William P. Thurston,1946–2012David Gabai and Steve Kerckhoff, Coordinating Editors

William P. Thurston, a geometric visionary and one of the greatest mathematicians ofthe twentieth century, died on August 21, 2012, at the age of sixty-five. This obituaryfor Thurston contains reminiscences by some of his many colleagues and friends. Whatfollows is the first part of the obituary; the second part will appear in the January 2016issue of the Notices.

William Paul Thurston, known univer-sally as Bill, was an extraordinarymathematician whose work andideas revolutionized many fields ofmathematics, including foliations,

Teichmüller theory, automorphisms of surfaces,3-manifold topology, contact structures, hyper-bolic geometry, rational maps, circle packings,incompressible surfaces, and geometrization of3-manifolds.

Bill’s influence extended far beyond his in-credible insights, theorems, and conjectures; hetransformed the way people think about and viewthings. He shared openly his playful, ever curious,near magical and sometimes messy approach tomathematics. Indeed, in his MathOverflow profilehe states, “Mathematics is a process of staring hardenough with enough perseverance at the fog ofmuddle and confusion to eventually break throughto improved clarity. I’m happy when I can admit, atleast to myself, that my thinking is muddled, and I

David Gabai is professor of mathematics at Princeton Uni-versity. His email address is gabai@math.princeton.edu

Steve Kerckhoff is professor of mathematics at Stanford Uni-versity. His email address is spk@math.stanford.edu.

Unless otherwise noted, all photographs are courtesy ofRachel Findley.

For permission to reprint this article, please contact:reprint-permission@ams.org.

DOI: http://dx.doi.org/10.1090/noti1300

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William Paul Thurston

try to overcome the embarrassment that I mightreveal ignorance or confusion. Over the years, thishas helped me develop clarity in some things, but Iremain muddled in many others. I enjoy questionsthat seem honest, even when they admit or revealconfusion, in preference to questions that appeardesigned to project sophistication.”

1318 Notices of the AMS Volume 62, Number 11

Margaret Thurston,with some of hercreations.

His work was notonly densely packedwith wonderful ideas,it was immensely richand deep. Studying anaspect of his work is of-ten like opening a boxto find two or more in-side. Opening each ofthose reveals two ormore boxes that oftenhave little tunnels con-necting to various othersystems of boxes. Withgreat effort one reachesan end, being simulta-neously rewarded withboth great illuminationand the realization that

one knows but a minuscule fraction of the whole.After a sufficient lapse of time, upon retracing theoriginal trail, one catches further insights on topicsthat one once thought one completely understood.

In a similar manner, this article hardly doesjustice to the deep and complicated person thatwas Bill Thurston. It gives but a few facts andvignettes from various dimensions of his life and,at various spots, points readers to other resourceswhere they might explore further.

Following this introduction are thirteen re-membrances from mathematicians connected withdifferent aspects of Thurston’s professional life. Itis difficult not to be overcome by how Bill affectedpeople and institutions in so many different andprofound ways.

FamilyBill’s father, Paul Thurston, had a PhD in physics andworked at Bell Labs doing physics and engineering.He was an expert at building things and was bold,smart, imaginative, and energetic. Once he showedBill how he could boil water with his bare hands.He took an ordinary basement vacuum pump andstarted it above the water so that the boiling pointwas just above the air temperature. Then he stuckhis hands in the water and it started boiling!

Bill’s mother, Margaret (nee Martt), was an expertseamstress who could sew intricate patterns thatwould baffle Paul and Bill. In later years, Bill’sfascination with hyperbolic geometry inspiredher to sew a hyperbolic hat-skirt, a seven-colortorus, and a Klein quartic (genus-3 surface witha symmetry group of order 168) made from 24heptagons that was designed by Bill and his sons,Nathaniel and Dylan. While at Ohio Wesleyan shewanted to be a math major but was told thatwomen don’t major in mathematics.

Bill was named after his father, Paul, andhis mother’s brother, William, who died in ahospital ship in the battle of Iwo Jima. He had an

Bill Thurston, 1952,Wheaton, MD.

older brother, Bob,and sister, Jean, anda younger brother,George, who marriedSarah, mathematicianHassler Whitney’sdaughter. He marriedhis college sweet-heart, Rachel Findley,and they had childrenNathaniel, Dylan, andEmily. He had chil-dren Jade and Liamfrom his second mar-riage with JulianThurston.

ChildhoodBill had a congenitalcase of strabismus and could not focus on an objectwith both eyes, eliminating his depth perception.He had to work hard to reconstruct a three-dimensional image from two two-dimensionalones. Margaret worked with him for hours whenhe was two, looking at special books with colors.His love for patterns dates at least to this time. Asa first-grader he made the decision “to practicevisualization every day.” Asked how he saw infour or five dimensions he said it is the sameas in three dimensions: reconstruct things fromtwo-dimensional projections.

To stop sibling squabbling Paul would ask thekids math questions. While driving, Paul asked Bill(when he was five), “What is 1 + 2 + · · · + 100?”Bill said, “5,000.” Paul said, “Almost right.” Billsaid, “Oh, I filled one square with 1, two squaresfor 2 and all the way up to 100, so that’s half of100× 100 = 10,000, but I forgot that the middlesquares are cut in two, so that’s 5,050!”

Paul was a scoutmaster, and Bill was veryinvolved with the Scouts. They did all sorts ofthings with ropes, e.g., making bridges. Bill wasexpert at making fires in the pouring rain. Thefamily loved camping and took many long trips.Music was a big part of their lives.

New College, FloridaBill was a member of the charter class starting in1964. According to alumnus and mathematicianJohn Smillie, the founders of New College decidedthat it was much easier to maintain quality than tobring it up, so a tremendous effort was made torecruit one hundred of the brightest young peoplefor the inaugural class. This included Bill’s futurewife, Rachel Findley.

New College was a three-year program withclasses eleven months of the year. There wastremendous freedom in how individual academicprograms were structured and how students were

December 2015 Notices of the AMS 1319

allowed to live. At various times Bill lived in a tentin adjacent woods or slept in academic buildings,

Bill Thurston, NewCollege.

playing hide and seek withthe janitor. The library wasminimal, and the college wouldbuy whatever math books Billwanted. Smillie said that nearlyall the books he took out hadBill’s name in them.

In Bill’s words: “I guess I wasdisenchanted with how highschool was and I wanted togo to a place where I’d havesome more freedom to workin my own way.” New Collegegave him that and more. Afterthe first year, half the facultyresigned and the library wason the brink of disaster. ButThurston says that he and manyother students benefited from

the turmoil. “It certainly taught us independenceand how to think for ourselves.”1 According toRachel, Bill could have left after two years for grad-uate school, but he really liked the independencethe school offered and chose to stay.

BerkeleyBill started Berkeley in 1967 in the thick of theVietnam War. “Many of us were involved in studentdemonstrations and student strikes. We weresprayed with tear gas, whether or not we protested.We had friends who were killed, others who refusedinduction and were convicted as felons,….”2

His concern about military activities continuedthroughout his life, particularly as it pertained to hisprofessional life. In 1984 he declined an invitationto become the first Fairchild Professor at PrincetonUniversity because of the donor’s involvement inthe business of military contracting.

Bill was a member of a committee that arguedagainst mathematicians accepting military fundingand that drafted five AMS resolutions related tothis and related issues, each of which receivedmajority votes from members of the AMS.3,4 (Seealso Epstein’s contribution.)

