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

Yair Minsky Some of my earliest memories of Bill Thurston—hands perched over a table in the Fine Hall common room, fingers moving to indicate the Teichmüller space of the 푛-times- punctured sphere—are filled with his sense of visual intuition and conceptual depth. I began working with Bill as an undergraduate more or less by accident. I had the notion that I could contribute more to mathematics at that point by programming computers, and people from video by Clay Mathematics told me “talk to Thurston. He’s been using computers to draw pictures of fractals.” In the end I did not do a Screenshot programming project; Bill suggested I work out a naive Institute. Courtesy of Martin Bridgeman. version of his theorem for critically finite rational maps, William P. Thurston died on August 21, 2012, at the using only the Brouwer fixed point theorem. As I struggled age of 65. This is the second part of a two-part with this, I began to be exposed to Bill’s style of doing obituary; the first appeared in the December 2015 mathematics, the importance of clear geometric intuition, issue of the Notices. and the beauty of topology. He did not push me hard or insist that I learn a lot of background, but provided a kind of gentle guidance and encouragement that I found from a vast conceptual landscape and the struggle as a very helpful. I remember feeling somewhat cowed by student to obtain a hint of these intuitions. When he fellow students who had mastered a lot of mathematical talked his eyes would be half-closed and a slight smile machinery and receiving a very clear signal from Bill that would be on his face as he described something that he this is not all there is to doing good mathematical work. could see very clearly. After a brief flirtation with computer science graduate Of course, in the end it was all mathematics, not school, I came back to mathematics and back to working mysticism, and the insights came down to proofs, clear with Bill in a more systematic way. His unique style of statements, and sometimes even a rare formula. When I talking about and explaining mathematics affected my didn’t understand a proof, Bill would rarely repeat the whole approach to the subject, for better and worse. Most details. Instead he would often just come up with another inspiring was his insistence on understanding everything proof. This gave a window into a system of interconnected in as intuitive and immediate a way as possible. A clear structure. mental image of a mathematical construction or proof was Bill’s writing had what I came to think of as the worth much more than a formalization or a calculation. “unique correct completion property”. He wrote clearly This worked well for him and for a lot of the mathematics but sparsely, and upon first (or second) reading it was he inspired, but it had its downsides. In his lectures there hard to know how to fill in the details in his arguments. I was a kind of 0–1 law: If you were able to follow the repeatedly had the experience of thinking for a long time images and the structure he was exploring, you obtained and realizing finally that there was a real missing point beautiful insights; but if you got lost, you were left with or condition that needed to be mentioned. Then, upon nothing: there was nothing on the board or in your notes reconsulting the text, I would discover that the point had at the end of the lecture but the scribbled overlay from a been written in there the whole time. dozen geometric arguments. I volunteered one time in Bill’s course to look up the I cannot remember many details from my meetings proof of something; to my detriment I cannot remember with Bill, but I recall the sense that he was drawing insights what it was. I dutifully studied the proof, understood the sequence of calculations, and reported back to the David Gabai is professor of mathematics at Princeton University. class, where I began working through them on the board. His email address is [email protected]. Bill looked on with a pained expression on his face and Steve Kerckhoff is professor of mathematics at Stanford Univer- eventually said, “I didn’t mean a formula.… ” sity. His email address is [email protected]. There was a year that Bill and Dennis Sullivan combined Yair Minsky is professor of mathematics at Yale University. His their groups of graduate students in a joint seminar that email address is [email protected]. alternated between Princeton and the CUNY Graduate For permission to reprint this article, please contact: Center. We learned a lot of mathematics from these [email protected]. seminars and from the conversations that took place DOI: http://dx.doi.org/10.1090/noti1324 around them and on the train rides. Bill’s interaction

January 2016 Notices of the AMS 31 32 and techniques and ideas of full was which course, enables Bill’s do mathematics.” about we clearly more what think whether and understand is success our (N.S.) his Soc., from [ Math. mathematics” sentence in progress summary and a proof “On in essay describe later as would mathematics, practices he Bill how of experienced benefit and the witnessed firsthand I activity. of through forms other and many spread discussions, seminars, ideas courses, notes, mathematical lecture papers, Bill’s and others. 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Thurston’s to because seemed the difficult that said was and Matt conversation to came talking Grayson and After Mr. father Thurston, graduation. with his Princeton 1983 between our encounter at the Thurston Grayson about Matt story attention. of a lack or tells his tracks be your to in seem would you what stop would that statements of strange unfolding quick the groups. automatic observe of to From theory exciting the automata. was compute deterministic it regular to there, finite processing able for using utility was UNIX he expressions the an saying: with group have was with that to he in group computerese what enough this of being just do But inkling knew first. can at I hear “I student to strange his me, his of sort in to was which sitting said grep,” while Bill Once, room, infectious. living was discovery ical attempted, in order and of semblance house, kitchen. some the their keep to at success, limited meals shopped with jointly cooked we which food, in and co-op Matt, Matt for food Rachel, of Princeton, Bill, kind a and at formed nearby, years I lived I two them, last with Dylan, our lived Nathaniel, During children, Emily. their and and lot Rachel a and out