FROM THE AMS SECRETARY 2020 Leroy P. Steele Prizes The 2020 Leroy P. Steele Prizes were presented at the 126th Annual Meeting of the AMS in Denver, Colorado, in Jan- uary 2020. The Steele Prize for Mathematical Exposition was awarded to Martin R. Bridson and André Haefliger; the Prize for Seminal Contribution to Research in Analysis/Probability was awarded to Craig Tracy and Harold Widom; and the Prize for Lifetime Achievement was awarded to Karen Uhlenbeck. Citation for Riemannian geometry and group theory, that the field of Mathematical Exposition: geometric group theory came into being. Much of the 1990s Martin R. Bridson was spent finding rigorous proofs of Gromov’s insights and André Haefliger and expanding upon them. Metric Spaces of Non-Positive The 2020 Steele Prize for Math- Curvature is the outcome of that decade of work, and has ematical Exposition is awarded been the standard textbook and reference work throughout to Martin R. Bridson and André the field in the two decades of dramatic progress since its Haefliger for the book Metric publication in 1999. Spaces of Non-Positive Curvature, A metric space of non-positive curvature is a geodesic published by Springer-Verlag metric space satisfying (local) CAT(0) condition, that every in 1999. pair of points on a geodesic triangle should be no further Metric Spaces of Non-Positive apart than the corresponding points on the “comparison Martin R. Bridson Curvature is the authoritative triangle” in the Euclidean plane. Examples of such spaces reference for a huge swath of include non-positively curved Riemannian manifolds, modern geometric group the- Bruhat–Tits buildings, and a wide range of polyhedral ory. It realizes Gromov’s vision complexes. of group theory studied via This book is the definitive text on these spaces and the geometry, has been the fun- groups associated with them. The theory is developed damental textbook for many carefully, in great generality. All the foundational theorems graduate students learning the are proved, and the important examples are covered. The subject, and paved the way for proofs are clear and comprehensive. The necessary density the developments of the subse- of such a work is offset by the inclusion of a large number quent decades. of exercises, making it invaluable both as a graduate text At the turn of the 20th cen- and as a reference for active researchers. tury, Max Dehn was interested in topological problems about Biographical Note: Martin R. Bridson André Haefliger closed surfaces. He translated Martin R. Bridson was born in the Isle of Man in 1964. these problems into algebraic He was an undergraduate at Hertford College Oxford and questions about the fundamental group and then solved received his PhD from Cornell University in 1991, advised them using the geometry of the action of the fundamental by Karen Vogtmann. He was an assistant professor at Princ- group on the universal cover. Subsequently, Dehn and eton until 1996, with extended leaves spent in Geneva and others used combinatorial properties of group presenta- Oxford. He was a tutorial fellow and professor of topology tions in place of geometric properties of spaces to develop at Oxford (Pembroke College), then professor of pure combinatorial group theory. It was only in the 1980s, mathematics at Imperial College London. Since 2007 he with Gromov’s seminal papers drawing parallels between has been the Whitehead Professor of Pure Mathematics at APRIL 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 563 FROM THE AMS SECRETARY the University of Oxford, where he served as head of the gave us an inspiring vision that melded the metric geometry Mathematical Institute 2015–18. He is now president of of the Russian school with numerous novel ideas that drew the Clay Mathematics Institute. on his unique insights into differential geometry, topol- Bridson’s research interests revolve around the interac- ogy, and group theory. Finitely generated groups, viewed tion of geometry, topology, and group theory. He has been as geometric objects, were at the heart of this vision, and awarded the Whitehead Prize of the London Mathematical the interaction of groups and geometry is correspondingly Society, the Forder Lectureship of the New Zealand Mathe- central to our book. matical Society, and a Royal Society Wolfson Research Merit It was a desire to extend Serre’s theory of graphs-of- Award. He gave an Invited Address to the Joint Mathematics groups to higher dimensions that led to our collaboration. Meetings in 2001 and was an Invited Speaker at the ICM André, who was developing a theory of complexes of in Madrid in 2006. He is a Fellow of the American Math- groups, visited John Stallings in Berkeley in 1989. Stall- ematical Society and was elected a Fellow of the Royal ings, working with Gersten on triangles of groups, was