The early years in Berkeley also saw the expan-sion of his and Rachel’s family. Rachel said theyhad a baby (Nathaniel) in part to keep Bill frombeing drafted. She went into labor the night beforehis qualifying exam, which couldn’t be rescheduled.Although Bill passed, it was not a smooth process,

1Home News, Lifestyle section, December 30, 1984.2Military funding in mathematics, Notices of the AMS 34(1987), 39–44.3Commentary on defense spending, Notices of the AMS 35(1988), 35–37.4Referendum results and letter, Notices of the AMS 35(1988), 554, 675.

as he gave answers that befuddled his examinersbut provided hints at the remarkable originality ofhis thinking.

Thurston’s graduate career contained an extra-ordinary collection of mathematical results. Heproved the striking theorem that the Godbillon-Veyinvariant takes on uncountably many values. Thisinvariant comes from a de Rham cohomology classthat he relates to the helical wobble of the leaves ofa foliation. His results show, among other things,that there exist uncountably many noncobordantcodimension-one foliations of the 3-sphere.

Bill’s (unpublished) thesis, written under thedirection of Moe Hirsch, provided a precise descrip-tion of a large class of C2 foliations on 3-manifoldsthat are circle bundles. It also gave a counterexam-ple to a result of S. P. Novikov that a certain partialorder associated to a foliation has a maximum. But,beyond any single theorem, this period providedthe foundation for a collection of ideas that wouldhelp answer many fundamental questions aboutfoliations and would ultimately lead to Thurston’srevolutionary work on surfaces and 3-manifolds.

MIT and the Institute for Advanced StudyBill was a member of the Institute for AdvancedStudy in 1972–73 and an assistant professor at MITin 1973–74. There he continued his extraordinarywork on foliations, for which he won the 1976Veblen Prize (shared with James Simons).5 Inparticular, it was during this period that he andMilnor began their work on the kneading invariantof piecewise monotone maps of the interval. Hisinterest in one-dimensional (real and complex)dynamics would stretch throughout his career,influencing his work on two- and three-dimensionalmanifolds. He would return to the topic once againduring the final years of his life.

PrincetonHe arrived at Princeton as a full professor in1974 and remained for almost twenty years. Therehe did his revolutionary and foundational workon Teichmüller theory, automorphisms of sur-faces, hyperbolic geometry, 3-manifolds, contactstructures, and rational maps.

One of the most striking aspects of this periodwas how Thurston intertwined so many types ofmathematics that had previously been viewed asdisparate and utilized them in shockingly originalways. His classification of mapping classes ofautomorphisms of surfaces utilized ideas fromtopology, hyperbolic geometry, complex analysis,Teichmüller theory, dynamics, and ergodic theory.

This work was followed by an explosion ofresults in three-dimensional topology, using hy-perbolic geometry as a primary tool. These results

5www.ams.org/profession/prizes-awards/pabrowse?purl=veblen-prize

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completely changed the landscape around the sub-ject. The notes from his 1978 hyperbolic geometrycourse (in part written and edited by Bill Floydand Steve Kerckhoff) were sent in installments (byregular mail) to over one thousand mathemati-cians. “The notes immediately circulated all overthe world. It is probably the opinion of all thepeople working in low-dimensional topology thatthe ideas contained in these notes have been themost important and influential ideas ever writtenon the subject.”6

The viewpoint represented by this work wasencapsulated by his geometrization conjecture,which states that all closed, orientable 3-manifoldshave a (classically known) decomposition intopieces, each of which has one of eight homogeneousmetric geometries. Although this was a trulydifferent way of viewing 3-manifolds, the conjecturesubsumed many long-standing conjectures inthe field, including the Poincaré Conjecture, thespherical space form problem, residual finitenessof their fundamental groups, and a classificationof their universal covering spaces. He solved theconjecture for a large class of manifolds (Hakenmanifolds). His 1982 AMS Bulletin article, in whichhe described his view of 3-manifolds, includedtwenty-four problems. Although he did not viewmathematics in terms of a list of problems to besolved, these received great attention and helpedfocus research in the field for the next threedecades. See Otal ([7]) for an account of progresson these problems through 2013.

His Princeton years also included seminal con-tributions to the theory of rational maps of the2-sphere, utilizing his ideas both from Kleiniangroups and from the earlier work on surfaces. Hisinterests in computing and group theory, bothfrom a practical and theoretical point of view, ledto his work (with Cannon, Epstein, and others) onautomatic groups in Word Processing in Groups.The outpouring of ideas from this period wasmonumental and cannot be adequately capturedin this space.

Bill was awarded the 1979 Waterman Prize “Inrecognition of his achievements in introducingrevolutionary new geometrical methods in thetheory of foliations, function theory and top-ology.”7 In 1982 he won the Fields Medal andin 1983 was elected to the National Academy ofSciences.

Generally very generous with his time andideas, Thurston had twenty-nine PhD studentsfrom his Princeton period, who, in turn, as ofthis writing have had 151 finishing students. Inaddition he influenced multitudes of visitors to

6A. Papadopoulos, MR1435975 (97m:57016) review ofThree-Dimensional Geometry and Topology.7www.nsf.gov/od/waterman/waterman_recipients.jsp

both the university and the Institute for AdvancedStudy. He was a social center too, often bringingout his volleyball net for an impromptu game aftertea in Fine Hall.

Beyond the extraordinary scope of his mathe-matical results, Bill was equally influential in theway he thought about and did mathematics. Hethought deeply about the process of doing math-ematics and methods for communicating it. Hisenthusiasm was infectious, and his truly uniquepoint of view provided many new avenues forincluding it in people’s lives.

ComputersBill was one of the first pure mathematicians toactively use computers in his research and was astrong proponent of all aspects of computing inthe mathematical community. In the late 1970s heinspired Jeff Weeks to develop his SnapPea programto compute and visualize hyperbolic structures.This program and a later version, SnapPy (by MarcCuller and Nathan Dunfield), and the related Snap(Goodman, Hodgson, Neumann) are essential toolsfor anyone working in the area. Weeks himselfdiscovered what is now known as the Weeksmanifold with this program. (This manifold wasindependently discovered by Matveev-Fomenkoand Józef Przytycki.)

According to Al Marden, Bill was the intellectualforce behind the creation and activities of theGeometry Supercomputer Project and the GeometryCenter, two NSF-funded programs centered at theUniversity of Minnesota, with Marden as thefounding director. Among many other things,these projects produced the wonderful videosNot Knot and Outside In. Not Knot is a computer-animated tour of hyperbolic space worlds that Billdiscovered, portions of which have appeared inGrateful Dead concerts.8 Outside In is an award-winning video of Bill’s proof of sphere eversion, anamazing result first discovered by Steven Smale in1957.

When a project by Gabai and Meyerhoff neededhigh-level computer expertise, they found it at theGeometry Center. This led to a computer-assistedproof of the log(3)/2-theorem with NathanielThurston (Bill’s son), a brilliant apprentice atthe Geometry Center who made full use of theircomputer resources. This result plays a crucial rolein the proof of the Smale conjecture for hyperbolic3-manifolds and the proof that the Weeks manifold

8Scientific American, October 1993, p. 101.

December 2015 Notices of the AMS 1321

Thurston and Ed Witten, Waterman Ceremony.

is the unique lowest-volume closed orientablehyperbolic 3-manifold.

EducationBill was always interested in education. He andfellow New College students, including RachelFindley, tutored disadvantaged students in math.At Berkeley he was part of a student committeethat helped reform the TA program. He wrote, “Ihelped organize a program for our new teachingassistants, which involves discussion groups, andvisiting of each others’ classes. The group of TAsI observed seemed to me to retain much of theinitial enthusiasm toward teaching which new TAsusually have, but older TAs frequently lose.”9

Each year at the science day program at hiskids’ elementary school class (at a Princeton publicschool), he would teach a “thing or two.” “It’s reallygratifying how open they are and how quickly theypick up things that seem to most adults far outand strange, the kinds of things that adults turnoff as you’re trying to explain it to them.”10

At Princeton, with John Conway and Peter Doyle,he developed the innovative Geometry and theImagination undergraduate course that was latergiven at the Geometry Center and UC Davis.

Thinking About and Doing MathematicsEvery student of mathematics needs to readThurston’s wide-ranging and well-thought-out es-say “On proof and progress in mathematics”.11

There he asks, How do mathematicians advancehuman understanding of mathematics? which hasthe subquestions: How do we understand and com-municate mathematics? What motivates us to do it?What is a proof? He closes with some personal

9Job application letter to Princeton while at Berkeley.10Home News, Lifestyle section, Sunday, December 30,1984.11Bull. Amer. Math. Soc. 30 (1994), 161–177.

experiences from his work on foliations, hyper-bolic geometry, and geometrization. In addition,he discusses how experts transmit new results toother experts and how he, at least, thinks aboutthings. This is far different from the way outsiders,or even many advanced students, think workingmathematicians function.

Variants of ideas expressed in the BAMS articleare given in other venues. He states, “Mathematicsis an art of human understanding.…Mathematicssings when we feel it in our whole brain.…You onlylearn to sing by singing.”12 “The most importantthing about mathematics is how it resides in the hu-man brain. Mathematics is not something we sensedirectly: it lives in our imagination and we sense itonly indirectly. The choices of how it flows in ourbrains are not standard and automatic, and canbe very sensitive to cues and context. Our mindsdepend on many interconnected special-purposebut powerful modules. We allocate everyday tasksto these various modules instinctively and sub-consciously.”13 Here are links to two other essaysalong related lines:

mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213

mathoverflow.net/questions/38639/thinking-and-explaining

Building Things, Models, and ClothingDesignBill enjoyed working with his hands and build-ing things. He had a workshop in his Princetonhome, and once, when guests came and therewere not enough beds, he bought lumber andin an afternoon built a bunk bed. His mother,Margaret, was a master seamstress who sewedmagnificent surfaces that he designed. He inspiredDaina Taimina to crochet magnificent pieces andwrite the exquisite Euler Award-winning CrochetingAdventures with Hyperbolic Planes. With Kelly Delphe developed methods for building nearly smoothmodels by gluing Euclidean discs together. Charac-teristically, in just two days he built a workshopwith a laminator, paper cutter, and riveter, andstarted constructing interesting models, eventuallyprogressing to beautiful no-tape foam constructionmodels that are marvels of engineering.

Fashion designer Dai Fujiwara contactedThurston after reading about his eight geometries.Inspired by Bill, as well as by a multitude ofgeometric materials that Bill provided, including

12From the forward to Crocheting Adventures with Hyper-bolic Planes, by Daina Taimina.13Excerpt from the forward to Teichmüller Theory and Ap-plications to Geometry, Topology, and Dynamics, Volume1: Teichmüller Theory, by John Hubbard.

1322 Notices of the AMS Volume 62, Number 11

a set of links whose 2-fold covers representedorbifolds of all eight geometries, Fujiwara and histeam at Issey Miyake, Inc., designed and created anarray of beautiful new women’s fashions that werepresented at a March 2010 Issey Miyake fashionshow.

Bill had expressed a connection between clothingdesign and manifold theory more than thirty-fiveyears earlier. His 1974 ICM proceedings papercommences with “Given a large supply of somesort of fabric, what kinds of manifolds can be madefrom it, in a way that the patterns match up alongthe seams? This is a very general question, whichhas been studied by diverse means in differentialtopology and differential geometry.” Quoting afront-page Wall Street Journal piece: “Mr. Thurstoncompares this discovery [the eight geometries] tofinding eight apparel outfits that can fit anybodyin the world, just as he hopes to prove someday that the eight geometric categories fit everythree-manifold imaginable.”14

MSRI, Berkeley, and DavisBill moved to Berkeley in 1992, serving as directorof MSRI from 1992 to 1997. He was on the facultyof UC Davis from 1996 to 2003. Carol Wood givesa detailed account of the MSRI years, pointing outthat, during that period, MSRI introduced manyinnovative education and outreach initiatives thatwere revolutionary in their time but are standard forresearch institutes today. At Davis he published hislong-awaited book, Three-Dimensional Geometryand Topology (edited by Silvio Levy), which grewout of a portion of his Princeton lectures andwon the 2005 AMS Book Prize. He also wrote theinfluential monograph Confoliations with YashaEliashberg. At Davis he taught a wide array ofcourses, both for undergraduates and for graduatestudents. During his two-year postdoc at Davis,Ian Agol was both a co-teacher and a student insome of these classes. For family reasons, Bill wasplanning to return to Davis in the fall of 2012.

CornellBill moved to Cornell in 2003. It was around thistime that Grigori Perelman announced a proofof the geometrization conjecture. Although thetechniques were primarily analytic and seeminglyvery different from his own work, Thurston feltthat the proof was fully in keeping with the spiritof his vision, and he was genuinely pleased withits solution.15

In 2011 Bill was diagnosed with a melanoma,and in April he underwent surgery to removea tumor, losing his right eye in the process.

14WSJ, March 18, 1983.15Perelman laudation at 2010 Clay Research Conference.

Bicycle repairs with Dylan and NathanielThurston.

Despite being under arduous medical treatments,he threw himself back into mathematics, attendingconferences, inspiring young people, and provingfundamental results in the theory of rational mapsthat harkened back to his work with Milnor in the1970s. He died on August 21, 2012, surroundedby his family.

In June 2014 a wide-ranging conference, What’sNext? The Mathematical Legacy of Bill Thurston,16

was held at Cornell. It was attended by his mother,brothers, and sister, Rachel and Julian, his children,and other family members, along with about threehundred mathematicians, including many studentsand recent PhDs. It was a true celebration of thetremendous future he left us.

Thurston Resources1) William Thurston at the Mathematics Geneal-

ogy Project:www.genealogy.ams.org/id.php?id=11749

2) Cornell tribute and remembrance page:www.math.cornell.edu/News/2012-2013/thurston.html

3) New College obituary:www.ncf.edu/william-thurston

4) AMS obituary:www.ams.org/news?news_id=1602

5) Mathematics Meets Fashion: Thurston’sConcepts Inspire Designer:www.ams.org/news/ams-news-releases/thurston-miyake

6) AMS Feature column:www.ams.org/samplings/feature-column/fc-2012-10

7) New York Times obituary:www.nytimes.com/2012/08/23/us/william-p-thurston-theoretical-mathematician-dies-at-65.html?_r=2

16www.math.cornell.edu/~thurston/index.php

December 2015 Notices of the AMS 1323

8) The Atlantic obituary:www.theatlantic.com/technology/archive/2012/08/remembering-bill-thurston-mathematician-who-helped-us-understand-the-shape-of-the-universe/261479/

9) AMS 2005 Book Prize citation for Three-Dimensional Geometry and Topology :www.ams.org/notices/200504/comm-book.pdf

10) AMS Steele Prize citation for “Seminal Con-tribution to Research”: www.ams.org/notices/201204/rtx120400563p.pdf

11) W. Thurston and J-P. Bourguignon, Interviewde William Thurston (in English), Gaz. Math. No.65 (1995), 11–18.

12) J. Hubbard, Bill Thurston (1946–2012),European Math. Soc. Newsletter, June 2014, 36–37.

Expository Accounts of Thurston’s Mathe-matical Work[1] W. Browder and W-C. Hsiang, The work of William P.