hung Bill I generous with and and Grayson Matt gracious time. also personal were their time, with that at wife time his ley, his with generous similarly I ideas. mathematical be years his the to and Over seminar Bill of Poenaru. French found and copy the often Laudenbach, by a Fathi, up his by written me including led been gave topic, had this surfaces, and homeomorphisms on of pseudo-Anosov office work of his his theory seminar. which magnificent to graduate in up the book out in the me blurted learned took I had curves I so He closed things something, simple on say about based to thoughts half-formed ought some I slight a that with with realized panic I me demeanor. at expectant looking and wandering direction, smiling a opposite came the Thurston in Hall, me the Fine towards down new wandering the my while of during day, corridor Thurston older one with when, all the semester at I first from much Meyerhoff. interacted Robert background I not and had of where Gabai David seminar, amount including student students, fair graduate a the mathematics. learned about attended thinking also of just I than way more a imparted but which mathematics, and tricks and theorems ilwsas eeoswt i epkoldeo math- of knowledge deep his with generous also was Bill of because either hard, be could Bill with Conversation mathemat- of stages early the during enthusiasm Bill’s Find- Rachel and Thurston years Princeton my During oue6,Nme 1 Number 63, Volume we had at first thought. We explained our troubles to Bill, who told us about a related but somewhat obscure phenomenon long known to complex analysts: a rigidity property for certain conformal maps of the hyperbolic plane, maps closely related to the quasi-isometries we were vainly attempting to construct. That was just the idea that we needed, and our project flipped over instead to developing a rigidity property for those groups. In our graduate student years Matt Grayson and I would marvel at the mysteriousness of Bill’s ideas, saying to each other, “He’s so weird.” We thought that Bill was obscurely acknowledging this point when, during our third year of graduate school, as part of Bill’s entourage visiting the University of Colorado, a bumper sticker appeared on his blue van with the word “Alien” superimposed over the

mountain silhouette familar from the Colorado license Courtesy of Princeton University. plate. Of course we also figured that it was really just an Thurston with some of his students, 2007. antichauvinistic response to the popular “Native” bumper sticker of the time. The marvel of Bill’s ideas continued to his last years. new ways to understand things. While many excellent His recent article “Entropy in dimension one” contains a mathematicians might understand a complicated situa- wonderful theorem characterizing the entropies of free tion, Bill could look at the same complicated situation group outer automorphisms. He includes an example that continues to boggle my mind of a free group outer and find simplicity. For example, his method for putting automorphism with entropy 3, in sharp contrast to the hyperbolic structures on knot complements is a straight- irrationality of the entropies of surface group outer forward exercise in cut-and-paste topology. When I first automorphisms. saw that method I was struck by its simplicity. My fellow The world is a richer place for Bill Thurston having graduate students and I could have discovered it years been in it. earlier. We had all the tools we needed, and the method was easy, almost obvious. But we didn’t see it. Bill did. Jeff Weeks Bill treated everybody with equal respect. Whether he was Benson Farb talking with the university president or with the janitor, Being a Thurston student was inspiring and frustrating— it didn’t matter; he treated all people with kindness. This often both at once. At our second meeting I told Bill that may or may not have been a direct result of Bill’s Quaker I had decided to work on understanding fundamental beliefs; it was certainly consistent with them. groups of negatively curved manifolds with cusps. In During my years as his graduate student, Bill gave me response I was introduced to the famous “Thurston much mathematical advice. Most of it I have forgotten, squint”, whereby he looked at you, squinted his eyes, and much of it I didn’t fully grasp even then. Yet one piece gave you a puzzled look, then gazed into the distance of advice made an impression on me at the time and has (still with the squint). After two minutes of this he turned stuck with me ever since: “Don’t make arbitrary choices. to me and said, “Oh, I see, it’s like a froth of bubbles, and Do only what you’re forced to do.” In other words, if the bubbles have a bounded amount of interaction.” Being you’re trying to prove a theorem and you find you need to a diligent graduate student, I dutifully wrote down in my make an arbitrary choice, then you’re probably looking at notes: “Froth of bubbles. Bounded interaction.” After our things the wrong way. You should resist the temptation meeting I ran to the library to begin work on the problem. to simply make the choice and push on. Instead, you I looked at the notes. Froth? Bubbles? Is that what he should stop, take a step back, and try to rethink the said? What does that mean? I was stuck. problem in a way that requires no arbitrary choice. That Three agonizing years of work later I solved the advice from Bill has served me well countless times over the years, and I’ve passed it on to all students willing to problem. It’s a lot to explain in detail, but if I were forced listen. It applies not only for proving theorems but also to summarize my thesis in five words or less, I’d go with for designing software: if a particular algorithm requires “Froth of bubbles. Bounded interaction.” you to make an arbitrary choice, it’s time to stop, step A Thurston lecture would typically begin by Bill drawing back, and find a better and more natural algorithm. a genus 4 surface, slowly erasing a hole, adding it back in, Bill’s gift, of course, was his vision, both in the di- futzing with the lines, and generally delaying things while rect sense of seeing geometrical structures that nobody he quickly thought up the lecture he hadn’t prepared. had seen before and in the extended sense of seeing Why did we all still attend? The answer is that once in

Jeff Weeks is a freelance geometer. His email address is jeffams@ Benson Farb is professor of mathematics at the University of geometrygames.org. Chicago. His email address is [email protected].

