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Paying Tribute to James Eells and Joseph H. Sampson: In Commemoration of the Fiftieth Anniversary of Their Pioneering Work on Harmonic Maps Yuan-Jen Chiang and Andrea Ratto James Eells was born in Cleveland, Ohio, in 1926 and passed away in Cambridge, England, in Feb- ruary 2007. He earned his PhD from Harvard University under the guidance of the topologist and analyst Hassler Whitney in 1954. He worked at the Institute for Advanced Study in Princeton and at the University of California in Berkeley from 1956–1958. He then returned to the East Coast where he accepted a position at Columbia University (1958–1964). He also taught at Churchill College, Cambridge, and Cornell University. Later, attracted by the atmosphere and potential at the University of Warwick, he joined the mathematics department there and became a professor of global analysis and geometry in 1969. Eells organized year-long highly successful Warwick symposia, namely, “Global Analysis” (1971–1972) and “Ge- ometry of the Laplace Operator” (1976–1977). He was the first director of the Mathematical Section of the International Centre for Theoretical Physics in Trieste from 1986 to 1992. As a prominent professor and an inventive mathematician, Eells’s mathematical influence in the field of harmonic ©ICTP Photo Archive. Professor J. Eells (1926–2007). Yuan-Jen Chiang is professor of mathematics at the Uni- versity of Mary Washington, VA. Her email address is [email protected]. Andrea Ratto is professor of mathematics at the University DOI: http://dx.doi.org/10.1090/noti1225 of Cagliari, Italy. His email address is [email protected]. 388 Notices of the AMS Volume 62, Number 4 maps became internationally recognized and wide- asking Joe the question, ‘What is the goal and ob- spread. According to the Mathematics Genealogy jective of your life?’ Joe replied, ‘Life is hostile, not Project, he advised thirty-eight PhDs, who, in turn, evil. Finance is very important, but mathematics is produced 198 descendants. One of Eells’s PhDs most important to me.’ Then I asked, ‘Would you students was Andrea Ratto, who describes his like to be my PhD advisor?’ Joe then asked, ‘Do experience as follows: you really want to study harmonic maps?’ ‘Yes!’ “When I first met Jim at Warwick in 1985, I replied. Then we started a long journey which he gave me a copy of his survey article [9] (with became an ordeal and Joe made me upset several L. Lemaire). He told me, ‘We look for a candidate for times. He was bombastic. At first he wanted me the best map in a given homotopy class.’ This is a to generalize (1) value distribution theory (VDT) fascinating problem, because it combines calculus for holomorphic maps of Riemann surfaces into of variations, differential equations, Riemannian harmonic maps and (2) VDT for harmonic maps of geometry and topology. A special instance concerns complex manifolds. Unfortunately, after approxi- the case of mappings between spheres, and Jim mately two years studying S. S. Chern’s articles and was intrigued by this topic, because in this context investigating other related papers, I realized that solutions are unstable and more difficult to obtain such generalizations would not work. Essentially I and thus less likely to be understandable. A few was able to show Joe that such a generalization had weeks after our first meeting, Jim told me that, an obstruction since the pull-back f ∗ commutes in his opinion, the right topic for my research with @ and @¯ for a holomorphic map, but not for activity could be the development of the beautiful a harmonic map. I remember that his face turned existence theory of R. T. Smith for equivariant gray suddenly, and he said, ‘Mathematics is diffi- harmonic maps of spheres. That was the beginning cult.’ Then we investigated another topic, which of my adventure in mathematics, and all that did not work well either. Finally, he suggested ‘Har- sprang from our conversations always remains, in monic maps of V-manifolds’ for me, and we both my mind, tightly tied to the extraordinary charisma experienced a happy ending. I was so fortunate to and exquisite mathematical taste of J. Eells.” be Joe’s only PhD student to study harmonic maps James Eells retired from the University of with him directly (his other student, E.