Thurston, Notices Amer. Math. Soc. 29 (1982), 501.[2] S. Friedl, Thurston’s vision and the virtual fibering

theorem, Jahresber. Deutsch. Math.-Verein. 111 (2014),223–241.

[3] D. Gabai and S. Kerckhoff, Thurston’s Geometriza-tion Conjecture, Clay Mathematics Institute, 2009Annual Report, pp. 32–38.

[4] M. Gromov, Hyperbolic Manifolds (accordingto Thurston and Jørgensen), Bourbaki Seminar,Vol. 1979/80, Springer Lecture Notes in Math., 842,1981, pp. 40–53.

[5] B. Lawson, Thurston’s work on foliations, NoticesAmer. Math. Soc. 26 (1979), 294–295.

[6] D. Margalit, Thurston’s work on Surfaces, BulletinAmer. Math. Soc. 51 (2014), 151–161.

[7] J. P. Otal, William P. Thurston: Three-dimensionalmanifolds, Kleinian groups and hyperbolic geometry,Jahresber. Deutsch. Math.-Verein. 116 (2014), 3–20.

[8] F. Sergeraert, BΓ , Seminaire Bourbaki 524 (1977–78),300–315.

[9] D. Sullivan, The new geometry of Thurston, NoticesAmer. Math. Soc. 26 (1979), 295–296.

[10] W. Thurston, How to see 3-manifolds, Class.Quantum Grav. 15 (1998), 2545–2571.

[11] , Three-dimensional manifolds, Kleinian groupsand hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.)6 (1982), no. 3, 357–381.

[12] C. T. C. Wall, On the Work of W. Thurston, Proceedingsof the International Congress of Mathematicians, vol.1, Warsaw, 1983, pp. 11–14 (PWN, Warsaw, 1984).

André HaefligerI first met Thurston at a conference dedicated tofoliation theory in March 1972 in Les Plans-sur-Bexin the Swiss mountains.

Let me first recall a few results in this areawhich played an important role. In 1969 Bott founda topological obstruction in terms of characteristic

André Haefliger is professor emeritus at the Université deGenève. His email address is Andre.Haefliger@unige.ch.

classes for deforming a plane field into an integrableone, i.e., tangent to a smooth foliation. It is thistheorem that renewed my interest in foliationtheory. In this year Gelfand and Fuks began topublish a series of articles on the cohomologyof the Lie algebra of various vector fields. In aconference which took place in Mont-Aigual (nearMontpellier) in 1969, I sketched the constructionof a classifying space called BΓq for Γq-structures,a notion I introduced in my thesis in 1958.

Here Γq stands for the topological groupoidof germs of smooth local diffeomorphisms ofRq . A Γq-structure on a topological space X canbe defined by an open cover U = {Ui}i∈I and a1-cocycle over U with values in Γq ; i.e., for eachi, j ∈ I a continuous map γij : Ui ∩Uj → Γq is givensuch that

γik(x) = γij(x)γjk(x), ∀x ∈ Ui ∩Uj ∩Uk.

So γii is a continuous map fi : Ui → Rq (identifiedwith the subspace of units of Γq).

Two cocycles γij and γ′ij are equivalent if thereexist continuous maps δi : Ui → Γq such thatγ′ij(x) = δi(x)γij(x)(δj(x))−1 for all x ∈ Ui ∩ Uj .A Γq-structure on X is an equivalence class of1-cocycles after taking the limit over open coversof X. If X is a smooth manifold and if the mapsfi are submersions, then the 1-cocycle γij definesa smooth foliation of codimension q on X. Theconstruction above works as well if Γq is replaced byany topological groupoid, for instance, the groupoidof germs of local analytic diffeomorphisms of Rqor a Lie group G. The homotopy classes of Γq-structures on X are in bijective correspondencewith the homotopy classes of continuous mapsfrom X to the classifying space BΓq .

In 1971 Harold Rosenberg gave a course onfoliation theory, and Thurston was among hisstudents. Harold realized immediately that hewas a brilliant student, and by the end of thecourse they wrote a paper together containing alot of interesting geometric results. Meanwhile,in that year, Godbillon explained in a meeting atOberwolfach the construction of the invariant GVof Godbillon and Vey, but at this time they did notknow if it was trivial or not (but Roussarie, who wasattending the meeting, almost immediately founda nontrivial example). When Lawson came backfrom this meeting he did not know the exampleof Roussarie; he explained the construction of theinvariant to Bill (extract from an email): “I showedhim the definition. The next day he knocked onmy door, and said: ‘The invariant is nontrivial.’The following day he knocked again, and this timesaid: ‘The invariant can assume any real number.’WHAT???!!!! Who is this fellow? I said to myself.”

Very quickly after that, connections of the

1324 Notices of the AMS Volume 62, Number 11

Gelfand-Fuchs cohomology with the cohomologyof various classifying spaces for foliations wereunderstood theoretically and independently bymany people.

Thurston got his PhD in Berkeley in early 1972.His thesis adviser was Moe Hirsch, and BlaineLawson was the examiner. Bill submitted his thesisentitled “Foliations on 3-manifolds which are circlebundles” to Inventiones. The referee suggestedthat the author should give more explanations. Asa consequence, Thurston, who was busy provingmore theorems, decided not to publish it.

Meanwhile, John Mather became independentlyinterested in discovering properties of classifyingspaces for foliations. In 1971 he wrote severalpapers and preprints proving deep theorems onBΓ1. His approach to foliation theory was not asgeometric as Thurston’s, but nevertheless was veryefficient.

When Rosenberg came back to Paris after hisstay in Berkeley, he invited Bill to Paris. When heheard that we organized, at the last moment, themeeting in Les Plans-sur-Bex, Rosenberg broughtBill with him. Also participating in this meeting wereA’Campo, Hector, Herman, Moussu, Siebenmann,Roger, Roussarie, Tischler, Vey, Wood, my studentBanyaga from Rwanda, and many others.

Under the suggestion of Lawson, Milnor invitedBill (as his “assistant”) and me for the academicyear 1972–73 to the IAS. Our two families wereneighbors in the housing project of the institute.We organized a seminar which met once a week.Several people participated in it, and Bill gaveseveral talks, always with new surprising resultsand ideas. In April 1973 I went back to Geneva.

On May 4, 1973, I wrote to William Browder aletter of recommendation for Bill for a position atPrinceton University. In this letter I summarizedthe impressive list of results so far obtained byThurston during this academic year: for instance,the deep connection between the homology of thegroup of diffeomorphisms of Rn with support incompact sets and the homology of BΓn; the fact thatany field of 2-planes on a manifold (in codimension> 1) is homotopic to a smooth integrable one; anobstruction theory for deforming a field ofp-planeson an n-manifold into a smooth integrable one(provided that n− p > 1), as a consequence that afield of p-planes whose normal bundle is trivial canbe deformed to a smooth integrable one; also that incontrast there is no obstruction to deform a field ofp-planes into one tangent to a C0- foliation, etc. Asa conclusion I wrote : “I am very much impressed bythe way he understands mathematics. He has a verydirect and original way of looking at geometricalquestions and an immediate understanding. Hehas sometimes difficulties to fill in the details of a

proof, but after a while you realize the few wordsthat he has written are just the essential ones.”

In a letter dated August 16, 1973, Thurston sentme, for publication in Comment. Math. Helv., themanuscript of his paper “The theory of foliationsof dimension greater than one.” In the same letter,he sent me the preprint of a paper about volume-preserving diffeomorphisms of Tn and said thathe would send me a paper on the construction offoliations on 3-manifolds.

On October 10 he sent me a letter of six pagesfrom Cambridge, Massachusetts, indicating thechanges he made in his manuscript to take accountof my remarks. In the last five pages he explainedto me in detail the proof of his generalization ofthe Reeb stability theorem. In addition, he sentthree pages of corrections of his manuscript forCMH.