January 2016 Notices of the AMS 33 a while we would receive a beautiful insight that was gives a geometric, “what it looks like” understanding of absolutely unavailable via any other source. these limits. Here’s an example. Consider a Tinker Toy set of rigid Bill was probably the best geometric thinker in the unit-length rods, bolts, and hinges. Rods can have one history of mathematics. Thus it came as a surprise when end bolted to a table and can be hinged to each other. I found out that he had no stereoscopic vision, that is, For any given Tinker Toy 푇, bolted down on a table no depth perception. Perhaps the latter was responsible at one point, we have the space 퐶(푇) of all possible somehow for the former? I once mentioned this theory configurations of 푇. If 푇 is a single rod, then 퐶(푇) is to Bill. He disagreed with it, claiming that all of his skill a circle. If one adds a hinged rod on the end of 푇, arose from his decision, apparently as a first-grader, to the resulting configuration space is the torus. What other “practice visualizing things” every day. smooth, compact manifolds can you get with this method? I think that there is a fundamental misunderstanding I still remember the communal thrill when Bill explained that many people have about Thurston’s work. In par- to us how to obtain all compact, smooth manifolds as ticular, the completeness of the proofs in his later work a component of some 퐶(푇). Further, every smooth map has sometimes been questioned. Such complaints are not between manifolds can actually be represented via some justified. One can point to Thurston’s occasional lack of rods connecting the two associated Tinker Toys. proper attributions and to some brevity in his mathemat- Thurston completely transformed several areas of ical arguments, but, for the most part, he gave complete, mathematics, including 3-manifold theory, foliation the- albeit concise, proofs. The skepticism seems to stem from ory, geometric group theory, and the theory of rational the frustration one can feel in not understanding what maps. His papers contain a dizzying array of deep, origi- Bill was trying to communicate and the desire for more nal, influential ideas. All of this is well known. However, in detail, only to realize, after understanding things, that my opinion Thurston’s influence is underrated; it goes far the details were there all along. beyond the (enormous) content of his mathematics. As I had an uneven relationship with Bill. However, like Bill wrote in his paper “On proof and progress in mathe- so many other people, my mathematical viewpoint was matics” [Bull. Amer. Math. Soc. (N.S.) 30 (1994), 161–177]: shaped by his way of thinking. In interacting with other “What mathematicians most wanted and needed from me mathematical greats, one gets the feeling that these was to learn my ways of thinking, and not in fact to learn people are like us but just 100 (OK, 500) times better. my proof of the geometrization conjecture for Haken In contrast, Thurston was a singular mind. He was an manifolds.” alien. There is no multiplicative factor here; Thurston We did learn his ways of thinking—or at least some was simply orthogonal to everyone. Mathematics loses a dimension with his death. approximation of them. Bill changed our idea of what it means to “encounter” and “interact with” a mathematical object. The phrase “I understand X” has taken on a whole Danny Calegari new meaning. Mathematical symbols and even pictures I counted Bill as my friend, as well as my mentor, and are not sufficient for true understanding, especially in I have many vivid and happy memories of time I spent geometry and topology. We must strive to live somehow with him. inside the objects we study, to experience them as three- I remember seeing Bill for the first time when I dimensional beings. I think that this change is now almost arrived at Berkeley in 1995. At the start of the academic invisible; it has become a structural feature of the way year, all the incoming graduate students were ushered 1 many of us do mathematics. This kind of pervasive into the colloquium room to meet some of the senior influence can be likened to the way that Grothendieck personnel. Bill was there in his capacity as director of changed the way many people think about mathematics, MSRI (the Mathematical Sciences Research Institute). He even on topics Grothendieck himself never touched. was wearing jeans with big holes at the knees. He made a The change in viewpoint described above was taken speech about MSRI, inviting us all to come up the hill and beyond topology by many of Thurston’s students, who interact with the visitors there. He also encouraged us to went out and “Thurstonized” a number of other areas of pronounce it as “emissary” rather than “misery”. It didn’t mathematics, changing those areas in a notable way. Oded work; we all called it “misery” (and still do). Schramm’s work is a case in point. Early in his career I remember actually taking the bus up the hill (maybe Schramm solved many of the major open problems about a few months later?) in the vague hope of running into circle packings. This theory gives a way to really under- Bill and asking him to be my advisor (people had warned stand (in the Thurstonian sense) the Riemann Mapping me against this, saying that Bill “wasn’t taking students,” Theorem as the limit of an iterative process. Schramm then because he was too busy running MSRI). I don’t think I moved on to apply his geometric insight to understand had a very clear plan about how this was going to work the scaling limit for many two-dimensional lattice models out. I walked in and saw Bill chatting with Richard Kenyon in statistical physics. The Schramm-Loewner evolution about the entropy of dimer tilings and hyperbolic volume. At this point I basically froze, turned around, and walked 1This reminds me of the story of the old fish who passes by two out again. young fish and says, “Morning, boys. How’s the water?” The two young fish look at each other, and one asks the other, “What the Danny Calegari is professor of mathematics at the Univer- heck is water?” sity of Chicago. His email address is [email protected].

