1996 Oswald Veblen Prize

comm-veblen.qxp 4/22/98 8:23 AM Page 325 1996 Oswald Veblen Prize The 1996 Oswald Veblen Prize in Geometry was upon receiving the prizes, and a brief bio- awarded at the Joint Mathematics Meetings in Or- graphical sketch of each recipient. lando in January 1996 to Richard Hamilton of the University of California, San Diego, and to Richard Hamilton Gang Tian of the Massachusetts Institute of Technology. Citation Oswald Veblen (1880–1960), who served as Richard Hamilton is cited for his continuing president of the Society in 1923 and 1924, was study of the Ricci flow and related parabolic well known for his mathematical work in geom- equations for a Riemannian metric and he is etry and topology. In 1961 the trustees of the So- cited in particular for his analysis of the singu- ciety established a fund in memory of Professor larities which develop along these flows. Veblen, contributed originally by former stu- The Ricci flow equations were introduced to dents and colleagues and later doubled by his geometers by Hamilton in 1982 (“Three mani- widow. Since 1964 the fund has been used for folds with positive Ricci curvature”, J. Differen- the award of the Oswald Veblen Prize in Geom- tial Geometry 17 (1982), 255–306). These equa- etry. Subsequent awards were made at five-year tions form a very nonlinear system of differential intervals. A total of ten awards have been made: equations (of essentially parabolic type) for the Christos D. Papakyriakopolous (1964), Raoul H. time evolution of a Riemannian metric on a Bott (1964), Stephen Smale (1966), Morton Brown smooth manifold. The equations assert simply and Barry Mazur (1966), Robion C. Kirby (1971), that the time derivative of the metric is equal to Dennis P. Sullivan (1971), William P. Thurston minus twice the Ricci curvature tensor. (The (1976), James Simons (1976), Mikhael Gromov Ricci curvature tensor is a symmetric, rank two (1981), Shing-Tung Yau (1981), Michael H. Freed- tensor which is obtained by a natural average of man (1986), and Andrew Casson and Clifford H. the sectional curvatures.) This flow equation Taubes (1991). At present, the award is supple- can be thought of as a nonlinear heat equation mented from the Steele Prize Fund, bringing the for the Riemannian metric. After an appropriate, value of the Veblen Prize to $4,000, divided time-dependent rescaling, the static solutions are equally between this year’s recipients. simply the Einstein metrics. In introducing the The 1996 Veblen Prize was awarded by the Ricci flow equations, Hamilton proved that com- AMS Council on the basis of a recommendation pact, three-dimensional manifolds with positive by a selection committee consisting of Jeff definite Ricci curvature are diffeomorphic to Cheeger, Peter Li, and Clifford Taubes (chair). spherical space forms. (These are quotients of The text that follows contains the committee’s the three-dimensional sphere by free, finite citation for each award, the recipients’ responses group actions.) MARCH 1996 NOTICES OF THE AMS 325 comm-veblen.qxp 4/22/98 8:23 AM Page 326 Over the sub- his recent and continuing work to uncover the sequent years, geometric and analytic properties of singulari- Hamilton has ties of the Ricci flow equation and related sys- continued his tems of differential equations. study of the Ricci flow equations Response and related equa- It is a great honor to receive the Oswald Veblen tions, delving ever Prize from the AMS. This award recognizes the deeper to under- tremendous growth of the whole field of non- stand the nature linear parabolic partial differential equations in of the singulari- geometry, of which my own work is but a small ties which arise part. Especial thanks are due to my parents, Dr. under the flow. and Mrs. Selden Hamilton, who provided me (Hamilton proved with every conceivable head start in education; that singularities my high school geometry teacher, Mrs. Becker, do not arise in for an enduring love of three-dimensional geom- three dimensions etry; my mentor, James Eells, Jr., whose work when the Ricci with Joseph Sampson on the Harmonic Map Heat curvature starts Flow originated and inspired the field; and my out positive.) colleagues S.-T. Yau and Richard Schoen, who Hamilton has suggested the neck-pinching phenomenon and Richard Hamilton come to under- encouraged me to study the formation of sin- stand the geo- gularities. metric constraints on the singularities which It is a pleasure to share the prize with Gang arise under the Ricci flow on a compact, three- Tian, whose work on Kähler manifolds is out- dimensional Riemannian manifold and under a standing. related flow equation (for the “isotropic curvature tensor”) on a compact, four-dimensional Biographical Sketch manifold. This understanding has allowed him, Professor Hamilton was born in Cincinnati, Ohio, in many cases, to classify all possible singular- in 1943. He received his B.A. from Yale Univer- ities of the flow. sity in 1963 and his Ph.D. from Princeton Uni- In the four-dimensional case, Hamilton was versity in 1966 under the direction of Robert recently able to give a topological characteriza- Gunning. He has held professorships at Cornell tion of the possible singularities which arise University and the University of California at from the isotropic curvature tensor flow if the Berkeley and visiting positions at the Univer- starting metric has positive isotropic curvature sity of Warwick, the Courant Institute, the In- tensor. The conclusion is as follows: If a singu- stitute for Advanced Study in Princeton, and larity arises, then it can be described as a length- the University of Hawaii. He is currently pro- ening neck in the manifold whose cross-section fessor of mathematics at the University of Cal- is an embedded spherical space form with in- ifornia, San Diego. jective fundamental group. Hamilton deduced from this fact that simply connected manifolds Gang Tian with positive isotropic curvature are diffeo- Citation morphic to the four-dimensional sphere. Gang Tian is cited for his contributions to geo- For the compact 3-manifold case, Hamilton, metric analysis and, in particular, for his work in a recent paper, analyzed the development of on the question of existence and obstructions singularities in the Ricci flow by studying the evo- for Kähler-Einstein metrics on complex mani- lution of stable, closed geodesics and stable, folds with positive first Chern class. minimal surfaces under their own, compatible, The basic Kähler-Einstein problem is to find geometric flows. This analysis of the flows of sta- necessary and sufficient conditions for the ex- ble geodesics and minimal surfaces leads to a istence of a Kähler metric on a given complex characterization of the developing singularities manifold whose Ricci curvature is a constant in terms of Ricci soliton solutions to the flow multiple of the metric itself. The sign of the equations along degenerating, geometric sub- constant is determined by the degree of the sets of the original manifold. (A Ricci soliton is manifold’s first Chern class. The case where the a solution whose motion in time is generated by sign is negative was solved independently by a 1-parameter group of diffeomorphisms of the Aubin and Yau, while the sign zero case (where underlying manifold.) the first Chern class vanishes) was solved by Yau The Oswald Veblen Prize in Geometry is in his celebrated solution to the Calabi Conjec- awarded to Richard Hamilton in recognition of ture. Applications of the zero (and non-posi- 326 NOTICES OF THE AMS VOLUME 43, NUMBER 3 comm-veblen.qxp 4/22/98 8:23 AM Page 327 tive) first Chern class results have been legion, search. It is surely and so progress on the positive first Chern class one of the most cases has been eagerly sought after. However, the stimulating case of positive first Chern class has remained places for mathe- mostly mysterious until the recent work of Tian matical research. (and others). Finally, I am very In particular, Tian completely settled the ex- happy to share istence question for Kähler-Einstein metrics on this prize with complex surfaces, showing that they exist if and R. Hamilton. only if the group of holomorphic transformations is reductive. Later, Tian (generalizing work with Biographical W. Y. Ding) found the first obstructions to the Sketch existence of Kähler-Einstein metrics which do not Gang Tian was require the existence of holomorphic vector born on Novem- fields. Subsequently, he was able to show that ber 24, 1958, in for hypersurfaces, the existence of a Kähler-Ein- the People’s Re- stein metric implies that the hypersurface is sta- public of China. ble in the geometric invariant theory sense. (This He received his constitutes a first big step in Yau’s program to B.S. from Nanking characterize manifolds with Kähler-Einstein met- University (1982), rics in geometric invariant theory terms.) Tian his M.S. from had previously developed some general criteria Peking University Gang Tian for the existence of Kähler-Einstein metrics, (1984), and his which he applied to complex hypersurfaces in Ph.D. from Harvard University (1988). After po- complex projective spaces. sitions at Princeton University and the State Uni- Tian has also proved various theorems which versity of New York at Stony Brook, he went to control the limiting behavior of sequences of the Courant Institute of Mathematical Sciences Kähler-Einstein metrics with bounded Ln- norm at New York University in 1991. In 1995 he on a complex n-dimensional manifold. And, he moved to the Massachusetts Institute of Tech- has classified the asymptotically locally Euclid- nology. He also holds professorships at the Math- ean Kähler-Einstein manifolds which result as ematics Institute of the Academia Sinica and at limits of such sequences.

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