On December 7, 1973, he sent me the correctedversion of his manuscript. In a letter of seven pages,he explained his generalization of the classifyingtheorem to codimension one.

During the summer of 1976 a big symposium wasorganized by the University of Warwick. Thurston,Milnor, Vey, Mather, and Hirsch were among theparticipants. With our two boys, at that time verymuch interested in folk music, we drove fromSwitzerland to Warwick, where we stayed for twoweeks. The enclosed picture shows a huge pileof straw built up by Bill and the youngsters. Onthe top were sitting my two boys and the olderdaughter of Milnor; on the side one can see Bill andmany others. Moe Hirsch played a lot of folk musicwith my boys, and Vey beautifully sang traditionalFrench songs.

From August 25 to September 4, 1976, asummer school organized by the CIME took placein Varenna in Italy. Thurston, Mather, and Igave expository courses on foliations. In addition,many participants gave talks, like Sergeraert;Banyaga spoke on his thesis concerning the groupof symplectic diffeomorphisms of a compactsymplectic manifold. This was a generalization ofa nonpublished preprint of Thurston on the groupof volume-preserving diffeomorphisms. VaughnJones, then a student in Geneva, was among theparticipants.

Thurston gave a beautiful course, but he didnot write it up. At that time he was not interestedanymore in foliations, but was planning to teachat Princeton a course on hyperbolic geometry.

December 2015 Notices of the AMS 1325

Photo includes: Rachel Findley (bottom right),Nathaniel Thurston (at Bill’s feet), Dylan

Thurston (peering out at top left).

David Epstein

Remembering BillIn December 1970 I gave a lecture at UCBerkeleyon my theorem that any foliation of a compact3-manifold by circles is a Seifert fibration. Afterthe lecture, Moe Hirsch, Bill’s PhD supervisor,introduced us, and Bill told me rather diffidentlythat he knew how to decompose R3 into roundplanar circles. He further explained that his circlesdid not form a foliation. Forty-three years later,I’m still wondering how this is possible.

I next heard from Bill when he wrote mea long letter (airmail; email did not yet exist)about foliations, classifying spaces, and Haefligerstructures, with lots of hand-drawn diagrams andspectral sequences—quite a bit more than I wasable to digest. Here’s one sentence to give youthe flavor: You’ll see the point if you squint at thisdiagram. I worked for a long time on Bill’s letter,eventually coming up with a counterexample toone of the statements, thinking to myself withrelief that now that I had found the error, I couldstop struggling. However, by return post, Bill fixedthe problem, at which point I had to reconcilemyself to incomprehension.

In an interesting essay, “On proof and progressin mathematics,” Bill distinguishes between the

David Epstein is professor emeritus at the University of War-wick. His email address isDavid.Epstein@warwick.ac.uk.

way he contributed to foliation theory and the wayhe contributed to 3-manifold theory. The huge anddaunting advances he made in foliation theorywere off-putting, and students stopped going intothe area, resulting in an unfortunate prematurearrest in the development of the subject whileit was still in its prime. (If someone writes abook incorporating Bill’s advances, it will takeoff again.) In contrast, Bill’s huge and dauntingcontributions to low-dimensional manifold theorywere buttressed by substantial notes written byBill and his students (not formally published butvery widely circulated) and by notes by others.These fleshed out the infrastructure of the subject,easing the task of those who wished to push Bill’swork further. Later, there was also a beautiful book,Three-Dimensional Geometry and Topology, by Billand Silvio Levy.

I knew Bill well during his marriage with RachelFindley, and my remarks and reminiscences relateto that period, ending around 1993. I used to visitPrinceton quite often, where I stayed in their large,untidy, warmly hospitable university-owned house,just a short walk from Fine Hall. Bill discoveredor invented interesting mathematics all the time,in diverse situations, and there was never time tofully formalize and make public all or even a majorpart of what he thought about. As just one example,he developed probability distributions on the setof mazes made from a grid by marking certainedges as impenetrable walls, as found in children’smagazines, and was delighted by his computerprogram that churned out random mazes of aprescribed level of difficulty.

Talking mathematics to Bill was interesting,inspiring, and frustrating. I often wished that I hada tape recorder: while listening to him, I was sureI understood. But when I tried to reconstruct theconversation, I almost always found difficulties.After some work, I would focus on one particularpoint where I asked urgently for clarification.Instead of answering, he would say, “Maybe youwould like this other proof better,” which he wouldthen explain to me. And the process would repeatitself. On the other hand, reading, understanding,and helping to smooth the rough edges in Bill’snotes was an enthralling and rewarding task fromwhich I learned an enormous amount, and I believethe same was true for his many other helpers.

The difficulties that people had in understandingBill’s mathematics made him very alive to theproblems of mathematical education. His views17

continue to be widely quoted. Even more strikingthan his writings were his experiments into adifferent way of teaching. Al Marden, the directorof the Geometry Center in Minneapolis, had madeit easy for me to visit frequently, so I arrived therein the summer of 1991, to find an atmosphere

17Mathematical Education, arXiv: math/0503081

1326 Notices of the AMS Volume 62, Number 11

of great intellectual ferment during a courseentitled Geometry and the Imagination, givenby Bill, John Conway, Peter Doyle, and JaneGilman. The audience was a mixture of schoolchildren, undergraduates, school teachers, andcollege teachers of mathematics. Some of thematerial for this course is available online,18 whereI strongly recommend any reader interested inmathematics or in mathematical education tospend time.

Some of the school children in the audienceexcitedly explained to me that Peter Doyle hadjust smeared the tires of his bicycle with mud,that enormous blank posters had been spread outon the floor of the Geometry Center, and thatPeter had then cycled around the huge room whilemembers of the audience were excluded. The firsttask of the audience, working in small groups, wasto demolish the reasoning of Sherlock Holmes,quoted in The Adventure of the Priory School : “Themore deeply sunk impression is, of course, thehind wheel, upon which the weight rests. Youperceive several places where it has passed acrossand obliterated the more shallow mark of thefront one. It was undoubtedly heading away fromthe school.” The audience’s second task was tocorrectly deduce from the mud-stained posters thedirection of travel. Peter’s ingenious solution isexplained in “Which way did the bicycle go?” 19 byStan Wagon and co-authors. Stan, from MacalesterCollege, ’round the corner from the GeometryCenter, was a contributor to this course and was, Iam told, in the audience at the time.

Bill arranged many courses like this, at manydifferent levels. Quite a few good mathematicianscurrently in post at universities in the US andaround the world were inspired by one or more ofhis courses.

I cannot omit mention of the two videos, NotKnot and Outside In, in which some of Bill’s ideaswere concretely realized by Geometry Center staff.These winners of major worldwide prizes are stillfreely available for your enjoyment and amaze-ment on YouTube. Each explains significant anddifficult mathematical ideas without recourse toformal mathematics. You can use them in outreachto nonmathematicians, and your undergraduateand graduate students would also benefit. Asyou do so, you will appreciate more deeply Bill’simaginative spirit and the cultural power of con-structive mathematics. (Each of the videos has anassociated booklet to make deeper study moreeasily accessible.)

Bill was a pioneer in the use of computers as atool in pure mathematics, an aspect of his strongly

18www.geom.uiuc.edu/docs/education/institute91/handouts/handouts.html19J. D. E. Kronhauser, D. J. Velleman, and Stan Wagon, MAA,1996.

constructivist point of view. In fact, he wonderedwhether to study logic rather than topology for hisPhD, but was dissuaded by Tarski, who told himthat the logicians at Berkeley were not interestedin intuitionism. Bill suggested to me the use of“hollow exist,” , to denote the result of a noncon-structive existence proof (like Cantor’s proof of theexistence of transcendental numbers). Bill wantedmore than constructivism, something like “rapidconstructivism,” with programs that complete ina reasonable amount of time. Jeff Weeks, one ofBill’s many talented students, developed SnapPea,a program that computes, if it exists, a completehyperbolic structure on a simplicial 3-manifold.This program, based on Bill’s ideas, has been usedin many significant projects. It can be used to givea quick proof that two simplicial 3-manifolds arenot homeomorphic or that two complicated knotsare not equivalent.

During the early 1980s, Bill decided that the FineHall pure mathematicians should be introduced tocomputing. The funding that he raised covered thepurchase of a computer (very expensive at that time)but not the cost of employing a person. I rememberBill going in to Fine Hall over the weekend to himselflay all necessary cables. He also became the localUNIX expert and an intrepid systems programmer.When the very large UNIX editing program vi didn’tbehave correctly, he decided to debug the code.Unsurprisingly, this venture failed. On anotheroccasion, a graduate student invented a processthat continually spawned clones of itself until themachine was full. It was a science fiction scenario: ifone of the clones was killed, one of the other cloneswould immediately construct a replacement. Billwas determined to find a humane method of gettingrid of the clones that would not entail switching offthe machine, and he eventually succeeded wheremost professional systems programmers wouldhave failed. Computer-related activity of this kindtook up a good deal of his time, time that themathematical community might have preferredhim to devote to writing up his mathematicaldiscoveries, but Bill placed a high priority onconverting his colleagues to the use of computers.

Bill and Rachel grew up during the VietnamWar and were strongly opposed to US militaryactivities. Rachel was particularly active. In theearly 1980s, as the usefulness of computers topure mathematicians increased (partly because ofthe spread of computer typesetting by LATEX), theDOD (US Department of Defense) started to fundgrants to pure mathematicians that were hard torefuse because alternative funding for computerswas not readily available. Some of these grantsdispensed with peer review and were funneledthrough the CIA. Mike Shub resigned from NewYork’s City University over the issue and started acampaign against the acceptance of military fundsby academics, in which Bill became involved. Those

December 2015 Notices of the AMS 1327

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Wall painting, Berkeley math department.

with access to the Notices of the AMS for 1987and 1988 will find interesting correspondenceon this issue, with many different points of viewrepresented, including letters from Bill. Eventually,in 1988, the AMS passed a resolution calling fora greater effort to decrease the proportion offunding for mathematics research coming fromthe DoD. At the same time, the AMS declaredskepticism about SDI, Reagan’s Star Wars program.Typically, Bill was not jubilant over the decision hehad fought for and won, and was rather concernedat the dismay of applied mathematicians, who hada tradition of accepting military funds and whodid not agree that this had a dangerous effect onthe values of civil society.

During the period when I knew him well, Billenjoyed life thoroughly. At conferences, and oftenat Fine Hall, he would organize and enthusiasticallyparticipate in games of volleyball. He had a greatsense of humor. When my wife, staying withthe Thurstons, asked about their clothes dryer,he puzzled her briefly by replying that it wassolar-powered. He earned a stiff reprimand fromPrinceton township officials for tapping a finemaple tree on the street outside his home for itssyrup, which he did carefully, with the properequipment. He was a passionately involved father,and I have vivid memories of him with his children.He was an excellent cook who would regularlyproduce both everyday meals and special treats,such as his unique peach ice cream. He lovedgames, picnics, socializing, and talking at a deeperlevel than chit chat. He was a very consideratefriend. Bill was a person of exceptional warmth,whose presence, once experienced, will never beforgotten.

Dennis P. Sullivan

A Decade of Thurston StoriesFirst Story

In December of 1971 a dynamics seminar ended atBerkeley with the solution to a thorny problem inthe plane which had a nice application in dynamics.The solution purported to move N distinct pointsto a second set of “epsilon near” N distinct pointsby a motion which kept the points distinct andmoved while staying always “epsilon prime near.”The senior dynamicists in the front row wereupbeat because the dynamics application up tothen had only been possible in dimensions at leastthree, where this matching problem is obvious bygeneral position, but now the dynamics theoremalso worked in dimension two.

A heavily bearded, long-haired graduate studentin the back of the room stood up and said hethought the algorithm of the proof didn’t work.He, Bill Thurston, went shyly to the blackboardand drew two configurations of about seven pointseach and started applying to these the methodat the end of the lecture. Little paths startedemerging and getting in the way of other emergingpaths, which, to avoid collision, had to get longerand longer. The algorithm didn’t work at all forthis quite involved diagrammatic reason. I hadnever seen such comprehension and such creativeconstruction of a counterexample done so quickly.This combined with my awe at the sheer complexityof the geometry that emerged.

Second Story

A couple of days later, the grad students invitedme (I was also heavily bearded with long hair) topaint math frescoes on the corridor wall separatingtheir offices from the elevator foyer. While millingaround before painting, that same graduate studentcame up to ask, “Do you think this is interestingto paint?” It was a complicated smooth one-dimensional object encircling three points in theplane. I asked, “What is it?” and was astonishedto hear, “It is a simple closed curve.” I said, “Youbet it’s interesting!”. So we proceeded to spendseveral hours painting this curve on the wall. It wasa great learning and bonding experience. For sucha curve to look good it has to be drawn in sectionsof short parallel slightly curved strands (like theflow boxes of a foliation) which are subsequentlysmoothly spliced together. When I asked how hegot such curves, he said, “By successively applyingto a given simple curve a pair of Dehn twists alongintersecting curves.” The “wall curve painting,” twometers high and four meters wide (see AMS Noticescover “Cave Drawings”), dated and signed “DPS and

Dennis P. Sullivan is professor of mathematics at SUNYStony Brook and Einstein Chair, CUNY Graduate Center.His email address is dennis@math.sunysb.edu.

1328 Notices of the AMS Volume 62, Number 11

BT, December, 1971” lasted on that Berkeley wallwith periodic restoration for almost four decadesbefore finally being painted over a few years ago.

Third Story

That week in December 1971 I was visiting Berkeleyfrom MIT to give a series of lectures on differentialforms and the homotopy theory of manifolds. Sincefoliations and differential forms were appearingeverywhere, I thought to use the one-forms thatemerged in my story describing the lower centralseries of the fundamental group to constructfoliations. Leaves of these foliations would covergraphs of maps of the manifold to the nilmanifoldsassociated to all the higher nilpotent quotients ofthe fundamental group. These would generalizeAbel’s map to the torus associated with the firsthomology torus. Being uninitiated in Lie theory,I had asked all the differential geometers at MITand Harvard about this possibility but couldn’tmake myself understood. It was too vague, tooalgebraic. I presented the discussion in my firstlecture at Berkeley, and to Bill privately, withoutmuch hope because of the weird algebra-geometrymixture. However, the next day Bill came up with acomplete solution and a full explanation. For himit was elementary and really only involved actuallyunderstanding the basic geometric meaning of theJacobi relation in the Elie Cartan dd = 0 dual form.

In between the times of the first two storiesabove I had spoken to old friend Moe Hirsch aboutBill Thurston, who was working with Moe and wasfinishing in his fifth year after an apparently slowstart. Moe or someone else told how Bill’s oralexam was a slight problem, because when askedfor an example of the universal cover of a spaceBill chose the surface of genus two and starteddrawing awkward octagons with many (eight)coming together at each vertex. This expositionquickly became an unconvincing mess on theblackboard. I think Bill was the only one in thatexam room who had ever thought about such anontrivial universal cover. Moe then said, “Lately,Bill has started solving thesis-level problems at therate of about one every month.” Some years later, Iheard from Bill that his first child, Nathaniel, didn’tlike to sleep at night, so Bill was sleep-deprived,“walking the floor with Nathaniel” for about a yearof grad school.