34 Notices of the AMS Volume 63, Number 1 I remember Bill running the “very informal foliations book The China Study with Bill while waiting for the seminar” at MSRI with Dave Gabai, Joe Christy, and a cafeteria people to make us our vegan burritos for few other people. This seminar was not advertised; I lunch. Bill’s wife happened to be very sick that week, basically wandered in off the street into the middle of a and in addition they were moving house, so Bill was three-hour lecture by Bill, explaining his new ideas about very distracted. When I left at the end of my visit, Bill universal circles and how they might be used to approach apologized for being distracted with so many other things the geometrization conjecture for 3-manifolds with taut but hoped that I’d visit again soon. Of course I told him foliations. By the time he was done, I had decided I wanted not to apologize, that I’d had a great visit (which was to work on foliations, and I more or less had my thesis true), and that I hoped I would come again soon when we problem. both had more free time. That was the last time I saw him. I remember when Bill moved to Davis. This was the only time I ever saw him in his office at Berkeley, when he Ian Agol was cleaning it out. I remember the little photo that used to be on the door—the one that’s on the cover of More My first encounter with Thurston was during a graduate Mathematical People [Harcourt Brace Jovanovich, Boston, student workshop at MSRI. He demonstrated to us how 1990]—of Bill as a child working at a desk. He saw me he could count in binary on his fingers and told us how watching him carrying his boxes out of his office and many steps we had taken on the hike to the picnic, which looking at the photo and gave a slightly embarrassed he had counted out using his method while hiking. Later smile. during the workshop, when I expressed my interest to I remember emailing Bill in early 1998 to explain a few him in finding a solution to the unknotting problem using of my tentative ideas about foliations, which had been grid diagrams, he immediately dismissed my approach inspired by his slitherings paper. He invited me to come and took me up to the computer lab to demonstrate out to visit him at Davis and talk to him in person. Over the program SnapPea (written by his former student Jeff the next year or so I drove out there perhaps a couple Weeks and others). This made an impression on me, since of times per month, struggling up the freeway in my his geometric approach clearly gave a more powerful third-hand lemon, with the wind rushing in through the way to study knots than the intrinsic three-dimensional bad seals in the door frame. We would have conversations route I was taking (it’s interesting to note, however, that that lasted for hours, stopping occasionally for lunch and grid diagrams have become an important tool recently coffee. Bill basically became my “unofficial advisor” (my in the study of knots and their invariants). Moreover, official advisor, Andrew Casson, was moving to Yale), and SnapPea enabled Thurston to take a deep mathematical perhaps because he did not have many “real” students at construction (namely, his proof of the geometrization Davis at the time, I got a lot of his attention. We spent conjecture for Haken manifolds) and make it into a very a lot of time working through the theory of universal concrete output that could be appreciated by nonexperts. circles; I learned a huge amount of mathematics, not I learned later that his approach to mathematics was only stuff obviously connected to foliations (or even low- to start with a simple model where things could be dimensional topology) but combinatorics, analysis, group understood very explicitly, and once this model was theory, and so on. And yet, Bill listened very carefully understood well, it would help him understand the more to my ideas and always gave them his full attention and general case. For example, he told me that he re-proved consideration. At the time I don’t think I appreciated how Andreev’s theorem for Haken reflection groups using rare this attitude is in a senior mathematician towards a the techniques that would generalize to his proof of graduate student. the geometrization conjecture for Haken manifolds. This I remember when we were trying to work out the details principle has guided my own research, where I always of some construction, Bill got very enthusiastic, and we ask myself when first considering a problem, “What is the went to the campus store to buy some enormous sheets simplest nontrivial special case of a given problem?” of paper and a few sets of colored pencils, bringing them My first meeting with Thurston at Davis was a bit all back to Bill’s office and laying the paper out on the of déja vu: I began to explain a result from my thesis floor. Bill was really excited by this episode; he remarked about volumes of Haken hyperbolic 3-manifolds using that he used to do this sort of thing “all the time” when the Gromov norm, and he immediately began thinking of he was at Princeton. I got the impression he hadn’t done an alternative approach using the new technology devel- it for a while. oped by Besson-Courtois-Gallot. Although this approach I remember working to try to get a project finished in didn’t work initially, it eventually led to a collaboration the week before Bill’s daughter was born (we didn’t make several years later when the more powerful techniques of it in time). My wife and I were thinking about having kids Perelman became available. at the time, and I shyly asked him about the experience. I sat in on a graduate seminar run by Thurston He became very emotional and tender and talked about each quarter called Experimental Mathematics, where he what it was like to hold a newborn and have them lie in would ask for questions that interested the students your arms, trusting you completely. and then would attempt to investigate the problem I remember visiting Bill in the winter of 2008. At the time my family and I were on a vegan kick, and Ian Agol is professor of mathematics at the University of Califor- I remember discussing veganism and Colin Campbell’s nia, Berkeley. His email address is [email protected].

