Tian Gang (; born November 24, 1958) is a Chinese mathematician known for his contributions to the mathematical fields of Kähler geometry, Gromov-Witten theory, and geometric analysis. He is a professor of mathematics at Peking University and the Higgins Professor Emeritus at Princeton University.
As of 2020, he is the Vice Chairman of the China Democratic League and the President of the Chinese Mathematical Society. From 2017 to 2019 he served as the Vice President of Peking University.
Biography Tian was born in Nanjing, Jiangsu, China, on November 24, 1958. In 1978, after the Cultural Revolution, he qualified in the second Gaokao exam. He graduated from Nanjing University with a Bachelor of Science in mathematics in 1982, earned a Master of Science in mathematics from Peking University in 1984, and earned his Ph.D. in mathematics from Harvard University in 1988 under the supervision of Shing-Tung Yau.
In 1998, he was appointed as a Cheung Kong Scholar professor at Peking University. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was a professor of mathematics at the Massachusetts Institute of Technology from 1995 to 2006 (holding the chair of Simons Professor of Mathematics from 1996). His employment at Princeton started from 2003, and was later appointed the Higgins Professor of Mathematics. Starting 2005, he has been the director of the Beijing International Center for Mathematical Research (BICMR);Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html from 2013 to 2017 he was the Dean of School of Mathematical Sciences at Peking University.History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html He and John Milnor are Senior Scholars of the Clay Mathematics Institute (CMI). In 2011, Tian became director of the Sino-French Research Program in Mathematics at the Centre national de la recherche scientifique (CNRS) in Paris. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy.
Tian has served on many committees, including for the Abel Prize and the Leroy P. Steele Prize. He is a member of the editorial boards of many journals, including Advances in Mathematics and the Journal of Geometric Analysis. In the past he has been on the editorial boards of the Annals of Mathematics and the Journal of the American Mathematical Society.
Among his awards and honors: Sloan Research Fellowship (1991-1993) Alan T. Waterman Award (1994) Oswald Veblen Prize in Geometry (1996) Elected to the Chinese Academy of Sciences (2001) Elected to the American Academy of Arts and Sciences (2004)
Since at least 2013 he has been heavily involved in Chinese politics, serving as the Vice Chairman of the China Democratic League, the second most populous political party in China.
Mathematical contributions The Kähler-Einstein problem Tian is well-known for his contributions to Kähler geometry, and in particular to the study of Kähler-Einstein metrics. Shing-Tung Yau, in his renowned resolution of the Calabi conjecture, had settled the case of closed Kähler manifolds with nonpositive first Chern class. His work in applying the method of continuity showed that control of the Kähler potentials would suffice to prove existence of Kähler-Einstein metrics on closed Kähler manifolds with positive first Chern class, also known as "Fano manifolds." Tian and Yau extended Yau's analysis of the Calabi conjecture to non-compact settings, where they obtained partial results. They also extended their work to allow orbifold singularities.
Tian introduced the "-invariant," which is essentially the optimal constant in the Moser-Trudinger inequality when applied to Kähler potentials with a supremal value of 0. He showed that if the -invariant is sufficiently large (i.e. if a sufficiently strong Moser-Trudinger inequality holds), then control in Yau's method of continuity could be achieved. This was applied to demonstrate new examples of Kähler-Einstein surfaces. The case of Kähler surfaces was revisited by Tian in 1990, claiming a complete resolution of the Kähler-Einstein problem in that context. The main technique was to study the possible geometric degenerations of a sequence of Kähler-Einstein metrics, as detectable by the Gromov–Hausdorff convergence. Tian adapted many of the technical innovations of Karen Uhlenbeck, as developed for Yang-Mills connections, to the setting of Kähler metrics. Some similar and influential work in the Riemannian setting was done in 1989 and 1990 by Michael Anderson, Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima.Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.Bando, Shigetoshi; Kasue, Atsushi; Nakajima, Hiraku. On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth. Invent. Math. 97 (1989), no. 2, 313–349.Anderson, Michael T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445. However, it was later discovered that Tian's paper contains serious gaps. In particular, in the degree 1 case, which is the most delicate, in Odaka-Spotti-Sun Odaka, Yuji; Spotti, Cristiano; Sun, Song. Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics, J. Differential Geom. 102(1): 127–172 (January 2016). (see Example 5.8) explained that Proposition 6.2 in Tian's paper is false by exhibiting an explicit counterexample. The proposition asserted that, for a degree 1 Gromov-Hausdorff limit, the linear system |-2K| is free of base points and the surface possesses at most one non-canonical singularity of specific type. It was shown in that there is an explicit toric Kähler-Einstein limit of Gorenstein index is 3, which shows these assertions fail. It therefore remains unclear whether Tian's strategy can be used to handle the degree 1 case.