Society in 2016. developing similar ideas. Martin, struggling to understand Gromov’s essay “Hyperbolic Groups” while a graduate Biographical Note: André Haefliger student at Cornell, had resolved a challenge in the geo- André Haefliger was born in Nyon, Switzerland, in 1929. metric foundations of polyhedral geometry that had been He received his PhD from Paris-Sorbonne in 1958; his obstructing the work of both André and Gersten–Stallings. thesis director was Charles Ehresmann, and the president When André learned of this from Stallings, he wrote to of the jury was Henri Cartan. From 1961, he spent two Martin and subsequently arranged a position for him in years at the Institute for Advanced Study in Princeton. With Geneva. It was there in 1992–93 that we decided to write the help of George de Rham, he created the Department our book, naively assuming that we would finish during of Mathematics at the University of Geneva, where he our stay at a chalet in the Swiss mountains in July 1993, remained as Professeur until retiring in 1996. He traveled allowing time for long walks in the afternoons. Our sense widely, visiting universities across Europe, the Americas, of what the book should contain expanded in the years that the Soviet Union, Japan, China, and (many times) India. followed, but as the field expanded we had to accept that He was honored by a Doctorat Honoris Causa from ETH there were many things we could not cover. We sent the Zurich in 1992 and the University of Dijon in 1997. final manuscript to Springer on the first day of spring 1998. His research interests have ranged widely, including: We were at opposite ends of our careers when we em- diverse aspects of the theory of foliations; differentiable barked on this project, and we came with our own tastes, maps—jet spaces, immersions and embeddings, knotting but it was a joy to explore the mathematics together and for high-dimensional spheres; complex analytic structures; to argue until we agreed on how to present each idea. The orbifolds and complexes of groups. For the past fifteen years structure of our profession does not reward the effort of he has concentrated his efforts on the archives of Armand writing a monograph as readily as it rewards theorems Borel (now in Geneva) and René Thom. He also initiated presented in discrete papers published promptly. There the publication of the complete mathematical works of are good reasons for this, but the enduring value of a book René Thom, with critical notes and significant unpublished that gives students and colleagues access to a coherent body documents. of ideas is something to be treasured, and we applaud the American Mathematical Society for recognising that value Response from Martin R. Bridson through the Steele Prize. and André Haefliger We are honoured and delighted to receive the Steele Prize The Steele Prizes are awarded by the AMS Council acting for Mathematical Exposition. We are particularly pleased on the recommendation of a selection committee. The that the Prize Committee commented on the value that members of the Steele Prize Subcommittee for Mathemat- students have found in our book; to see it used widely ical Exposition were: as a textbook has been immensely gratifying. It has also • Charles Fefferman • Alice Guionnet been rewarding to see it serve as a reference for the many • Eric Friedlander • Michael Jordan colleagues who have advanced geometric group theory so (Chair) • Dusa McDuff spectacularly over the past twenty years. • Mark Green • Victor Reiner We wrote, “The purpose of this book is to describe the • Benedict Gross • Thomas Scanlon global properties of complete simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov Citation for Seminal Contribution to Research: and to examine the structure of groups that act on such Craig Tracy and Harold Widom spaces by isometries.” Misha Gromov brought many ideas The 2020 Steele Prize for a Seminal Contribution to Re- from the Alexandrov school to prominence in the West, and search in Analysis/Probability Theory is awarded to Craig his contribution goes far beyond an act of transmission: he Tracy and Harold Widom for the paper “Level-Spacing 564 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 4 FROM THE AMS SECRETARY Distributions and the Airy Biographical Sketch: Craig Tracy Kernel,” published in 1994 in Craig Tracy was born in England on September 9, 1945, the Communications in Mathemati- son of Eileen Arnold, a British subject, and Robert Tracy, cal Physics. an American serving in the U.S. Army. After immigrating to In this work, Tracy and the United States as an infant, Tracy grew up in Missouri, Widom found the exact as- where he attended the University of Missouri at Colum- ymptotics of the nth largest bia, graduating in 1967 as an O. M. Stewart Fellow with a BS degree in physics.