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walking around with a bag full of perfectly round key of Carol Wood. rings. But I still didn’t know what I was supposed to DO with them. I asked Joel Hass for some mathematical direction one time, to which he replied, “Do whatever it Courtesy is that Bill tells you to do!” Great circle links did turn Thurston and Bob Osserman at MSRI, 1997. out to be beautiful examples of several phenomena, but I Sculpture by Helaman Ferguson. didn’t figure that out until later. They are also just plain fascinating, as are so many of his suggestions. Years later I went to Cornell to give a talk and had director, 1992–97, although anyone involved with MSRI the opportunity to meet with Bill Thurston again. He was in that era would know that the genius touch for public very ill and seemed tired as we talked in his small house events was Bob Osserman’s. one afternoon by the fire. Bill was wearing a big wool Under Bill’s leadership, MSRI’s reach broadened dra- sweater that dwarfed him, and he could barely speak. But matically. Here are some examples: Bill asked what I was doing mathematically, and I wanted • Mathematical Conversations, in which mathemat- to talk about it. So I asked him about the space of acute ics teachers and researchers meet as equals, as triangulations of the 2-sphere that are combinatorially well as workshops and conferences about math- equivalent to some specific triangulation. In particular, is ematics education, such as one on the “calculus this space connected? We talked about it, and he turned wars”. the problem over in his head, thinking about it from • Programs for students and young researchers, different points of view. He thought it was interesting, including introductory workshops and the admis- and at one point he nodded and said, “I think it probably sion of graduate students to research programs. is.” (In 1989–90 we had to smuggle students into the building!) Carol Wood • Public events, most famously the Fermat Fest in San Francisco in October 1993, but several other Bigger than life and smarter than I could have imagined, events aimed at bringing mathematically relevant Bill Thurston brought a generation of mathematicians to topics to the general public. see things that they might otherwise have missed: math- • Introduction of the latest forms of technology ematical things, of course, but not only. In remembering and communication, including video streaming, Bill, I would like to point to things he saw and did at MSRI, Mbone, and Elmo projection. I recall a demonstra- resulting in changes in the fabric of the institution. It has been my privilege over time to meet all the MSRI tion of 3D printing in 1997, something only now directors and their teams. Each has put an imprint on poised for widespread use. MSRI, and each has worked hard and smart, to the benefit • Promoting diversity, with the establishment of of mathematics and mathematicians. My perspective is the Human Resources Advisory Committee, by that of someone who was a member in 1989–90, a member hosting the first-ever CAARMS (Conference for of Bill’s team in 1996–97, and, subsequently, a program African-American Researchers in the Mathemati- organizer and trustee. Bill was director of MSRI from cal Sciences), by coorganizing the Julia Robinson 1992 to 1997; his deputies were Bob Osserman, Lenore Conference with the Association for Women in Blum, T. Y. Lam, and, for the final year, me. Bill thought Mathematics. deeply about things. He formulated a vision of the role • Opening sponsorship to departments beyond the of MSRI and of himself as director, some of which was eight in the original proposal, so that there were articulated in a document called “Possibilities”. I cannot thirty in 1997. (At present there are over ninety assign individual credit to each event of Bill’s years as sponsors.) One change, a cosmetic one, was to trade the MSRI Carol Wood is professor of mathematics at Wesleyan University. nickname “misery”, with its negative connotations, for Her email address is [email protected]. “emissary”, to indicate the outward-looking institute it

January 2016 Notices of the AMS 37 was to become. Ironically, this new nickname did not than Bill Thurston. For his leadership at MSRI, as well as take except as the name of the newsletter: some of the for his mathematics, the mathematics community owes old guard persist with “misery”, but most often the four him an enormous debt of gratitude. initials are articulated, in the mouthful “em-ess-are-eye”. I admit I never cottoned to either nickname, so I’m happy with the end result. Tan Lei Most, if not all, of the changes Bill achieved may seem I was visiting John Hubbard at Cornell during the winter less original or revolutionary today, but they were sea of 1986. His student Ben Wittner and I were among changes in the mathematical culture of the day, all the the first generation of students exploring Thurston’s more so since they took place in an institution devoted newly found characterization theorem for rational maps. to the best of mathematics, and these changes put MSRI This powerful theorem gives a necessary and sufficient well ahead of the curve. Bill’s initiatives were developed in condition for a flexible object, dynamics on a branched subsequent years under the leadership of David Eisenbud covering, to represent a rigid object, the dynamics of a and Robert Bryant, who added their own ideas as well. rational map. It is one of the fundamental theorems in Today, mathematics students, educators, departments, the theory