally generous instepping back from credit. His gentleness and kindness are renowned, and he hasbeen a unique role model to generations of those who have learned from him,listened to his lectures and seminars2, or had the privilege of being party to one ofhis many informal asides, questions or car-ride reflections.

The editors wish to thank Zak Turcinovic for help with the typesetting.

Vicente Munoz, Ivan Smith, Richard Thomas

2In the early 1990s, at his “Geometry and Analysis” seminar at Oxford, instead of invitinga speaker, Donaldson would sometimes talk about a result which excited him, outlining the proofhe imagined the author had given. Often this had no resemblance to the actual work, and openedup an entirely new perspective.

PUBLISHED TITLES IN THIS SERIES

99 Vicente Munoz, Ivan Smith, and Richard P. Thomas, Editors, Modern Geometry,2018

96 Si Li, Bong H. Lian, Wei Song, and Shing-Tung Yau, Editors, String-Math 2015,2017

95 Izzet Coskun, Tommaso de Fernex, and Angela Gibney, Editors, Surveys onRecent Developments in Algebraic Geometry, 2017

94 Mahir Bilen Can, Editor, Algebraic Groups: Structure and Actions, 2017

93 Vincent Bouchard, Charles Doran, Stefan Mendez-Diez, and Callum Quigley,Editors, String-Math 2014, 2016

92 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, LieAlgebras, Lie Superalgebras, Vertex Algebras and Related Topics, 2016

91 V. Sidoravicius and S. Smirnov, Editors, Probability and Statistical Physics in St.Petersburg, 2016

90 Ron Donagi, Sheldon Katz, Albrecht Klemm, and David R. Morrison, Editors,String-Math 2012, 2015

89 D. Dolgopyat, Y. Pesin, M. Pollicott, and L. Stoyanov, Editors, HyperbolicDynamics, Fluctuations and Large Deviations, 2015

88 Ron Donagi, Michael R. Douglas, Ljudmila Kamenova, and Martin Rocek,Editors, String-Math 2013, 2014

87 Helge Holden, Barry Simon, and Gerald Teschl, Editors, Spectral Analysis,Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s60th Birthday, 2013

86 Kailash C. Misra, Daniel K. Nakano, and Brian J. Parshall, Editors, RecentDevelopments in Lie Algebras, Groups and Representation Theory, 2012

85 Jonathan Block, Jacques Distler, Ron Donagi, and Eric Sharpe, Editors,String-Math 2011, 2012

84 Alex H. Barnett, Carolyn S. Gordon, Peter A. Perry, and Alejandro Uribe,

Editors, Spectral Geometry, 2012

83 Hisham Sati and Urs Schreiber, Editors, Mathematical Foundations of QuantumField Theory and Perturbative String Theory, 2011

82 Michael Usher, Editor, Low-dimensional and Symplectic Topology, 2011

81 Robert S. Doran, Greg Friedman, and Jonathan Rosenberg, Editors,Superstrings, Geometry, Topology, and C∗-algebras, 2010

80 D. Abramovich, A. Bertram, L. Katzarkov, R. Pandharipande, and M.Thaddeus, Editors, Algebraic Geometry, 2009

79 Dorina Mitrea and Marius Mitrea, Editors, Perspectives in Partial DifferentialEquations, Harmonic Analysis and Applications, 2008

78 Ron Y. Donagi and Katrin Wendland, Editors, From Hodge Theory to Integrabilityand TQFT, 2008

77 Pavel Exner, Jonathan P. Keating, Peter Kuchment, Toshikazu Sunada,and Alexander Teplyaev, Editors, Analysis on Graphs and Its Applications, 2008

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/pspumseries/.

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