Proceedings of Symposia in PURE MATHEMATICS Volume 99 Modern Geometry: A Celebration of the Work of Simon Donaldson Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors Volume 99 Modern Geometry: A Celebration of the Work of Simon Donaldson Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors Nathalie Wahl, 2017 Proceedings of Symposia in PURE MATHEMATICS Volume 99 Modern Geometry: A Celebration of the Work of Simon Donaldson Vicente Munoz˜ Ivan Smith Richard P. Thomas Editors 2010 Mathematics Subject Classification. Primary 32J25, 32L05, 53C07, 53C44, 53D35, 53D40, 53D50, 57R55, 57R57, 57R58. Library of Congress Cataloging-in-Publication Data Names: Mu˜noz, 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 Mu˜noz, Ivan Smith, 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 analytic spaces – Transcendental methods of algebraic geometry. msc | Several complex variables and analytic spaces – Holomorphic fiber spaces – Holomorphic bundles and generalizations. msc | Differential geometry – Global differential geometry – Special connections and metrics on vector bundles (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 – Applications of 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 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 Contents Preface vii Graded linearisations Gergely Berczi,` Brent Doran, and Frances Kirwan 1 Atiyah-Floer conjecture: A formulation, a strategy of proof and generalizations Aliakbar Daemi and Kenji Fukaya 23 Weinstein manifolds revisited Yakov Eliashberg 59 Remarks on Nahm’s equations Nigel Hitchin 83 Conjectures on counting associative 3-folds in G2-manifolds Dominic Joyce 97 Toward an algebraic Donaldson-Floer theory Jun Li 161 Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories Hiraku Nakajima 193 An overview of knot Floer homology Peter Ozsvath´ and Zoltan´ Szabo´ 213 Descendants for stable pairs on 3-folds Rahul Pandharipande 251 The Dirichlet problem for the complex homogeneous Monge-Amp`ere equation Julius Ross and David Witt Nystrom¨ 289 K¨ahler-Einstein metrics Gabor´ Szekelyhidi´ 331 Donaldson theory in non-K¨ahlerian geometry Andrei Teleman 363 Two lectures on gauge theory and Khovanov homology Edward Witten 393 v Preface Simon Donaldson has been one of the central figures in modern geometry for thirty-five years, and remains as active today as ever. His work has revolutionised numerous fields; the breadth of the essays in this volume are testament to his profound influence across different areas of differential and algebraic geometry, and its 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 under the joint supervision of Michael Atiyah and Nigel Hitchin at Oxford. After his DPhil 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, then spent one year at Stanford, California, before returning to the U.K. with a Chair at Imperial College, London. In 2014 he joined the Simons Center for Geometry and Physics at Stony Brook, and now divides his time between there and Imperial. Donaldson was an invited speaker at the 1982 ICM in Warsaw, and was awarded the Fields Medal at the 1986 ICM in Berkeley. Amongst his many other awards are the King Faisal International Prize (2006), the Nemmers Prize (2008), the Shaw Prize (2009, joint with Cliff Taubes), and the Breakthrough Prize (2015). He was knighted in the 2012 New Year Honours list for services to mathematics. Whilst still a graduate student, in 1982, Donaldson overturned the world of low-dimensional topology, bringing to bear methods from classical gauge theory and the Yang-Mills equations – ideas later recast by Witten in terms of quantum field theory – to prove new constraints on the topology of smooth four-dimensional manifolds, 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 invariants of four-manifolds; the Donaldson-Uhlenbeck-Yau (DUY) theorem describing the solutions of the Hermitian-Yang-Mills equations on K¨ahler manifolds; and his work on Nahm’s equations and monopoles. Whilst his work in low-dimensional topology dominated four-manifold theory from 1982–1994, Donaldson later made profound contributions to three quite differ- ent areas. In 1996 he introduced Lefschetz pencils into symplectic topology, proving the 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 passion for sailing with an abstracted version of the renowned image of the cobordism underlying the diagonalisability 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 a novel h-principle based on analytical methods of approximately holomorphic geom- etry. In an attempt to fit Floer’s symplectic-geometric invariants into the formalism of topological quantum field theory, in analogy with the expected and known struc- tures for gauge-theoretic Floer homology, he introduced the triangle product in Lagrangian Floer cohomology, and the quantum category of a symplectic manifold – the cohomological version of which became the Fukaya category, central in mirror symmetry. At around the same time, Donaldson laid out a program in higher- dimensional gauge theory suggesting generalisations of both instanton theory and Lagrangian Floer theory to G2 and Spin(7)-manifolds, a program in rapid current development. In the mid 1990s, Donaldson began studying the existence question for con- stant scalar curvature K¨ahler metrics – the higher-dimensional analogue of the constant curvature metrics on Riemann surfaces provided by the uniformisation theorem. Over the following two decades, he introduced a huge array of new ideas into this part of complex differential geometry, partly based on intuitions derived from 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 K¨ahler-Einstein metrics on Fano manifolds, as conjec- tured by Yau and Tian – a landmark achievement, once again binding together ideas from algebraic geometry and from infinite-dimensional analysis. Whilst the DUY theorem relied essentially on the link between stability of bundles and the existence of special-curvature connections, the results in complex geometry