of iterations of rational maps. and researchers worldwide claim ownership of MSRI. The Ben and I decided to make a one-day trip to Princeton visionary Bill Thurston had indeed thrown open the doors, to meet Bill Thurston in person. That was the first and good luck to anyone who tries to pull them shut again! time I met Bill. I was very impressed by his highly In 1996–97, Bill’s final year as director of MSRI, I joined animated seminar, scheduled at lunchtime, full of young the team as deputy director, a job offered to me out of people eating burgers and drinking cokes while discussing the blue. This was not the first time since we met in the mathematics around him. early ‘90s that Bill had proposed a role for me that I could Many years later I mentioned this to Bill, and he said barely imagine for myself, but it was the biggest role by that a lunchtime seminar was not a common practice, far. In conversations with others that spring about the but they were trying to get a more informal atmosphere job, I heard complaints, grumbles, even predictions of to lower people’s defenses so that they could discuss disaster for MSRI. I trusted Bill’s judgment, buoyed by more naturally what they were actually thinking. Bill also Alberto Grunbaum’s saying he too thought I could do the expressed that he was spoiled in Princeton, where he job. When I arrived at MSRI, Bill greeted me, as he did was surrounded by a big group of graduate students all visitors at that time, by displaying his handmade tape and junior mathematicians, enabling him to take the model showing the great similarities between human and role of queen bee. I think that around this time he was pig DNA. I witnessed then and many times thereafter his supervising several PhD students simultaneously. childlike capacity for sheer delight. That was my first hint Despite his fundamental role in the theory of holo- that this daunting job was also going to be fun. morphic dynamics, Bill Thurston rarely participated in Bill was the kindest, least pretentious, and most affable the activities of the community, probably because he person I have ever worked alongside. Moreover, he had was totally absorbed by his other interests (and what assembled a staff remarkably talented and dedicated to interests!). Most of the newcomers in the field did not the concept of an institute which served its community. have the chance to meet him in person. My role was straightforward: to make timely decisions so Luckily for us, Bill renewed his interest in rational that others could do their jobs. Bill was untidy and Bill dynamics during the last two or three years of his life was a perfectionist, a combination that sometimes chal- and started to participate in our conferences. The reason lenged this practical, not-so-visionary deputy. However, for this comeback was explained in his talk at the Banff during what he might easily have considered his “annus conference in honor of John Milnor’s eightieth birthday in horribilis”, I never heard a complaint from Bill nor an February 2011: he was led by his investigation of realizing unkind word from him about anyone. (I cannot say the Perron numbers as various dynamical growth factors. A same for me.) Seeing that the MSRI train ran on time video record of this talk can be viewed on the conference was—and still is—a 24-7 job. But I regretted then, and webpage.2 regret all the more with his untimely death, that I didn’t This video, as well as many other video records of stop more often just to enjoy being around Bill. When I Bill’s talks, illustrates very well Bill’s working style: he did, it was delightful. definitely preferred geometrical visions to formulae, he The last time I saw Bill was in Boston at the mathematics used computers intensively to do calculations and illus- meetings in January 2012. He spoke of his plan to visit trations for him, and in his descriptions of mathematical Berkeley to be near family. He intended to come to objects he often put himself inside, like the moment MSRI that year for the first time since 1997. I know he was talking about projective space and looking up Robert Bryant shares my regret that fate intervened. Bill’s towards the line at infinity. Later on I experienced many remarks in accepting the Steele Prize were poignant: he face-to-face and email conversations with him in that expressed gratitude to the mathematics community for style. I must say it wasn’t always easy (actually quite often its acceptance of him and his ideas. In our finest moments, mathematicians are tolerant of each other’s quirks, a fact Tan Lei is professor of mathematics at the Université d‘Angers. about which I have always felt proud. No one achieved this His email address is [email protected]. acceptance of others more effectively and more widely 2www.math.sunysb.edu/jackfest/Videos/Thurston/

38 Notices of the AMS Volume 63, Number 1 difficult) to understand him: either I would have the right cubic polynomial with no multiple roots? This idea put intuition and get it all or I would get nothing. Sometimes him on the right track. He soon saw that the construction I preferred email conversations, which would allow me could easily be made, be reversed, and even worked for to slowly build up correct intuitions represented by his any degree. Here is what he wrote on April 1: descriptive words. And once you realized what he meant, Take any degree d polynomial P with no multiple zeros, the clouds would disappear and things became just clear and look at log(P), thought of as a map from ℂ\{푟표표푡푠} to and beautiful. an infinite cylinder. For each critical point, draw the two Bill’s geometric insight was truly amazing. Even a separatrices going upward (i.e., this is the curve through classical result such as the Gauss-Lucas Theorem, when it each critical point of P that maps by P to a vertical half-line passed through Bill’s hands, gained immediately a deeper on the cylinder, a ray in ℂ pointed opposite the direction geometric meaning that I never saw before. Later on, to the origin). Arnaud Chéritat and I wrote a tribute to Thurston (on the Then make the finite lamination in a disk that joins French website Images des mathématiques) illustrating the ending angles of these separatrices. This is a degree-d his point of view of this theorem. major set. Conversely, for each major set, there is a con- Here are some of Bill’s own words about his being a tractible family of polynomials whose separatrices end at geometer: the corresponding pairs of angles. I’ve always taken a “lazy” attitude toward calculations. To pick a canonical representative of each of these fam- I’ve often ended up spending an inordinate amount of time ilies: look at polynomials whose critical values are on the trying to figure out an easy way to see something, prefer- unit circle. This forms a spine for the complement of the ably to see it in my head without having to write down discriminant locus for degree d polynomials. a long chain of reasoning. I became convinced early on This is it! A beautiful theorem is born; a surprising that it can make a huge difference to find ways to take a bridge is built. step-by-step proof or description and find a way to par- Bill was obviously very proud of this discovery, as we allelize it, to see it all together all at once—but it often can see by how he presented it one year later.3 takes a lot of struggle to be able to do that. I think it’s It turns out that the set of all “critical” degree d poly- much more common for people to approach long case-by- nomials can be approximately described by collections of case and step-by-step proofs and computations as tedious arcs of the disk whose endpoints have angle between them but necessary work, rather than something to figure out a of the form 푘/푑. It took me a while to realize that the set way to avoid. By now, I’ve found lots of “big picture” ways of all such arcs, along with the limiting cases where some to look at the things I understand, so it’s not as hard. endpoints coincide and there are additional implicit arcs, Another deep conviction of Bill Thurston’s was that all describe[s] a spine for sets of 푑 disjoint points in ℂ, that mathematics is connected: The more you make connec- is, its fundamental group is the braid group and higher tions, the more you see things as interconnected and the homotopy groups are trivial. more you expect these connections. People who had the chance to meet Bill Thurston I once witnessed his exceptional connecting skills in know that he was a very caring and friendly person. You action, a truly remarkable experience. In March 2011, Bill could feel his respect for others and his keenness to was investigating the space of iterated cubic polynomials. make everybody comfortable. I remember once, over a Each such polynomial can be combinatorially described conference dinner table, he asked us one by one, with a by a finite lamination in the unit disc called “a primitive gentle smile and with full attention, “So how did you come cubic major”: it is either an equilateral triangle inscribed to mathematics?” One of us, Pascale Roesch, said that she in the circle or a pair of chords that each cuts off a initially hesitated between psychology and mathematics, segment of angle-length 2휋/3. While trying to visualize and that response provoked a vivid conversation around the space of such laminations, he recognized a familiar the table for a long while. pattern. Here is how he described the discovery on the At some point Bill was invited to a brain study about 26th of March: creative people. He got his brain tested, scanned, and This figure can be embedded in 푆3, which should some- modeled. During one of my visits to Cornell, while I was how connect to the parameter space picture… . This (fig- deeply absorbed by some hard thinking, he checked his ure) is also a spine for the complement of the discriminant email and said, out of nowhere, “My brain will arrive locus for cubic polynomials, but I’m not sure how that de- tomorrow.” After a long moment of total confusion, I scription fits in. finally realized that he was only talking about the clay Having sensed a connection between the new object model of his brain which was sent to him. and the old familiar one, he obviously set off in search of Later on Bill showed off proudly his “brain” to every- a genuine link. The complement of the discriminant locus body, and I must say it provoked quite a sensation in me, consists of polynomials with no multiple roots. However, holding in my hands this exceptional “brain”. it is not obvious at all how such a polynomial might Sadly, we have since then lost this brain, leader of appear from a pair of disc chords, but they should be thoughts, and the many theorems he was about to prove. connected somehow. At some moment Bill decided to try But, above all, we have lost a very dear friend. Bill, William the opposite direction: why not try to put oneself in the Paul Thurston, will always be missed. familiar object and try to reach the new one from there? Why not try first to construct a pair of chords from a 3Topology Festival, Cornell, May 2012, from Kathryn Lindsey.

January 2016 Notices of the AMS 39 Curtis McMullen Thurston at Princeton While a professor at Princeton, Thurston served as my Simple Curves NSF postdoctoral supervisor, 1987–90. As I soon realized, Thurston actually had an uncanny ability to turn his Thurston was a master at finding fresh and novel ways of insights into transparent logical proofs which integrated, looking at things. rather than disguised, the underlying geometric intuition. What could be simpler than a loop on a surface? The audience for Thurston’s course at Princeton in- But in Thurston’s hands, the collection of all simple cluded Peter Doyle and me in the front row, a coterie of loops (once completed) became a geometric object in graduate students, and a back row of visitors from the its own right—the space ℙℳℒ of projective measured IAS. During my first year he lectured on his geometriza- laminations. It now plays a central role as the boundary tion theorem for Haken 3-manifolds, a tour de force with of Teichmüller space and stands as one of Thurston’s another iteration on Teichmüller space as its engine: the most widely used inventions/discoveries (anticipated in famous skinning map. the work of Nielsen). Yet it emerges from elementary The course began with a discussion of the boundary topology as directly as the real numbers emerge from ℚ. of the convex hull of the limit set of a Kleinian group: The accompanying figure, taken from his Princeton it is a hyperbolic surface bent along a measured lamina- notes The Geometry and Topology of 3-Manifolds (1979), tion. Similar-looking pleated surfaces interpolate between gives part of the construction of stereographic coordinates the faces of the convex core, and the area of each sur- on ℙℳℒ, showing this space is a sphere (of dimension face is controlled by its Euler characteristic. These ideas 6푔 − 7 for a surface of genus 푔 > 1). then flowed into a series of compactness results (ap- proximate Mostow rigidity) which underpin convergence of the skinning iteration: boundedness yields uniform contraction. When asked a question, Thurston would usually fix his gaze on the middle-distance as if to grasp some private, kinesthetic-geometric model. He almost always worked things out live, on the spot, especially in class. His ideas seemed to come out of nowhere. It was as if Thurston had started off on a different track at an early age and had looked at everything since then with fresh eyes; to fully appreciate his work might require a complete reeducation. To explain the idea of orbifolds, Thurston once brought Figure 1. two mirrors and a toy Smurf to a kindergarten class. By changing the angle between mirrors, the children could see first three, and then four, and then more copies View from Harvard, 1980s of the original stuffed figure. He then added a third I first heard about Thurston’s work when I was a graduate mirror, making a triangle. The children crowded around student at Harvard, 1981–85. He had recently discovered and peered into it from above, to see an infinite Smurf 2 a characterization of the branched covers of 푆 that arise universe, with the figures repeating at different angles in the dynamics of rational maps, but no detailed write- forever. up was available. In this exciting atmosphere, Hubbard On another occasion the course started with a dis- (who was visiting for a semester) lectured to a small cussion of the “pentagon problem” from the 1986 high 5 group on the audacious idea of the proof: an iterative school math Olympiad. Numbers (푥푖)푖=1 are assigned to scheme on Teichmüller space whose fixed point would the vertices of a pentagon, with ∑ 푥푖 > 0. A move consists give the desired rational map. In those days Thurston’s of picking a vertex with 푥푖 < 0, changing its sign, and then mimeographed Princeton notes, whose typed pages were subtracting the same amount from its two neighbors. Will covered with drawings that sometimes crossed the text this process eventually make all the numbers positive? itself, sat on the library reserve shelf next to more Soon the blackboard was covered with cone manifolds forbidding texts that assumed background in schemes, and polygons and butterfly moves. Although he didn’t 퐿-functions, symbols of operators, etc., and were often mention it in class, by the end of the semester Thurston had connected this Olympiad problem to the work of unencumbered by pictures, intuitions, or even examples. Picard and Deligne–Mostow on hypergeometric functions One sometimes heard that Thurston’s notes were full and rediscovered their constructions of nonarithmetic of “great ideas,” with a hint that rigorous arguments lattices in PU(1, 푛), 푛 > 1. were missing and the work might be impounded for mathematical misdemeanor. Thurston at MSRI Figure 2 is taken from “The theory of foliations of Curtis McMullen is professor of mathematics at Harvard Uni- codimension greater than one” [Comment Math. Helv. 49 versity. His email address is [email protected]. (1974), 214–231]. Thurston’s work in this area is an

40 Notices of the AMS Volume 63, Number 1 example of the power of the ℎ-principle: by exploiting the flexibility of smooth constructions with sufficient imagination, one can construct a foliation that realizes any given homotopy data. (The same methods can be used for sphere eversion, as illustrated in his movie Outside In.) While director at MSRI (1992–97), Thurston once lec- tured on a special case: any manifold with zero Euler characteristic admits a smooth codimension-one folia- tion. The lecture involved parking garages and ramps running up and down, as well as a preliminary triangula- tion of the manifold. Bott was in the audience, somewhat dismayed by these hands-on constructions. He asked at the end: Can’t one do this sort of thing using an evolution equation from differential geometry that would gradually deform a field of tangent hyperplanes until it becomes integrable? Thurston’s answer was immediate: The solu- tion to a parabolic equation would be real-analytic, but it is well known (Haefliger) that real-analytic foliations are much harder to construct than smooth ones (e.g., there are none of codimension one on 푆3).

Thurston in Banff The last illustration (Figure 3) comes from Thurston’s Figure 2. Opening the window. final paper, “Entropy in dimension one” [Princeton Math. Soc., 51, Princeton Univ. Press, 2014, pp. 339–384]; it shows the Galois conjugates in ℂ of the expansion factors for critically finite quadratic maps of the interval. These expansion factors are simply the values of 휆 > 1 such that 푥 = 0 has a finite forward orbit under the tent map 푥 ↦ 휆(1 − |푥|). This paper, which returns to the work begun by Milnor and Thurston in the 1970s, characterizes the entropies of multimodal maps and free-group automorphisms. The latter, when realized on train tracks, also become one- dimensional dynamical systems, and the essential unity between these two subjects and the mapping-class group of a surface emerges. Thurston spoke on this work at a conference in Banff in honor of Milnor in 2011. By that time he had been Figure 3. energetically networking, via email and dropbox, with a group of younger mathematicians from around the world, many of whom were present. After Banff, and more than thirty years after his mimeographed notes had arrived, graduate students and postdocs at Harvard were holding a weekly reading group on this new paper by Thurston. Bill’s radical way of looking at things continues to shape mathematics as much as his deepest theorems.

January 2016 Notices of the AMS 41