That week of math at Berkeley was life-changingfor me. I was very grateful to be able to seriouslyappreciate the Mozart-like phenomenon I had beenobserving, and I had a new friend. Upon returningto MIT after the week in Berkeley, I related my newsto the colleagues there, but I think my enthusiasmwas too intense to be believed: “I have just metthe best graduate student I have ever seen or everexpect to see.” It was arranged for Bill to give atalk at MIT, which evolved into a plan for him tocome to MIT after going first to IAS in Princeton. It

Horocycles.

turned out that he did cometo MIT, for just one year1973–74. (That year I vis-ited IHES, where I ultimatelystayed for twenty-odd years,while Bill was invited back toPrinceton, to the University.)

Fourth StoryIAS Princeton, 1972–73

When I visited the environsof Princeton from MIT in1972–73, I had chances tointeract more with Bill. Oneday, walking outside towardslunch at IAS, I asked Billwhat a horocycle was. Hesaid, “You stay here,” andhe started walking away intothe institute meadow. Aftersome distance he turned and stood still, saying,“You are on the circumference of a circle withme as center.” Then he turned, walked muchfarther away, turned back, and said somethingwhich I couldn’t hear because of the distance.After shouting back and forth to the amusementof the members, we realized he was saying thesame thing, “You are on the circumference of acircle with me as center.” Then he walked evenfarther away, just a small figure in the distanceand certainly out of hearing, whereupon he turned,and started shouting presumably the same thingagain. We got the idea what a horocycle was.

Atiyah asked some of us topologists if we knewif flat vector bundles had a classifying space (he hadconstructed some new characteristic classes forsuch). We knew it existed from Brown’s theorembut didn’t know how to construct it explicitly.The next day, Atiyah said he asked Thurston thisquestion, who did it by what was then a shockingconstruction: take the Lie structure group of thevector bundle as an abstract group with the discretetopology and form its classifying space. Later, Iheard about Thurston drawing Jack Milnor a pictureproving any dynamical pattern for any unimodalmap appears in the quadratic family x → x2 + c.Since I was studying dynamics, I planned to spenda semester with Bill at Princeton to learn aboutthe celebrated Milnor-Thurston universality paperthat resulted from this drawing.

Fifth StoryPrinceton University, Fall 1976

I expected to learn about one-dimensional dynamicsupon arriving in Princeton in September 1976, butThurston had already developed a new theoryof surface transformations. In the first few days,he expounded on this in a wonderful three-hourextemporaneous lecture at the institute. Luckily

December 2015 Notices of the AMS 1329

for me, the main theorem about limiting foliationswas intuitively clear because of the painstakingBerkeley wall curve painting described above. Atthe end of the stay that semester, Bill told mehe believed the mapping torus of these carriedhyperbolic metrics. When I asked why, he toldme he couldn’t explain it to me because I didn’tunderstand enough differential geometry. A fewweeks after I left Princeton, with more time to workwithout my distractions, Bill essentially understoodthe proof of the hyperbolic metric for appropriateHaken manifolds. The mapping torus case tooktwo more years, as discussed below.

During the semester graduate course that Billgave, the graduate students and I learned severalkey ideas:

(1) The quasi-analogue of “hyperbolic geometryat infinity becomes conformal geometry onthe sphere at infinity.” (A notable memoryhere is the feeling that Bill conveyed aboutreally being inside hyperbolic space ratherthan being outside and looking at a particularmodel. For me this made a psychologicaldifference.)

(2) We learned about the intrinsic geometry ofconvex surfaces outside the extreme points.(Bill came into class one day, and, for manyminutes, he rolled a paper contraption he hadmade around and around on the lecturer’stable without saying a word until we felt theflatness.)

(3) We learned about the thick-thin decompo-sition of hyperbolic surfaces. (I rememberhow Bill drew a fifty-meter-long thin partwinding all around the blackboard near thecommon room, and suddenly everything wasclear, including geometric convergence to thepoints of the celebrated DM compactificationof the space of Riemann surfaces.)

During that fall 1976 semester stay at Princeton,Bill and I discussed understanding the Poincaréconjecture by trying to prove a general theoremabout all closed 3-manifolds, based on the ideathat three is a relatively small dimension. Weincluded in our little paper on “Canonical coordi-nates…Commentarii” the sufficient for Poincaréconjecture possibility that all closed 3-manifoldscarried conformally flat coordinates. (However, anundergrad, Bill Goldman, who was often aroundthat fall, disproved this a few years later.) Wedecided to try to spend an academic year togetherin the future.

In the next period, Bill developed limits ofquasi-Fuchsian Kleinian groups and pursued themapping torus hyperbolic structure in Princeton,while I pursued the Ahlfors limit set measureproblem in Paris. After about a year, Bill hadmade substantial positive progress (e.g., closingthe cusp), and I had made substantial negativeprogress (showing all known ergodic methods

coupled with all known Kleinian group informationwere inadequate: there was too much potentialnonlinearity). We met in the Swiss Alps at thePlan-sur-Bex conference and compared notes. Hismapping torus program was positively finished butvery complicated, while my negative informationhad revealed a rigidity result extending Mostow’s,which allowed a considerable simplification ofBill’s fibring proof. (See the Bourbaki report onThurston’s work during the next year.)

Sixth StoryThe Stonybrook Conference, Summer 1978

There was a big conference on Kleinian groups atStonybrook, and Bill was in attendance but not asa speaker. Gromov and I got him to give a lengthyimpromptu talk outside the schedule. It was awonderful trip out into the end of a hyperbolic3-manifold, combined with convex hulls, pleatedsurfaces, and ending laminations…. During thelecture, Gromov leaned over and said watchingBill made him feel like “this field hadn’t officiallystarted yet.”

Seventh StoryColorado, June 1980 to August 1981

Bill and I shared the Stanislaus Ulam VisitingProfessorship at Boulder and ran two seminars:a big one drawing together all the threads forthe full hyperbolic theorem and a smaller one onthe dynamics of Kleinian groups and dynamics ingeneral. All aspects of the hyperbolic proof passedin review with many grad students in attendance.One day in the other seminar, Bill was late. DanRudolph was very energetically explaining in justone hour a new, shorter version of an extremelycomplicated proof. The theorem promoted anorbit equivalence to a conjugacy between twoergodic measure-preserving transformations ifthe discrepancy of the orbit equivalence wascontrolled. The new proof was due to a subset ofthe triumvirate Katznelson, Ornstein, and Weissand was notable because it could be explained inone hour, whereas the first proof took a minicourseto explain. Thurston at last came in and askedme to bring him up to speed, which I did. Thelecture continued to the end with Bill wonderingin loud whispers what the difficulty was and withme shushing him out of respect for the context.Finally, at the end, Bill said, just imagine a bi-infinite string of beads on a wire with finitely manymissing spaces and (illustrated by full sweep of hisextended arm) just slide them all to the left, say.Up to some standard bookkeeping this gave a newproof. Later that day an awe-struck Dan Rudolphsaid to me he never realized before then just howsmart Bill Thurston really was.

1330 Notices of the AMS Volume 62, Number 11

Eighth StoryLa Jolla and Paris, End of Summer 1981

The Colorado experience was very good, relaxingin the Thurston seminar with geometry (one daywe worked out the eight geometries and anotherday we voted on terminology “manifolded” or“orbifold”) and writing several papers of my ownon Hausdorff dimension, dynamics, and measureson dynamical limit sets. Later that summer I wasflying from Paris to La Jolla to give a series of AMSlectures on the dynamics stuff when I changedthe plan and decided instead to try to expose theentire hyperbolic theorem “for the greater good”and as a self-imposed Boulder final exam for me.I managed to come up with a one-page sketchwhile on the plane. There were to be two lectures aday for four or five days. The first day would beokay, I thought: just survey things and then try toimprovise for the rest, but I needed a stroke ofluck. It came big time.

There is a nine-hour time difference betweenCalifornia and Paris, and the first day, awakingaround midnight local time, I went to my assignedoffice to prepare. After a few hours I had generatedmany questions and fewer answers about thehyperbolic argument. I noticed a phone on the deskthat miraculously allowed long-distance calls, and,by then, it was around 4:00 a.m. California timeand 7:00 a.m. in Princeton. I called Bill’s house, andhe answered. I posed my questions. He gave quickresponses, I took notes, and he said call back afterhe dropped the kids at school and got to his office.I gave my objections to his answers around 9:30a.m. his time, and he responded more fully. Weended up with various alternate routes that all inall covered every point. By 8:00 a.m. my time, I hada pair of lectures prepared. The first day went well:lecture, lunch/beach/swim, second lecture, dinner,then goodbye to colleagues and back to bed. Thistook some discipline, but as viewers of the videoswill see, the audience was formidable (Ahlfors, Bott,Chern, Kirby, Siebenmann, Edwards, Rosenberg,Freedman, Yau, Maskit, Kra, Keen, Dodziuk,…).

Bill and I repeated this each day, perfecting theback and forth, so that by 8:00 a.m. Californiatime each day, I had my two lectures prepared,and they were getting the job done. The climaxcame when presenting Bill’s delicious argumentthat controls the length of a geodesic representingthe branching locus of a branched pleated surfaceby the dynamical rate of chaos or entropy createdby the geodesic flow on the intrinsic surface. Oneknows that this is controlled by the area growth ofthe universal cover of the branched surface, whichby negative curvature is controlled by the volumegrowth of the containing hyperbolic 3-space. QED.There was in addition Bill’s beautiful exampleshowing the estimate was qualitatively sharp. Thissplendid level of lecturing was too much forHarold Rosenberg, my astute friend from Paris,

who was in the audience. He came to me afterwardsand asked frustratingly, “Dennis, do you keepThurston locked up in your office upstairs?” Thelectures were taped by Michael Freedman, and Ihave kept my lips sealed until now. The taped(Thurston)-Sullivan lectures are available online.20

Ninth StoryParis, Fall 1981

Bill visited me in Paris, and I bought a comfy sofabed for my home office where he could sleep. Hepolitely asked what would I have talked abouthad I not changed plans for the AMS lectures and,in particular, what had I been doing in detail inColorado beyond the hyperbolic seminar. Therewere about six papers to tell him about. One ofthe most appealing ideas I had learned from him:namely, that the visual Hausdorff-dimensionalmeasure of an appropriate set on the sphere atinfinity, as viewed from a point inside, definesa positive eigenfunction for the hyperbolic 3-space Laplacian with eigenvalue f (2− f ). I startedgoing through the ideas and statements. I madea statement, and he either immediately gave theproof or I gave the idea of my proof. We wentthrough all the theorems in the six papers inone session, with either him or me giving theproof. There was one missing result: that thebottom eigenfunction when f was > 1 would berepresented by a normalized eigenfunction whosesquare integral norm was estimated by the volumeof the convex core. Bill lay back for a momenton his sofa bed, his eyes closed, and immediatelyproved the missing theorem. He produced theestimate by diffusing geodesics transversally andaveraging. Then we went out to walk through Parisfrom Port d’Orleans to Port de Clignancourt. Ofcourse, we spoke so much about mathematicsthat Paris was essentially forgotten, except maybethe simultaneous view of Notre Dame and theConciergerie as we crossed over the Seine.

Tenth StoryPrinceton-Manhattan, 1982–83

I began splitting time between IHES and theCUNY Grad Center, where I started a thirteen-year-long Einstein chair seminar on dynamics andquasi-conformal homeomorphisms (which changedthen to quantum objects in topology), while Billcontinued developing a cadre of young geometersto spread the beautiful ideas of negatively curvedspace. Bill delayed writing a definitive text on thehyperbolic proof in lieu of letting things developalong many opening avenues by his increasinglyinformed cadre of younger/older geometers. Hewanted to avoid in hyperbolic geometry what hadhappened when his basic papers on foliations“tsunamied” the field in the early 1970s. Once,

20www.math.sunysb.edu/Videos/Einstein

December 2015 Notices of the AMS 1331

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Thurston lecturing at “Jackfest”, Banff, 2011.

we planned to meet in Manhattan to discussholomorphic dynamics in one variable and theanalogy with hyperbolic geometry and Kleiniangroups that I had been preoccupied with. We werenot disciplined and began talking about otherthings at the apartment; we finally got aroundto our agenda about thirty minutes before Billhad to leave for his train back to Princeton. Isketched the general analogy: Poincaré limit set,domain of discontinuity, deformations, rigidity,classification, Ahlfors finiteness theorem, the workof Ahlfors-Bers,…to be compared with Julia set,Fatou set, deformations, rigidity, classification,non-wandering domain theorem, the work ofHubbard-Douady…, which he perfectly and quicklyabsorbed until he had to leave for the train. Twoweeks later we heard about his reformulation ofa holomorphic dynamical system as a fixed pointon Teichmüller space, analogous to part of hishyperbolic theorem. There were many new results,including those of Curt McMullen some years later,and the subject of holomorphic dynamics wasraised to another higher level.

Postscript

Bill and I met again at Milnor’s eightieth fest atBanff—Bill’s "Jackfest" lecture is pictured above—after essentially thirty years and picked up wherewe had left off. (I admired his checked green shirtthe second time it appeared, and he presentedit to me the next day.) We promised to tryto attack together a remaining big hole in theKleinian group/holomorphic dynamics dictionary:“the invariant line field conjecture.”It was a good idea, but unfortunately turned outto be impossible.

Applications are invited for:-

Department of MathematicsResearch Assistant Professor(Ref. 1516/036(576)/2)The Department invites applications for a Research Assistant Professorship in all areas of mathematics. Applicants should have (a) a relevant PhD degree; and (b) good potential for research and teaching.Appointment will initially be made on contract basis for up to two years, renewable subject to mutual agreement.Applications will be accepted until the post is fi lled.Salary and Fringe Benefi tsSalary will be highly competitive, commensurate with qualifi cations and experience. The University offers a comprehensive fringe benefi t package, including medical care, plus a contract-end gratuity for an appointment of two years and housing benefi ts for eligible appointee. Further information about the University and the general terms of service for appointments is available at https://www2.per.cuhk.edu.hk/. The terms mentioned herein are for reference only and are subject to revision by the University.Application ProcedureApplication forms are obtainable (a) at https://www2.per.cuhk.edu.hk/, or (b) in person/by mail with a stamped, self-addressed envelope from the Personnel Offi ce, The Chinese University of Hong Kong, Shatin, Hong Kong.Please send the completed application form and/or full curriculum vitae, together with copies of qualifi cation documents, a publication list and/or abstracts of selected published papers, and names, addresses and fax numbers/e-mail addresses of three referees to whom the applicants’ consent has been given for their providing references (unless otherwise specifi ed), to the Personnel Offi ce by post or by fax to (852) 3942 0947.Please quote the reference number and mark ‘Application – Confi dential’ on cover. The Personal Information Collection Statement will be provided upon request.

1332 Notices of the AMS Volume 62, Number 11