-B. Lin, went Warwick in 1992. Then he moved to Cambridge to MIT to study geometric quantization and came and continued to work on harmonic maps for the back for defense). I was very pleased to learn the remainder of his life. beautiful tensor techniques from Joe through Eisen- Joseph H. Sampson was born in Philadelphia in hart’s book. I admired his penetrating insight and 1926 and passed away in the South of France in impeccable taste which characterized the precious July 2003. He earned his PhD from Princeton Uni- guidance that he provided over those years.” versity under Salomon Bochner in 1951 (we often For more than four utilize Bochner’s techniques in harmonic maps). decades, Eells and Samp- Sampson then worked as a Moore Instructor at MIT. son were very good He was appointed as visiting assistant professor at friends; both were the Johns Hopkins University in 1955 and then as outstanding mathemati- assistant professor and was promoted to associate cians, brilliant teachers, professor in 1963. He finally became full professor and great experts in in 1965. He was an editor of the American Journal analysis, geometry, and of Mathematics from 1978 to 1992 and the chair topology. Sampson in- of the mathematics department at Johns Hopkins vited Eells to speak on from 1969 to 1979. Sampson retired from Johns harmonic maps at a Hopkins in 1990. He was Yuan-Jen Chiang’s advi- mathematics department sor at Johns Hopkins, where she earned her PhD colloquium at Johns Hop- in 1989. For insights into her interactions with kins around 1985. The Sampson, Chiang provides the following: joint paper of Chiang and “In 1982, when I first met Joe, I had only a Bach- Ratto [5], published in elor of Science degree. The requirements for a PhD the Bulletin of the French included passing three oral examinations (each Mathematical Society in with two professors). For my three exams I chose 1992, was dedicated to real analysis, differential geometry, and algebraic Eells and Sampson. On ©ICTP Photo Archive. topology. When I took Differential Geometry, Joe July 20, 1992, Chiang Professor J. H. Sampson and I had arguments, which caused me to liter- received a letter from (1926–2003). ally cry. Later, however, I recalled the questions Eells: Dear Dr. Chiang, and then realized that he was exceptionally bril- I have just received my copy of Bull. SMF 120 liant. In 1984, and even to this day, I remember and was most surprised and pleased to read a April 2015 Notices of the AMS 389 dedication by you and our excellent friend Andrea. expect that the system evolves towards a configu- What a nice idea! Thank you very much, and all ration of minimum energy (i.e., a harmonic map). my best. Yours cordially, James Eells. That was the Note that this is the first example of a significant style of Jim: genuine, simple, and direct. application of a nonlinear heat flow, where the non- The theory of harmonic maps between Riemann- linearity has a strong geometrical meaning because ian manifolds was first established by Eells and it depends on the curvature of N. The novelty of Sampson [14]. However, the notion of harmonic this approach is that its success is strictly related map was introduced by Sampson in the hope of ob- to the curvature itself (and so, to the topology) of taining a homotopy version of the highly successful the target manifold N. In this sense, it is important Hodge theory for cohomology in 1952. Not long to say that Eells-Sampson’s Theorem was the first after that, his then colleague John Nash (one of the instance where curvature played a basic role in three Nobel laureates in economics in 1994) pro- analysis, a fact which was a crucial milestone for posed a quite different but equivalent definition— global analysis. More specifically, Eells and Samp- both of them were Moore Instructors at MIT at son [14] were able to prove the following assertion: that time. Fuller [15] also came upon harmonic Let N have nonpositive Riemannian curvature and maps in 1954. Later, when both were working at let f : M ! N be a continuously differentiable map. the Institute for Advanced Study at Princeton, Eells Let ft be the solution of the heat equation associ- and Sampson wrote their famous and outstanding @f ated to (1); i.e., τ(ft ) = @t ; which reduces to f at paper “Harmonic mappings of Riemannian man- t = 0. If ft is bounded as t ! 1, then f is homotopic ifolds” [14]. This article is considered as the pio- to a harmonic map f 0 for which E(f 0) ≤ E(f ). In neering work in the theory of harmonic maps and particular, if N is compact, then every continuous has provided the seeds for many further develop- map M ! N is homotopic to a harmonic map. ments. As Eells pointed out in the preface of [7], In order to study geometric and topological “Harmonic maps pervade differential geometry and problems, Hamilton introduced the Ricci flow with mathematical physics: they include geodesics, mini- the aim of attacking the Thurston geometrization mal surfaces, harmonic functions, abelian integrals, conjecture and the Poincar´e conjecture. At the Riemannian fibrations with minimal fibres, holo- 2006 International Congress of Mathematicians morphic maps between Ka¨hler manifolds, chiral in Madrid, he said that his initial inspiration came models, and strings.” in the late 1960s, when he attended the seminars m n A harmonic map f : (M ; g) ! (N ; h) from of Eells and Sampson on harmonic maps, who first an m-dimensional Riemannian manifold into an suggested that one might be able to utilize evolu- n-dimensional Riemannian manifold is defined as tion equations to study the Poincar´e conjecture.

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