Tian's most renowned contribution to the Kähler-Einstein problem came in 1997. Yau had conjectured in the 1980s, based partly in analogy to the Donaldson-Uhlenbeck-Yau theorem, that existence of a Kähler-Einstein metric should correspond to stability of the underlying Kähler manifold in a certain sense of geometric invariant theory. It was generally understood, especially following work of Akito Futaki,Futaki, A. An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73 (1983), no. 3, 437–443. that the existence of holomorphic vector fields should act as an obstruction to the existence of Kähler-Einstein metrics. Tian and Wei Yue Ding established that this obstruction is not sufficient within the class of Kähler orbifolds. Tian, in his 1997 article, gave concrete examples of Kähler manifolds (rather than orbifolds) which had no holomorphic vector fields and also no Kähler-Einstein metrics, showing that the desired criterion lies deeper. Yau had proposed that, rather than holomorphic vector fields on the manifold itself, it should be relevant to study the deformations of projective embeddings of Kähler manifolds under holomorphic vector fields on projective space. This idea was modified by Tian, introducing the notion of K-stability and showing that any Kähler-Einstein manifold must be K-stable.
Simon Donaldson, in 2002, modified and extended Tian's definition of K-stability.Donaldson, S.K. Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, 289–349. The conjecture that K-stability would be sufficient to ensure the existence of a Kähler-Einstein metric became known as the Yau-Tian-Donaldson conjecture. In 2015, Xiuxiong Chen, Donaldson, and Song Sun, published a proof of the conjecture, receiving the Oswald Veblen Prize in Geometry for their work.Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (2015), no. 1, 183–197.Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π. J. Amer. Math. Soc. 28 (2015), no. 1, 199–234.Chen, Xiuxiong; Donaldson, Simon; Sun, Song. Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Amer. Math. Soc. 28 (2015), no. 1, 235–278. Tian published a proof of the conjecture in the same year, although Chen, Donaldson, and Sun have accused Tian of academic and mathematical misconduct over his paper.Xiuxiong Chen, Simon, Donaldson, and Song Sun. On some recent developments in Kähler geometry.Gang Tian. Response to CDS. In Section 6 of , Tian outlined two approaches to finish the proof of Theorem 1.1 (the main theorem), the second approach is similar to the one used in, and the first approach follows Tian's old work , involving CM-stability. The first approach is now known to contain a gap. Specifically, Theorem 6.6 asserted that K-stability implies CM-stability. The proof relies on the generalised numerical criterion for pairs claimed in the preprint works of S. Paul. However, it was subsequently observed that Paul's proof of the generalized numerical criterion contains a gap (see Section 5.3 of ). Several subsequent works attempted to repair this gap. More recently, however, Paul, Sun, Zhang constructed explicit counterexamples to the generalized numerical criterion (see Theorem 1.3 and Remark 1.4 in ), showing that the criterions, as originally stated, is false. Consequently, the arguments establishing Theorem 6.6 in is no longer valid.
Kähler geometry In one of his first articles, Tian studied the space of Calabi-Yau metrics on a Kähler manifold. He showed that any infinitesimal deformation of Calabi-Yau structure can be 'integrated' to a one-parameter family of Calabi-Yau metrics; this proves that the "moduli space" of Calabi-Yau metrics on the given manifold has the structure of a smooth manifold. This was also studied earlier by Andrey Todorov, and the result is known as the Tian−Todorov theorem.Todorov, Andrey N. The Weil-Petersson geometry of the moduli space of SU(n ≥ 3) (Calabi-Yau) manifolds. I. Comm. Math. Phys. 126 (1989), no. 2, 325–346. As an application, Tian found a formula for the Weil-Petersson metric on the moduli space of Calabi-Yau metrics in terms of the period mapping.Huybrechts, Daniel. Complex geometry. An introduction. [Chapter 6.] Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp.