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Tian Gang
Tian atOberwolfach in 2005
Born	(1958-11-24)24 November 1958 (age 67) Nanjing,Jiangsu, China
Alma mater	Nanjing University (BS) Peking University (MS) Harvard University (PhD)
Known for	Yau-Tian-Donaldson conjecture K-stability K-stability of Fano varieties
Awards	Veblen Prize (1996) Alan T. Waterman Award (1994)
Scientific career
Fields	Mathematics
Institutions	Princeton University Peking University
Thesis	Kähler Metrics on Algebraic Manifolds (1988)
Doctoral advisor	Shing-Tung Yau
Doctoral students	Aaron Naber Nataša Šešum Wei Dongyi
Chinese name
Traditional Chinese	田剛
Simplified Chinese	田刚
Transcriptions Standard Mandarin Hanyu Pinyin Tián Gāng

Tian Gang (Chinese:田刚; born November 24, 1958)^[1] is a Chinese mathematician. He is a professor of mathematics atPeking University and Higgins Professor Emeritus atPrinceton University. He is known for contributions to themathematical fields ofKähler geometry,Gromov-Witten theory, andgeometric analysis.

As of 2020, he is the Vice Chairman of theChina Democratic League and the President of theChinese Mathematical Society. From 2017 to 2019 he served as the Vice President ofPeking University.

Tian was born inNanjing,Jiangsu, China. He qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated fromNanjing University in 1982, and received amaster's degree from Peking University in 1984. In 1988, he received aPh.D. in mathematics fromHarvard University, under the supervision ofShing-Tung Yau.

In 1998, he was appointed as aCheung Kong Scholar professor at Peking University. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was a professor of mathematics at theMassachusetts 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);^[2] from 2013 to 2017 he was the Dean of School of Mathematical Sciences at Peking University.^[3] He andJohn Milnor are Senior Scholars of theClay Mathematics Institute (CMI). In 2011, Tian became director of the Sino-French Research Program in Mathematics at theCentre national de la recherche scientifique (CNRS) inParis. In 2010, he became scientific consultant for theInternational Center for Theoretical Physics inTrieste, Italy.^[4]

Tian has served on many committees, including for theAbel Prize and theLeroy P. Steele Prize.^[5] He is a member of the editorial boards of many journals, includingAdvances in Mathematics and the Journal of Geometric Analysis. In the past he has been on the editorial boards ofAnnals of Mathematics and theJournal 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 theChinese Academy of Sciences (2001)
• Elected to theAmerican Academy of Arts and Sciences (2004)

Since at least 2013 he has been heavily involved in Chinese politics, serving as the Vice Chairman of theChina Democratic League, the second most populouspolitical party in China.

Tian is well-known for his contributions toKähler geometry, and in particular to the study ofKähler-Einstein metrics.Shing-Tung Yau, in his renowned resolution of theCalabi conjecture, had settled the case ofclosed Kähler manifolds with nonpositive firstChern class. His work in applying themethod of continuity showed thatC⁰ control of the Kähler potentials would suffice toprove 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 tonon-compact settings, where they obtained partial results.^[TY90] They also extended their work to allow orbifold singularities.^[TY91]

Tian introduced the "α-invariant," which is essentially the optimal constant in theMoser-Trudinger inequality when applied to Kähler potentials with a supremal value of 0. He showed that if theα-invariant is sufficiently large (i.e. if a sufficiently strong Moser-Trudinger inequality holds), thenC⁰ control in Yau's method of continuity could be achieved.^[T87b] 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.^[T90b] The main technique was to study the possible geometric degenerations of a sequence of Kähler-Einstein metrics, as detectable by theGromov–Hausdorff convergence. Tian adapted many of the technical innovations ofKaren Uhlenbeck, as developed for Yang-Mills connections, to the setting of Kähler metrics. Some similar and influential work in theRiemannian setting was done in 1989 and 1990 byMichael Anderson, Shigetoshi Bando, Atsushi Kasue, andHiraku Nakajima.^[6][7][8] However, certain incorrect statements in Tian's work, owing to the highly technical nature of the paper, went unnoticed until after its publication.^[9]

Tian's most renowned contribution to the Kähler-Einstein problem came in 1997. Yau hadconjectured in the 1980s, based partly in analogy to theDonaldson-Uhlenbeck-Yau theorem, that existence of a Kähler-Einstein metric should correspond to stability of the underlying Kähler manifold in a certain sense ofgeometric invariant theory. It was generally understood, especially following work of Akito Futaki,^[10] that the existence of holomorphicvector 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 Kählerorbifolds.^[DT92] 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.^[T97] 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 ofK-stability and showing that any Kähler-Einstein manifold must beK-stable.^[T97]

Simon Donaldson, in 2002, modified and extended Tian's definition of K-stability.^[11] The conjecture that K-stability would be sufficient to ensure the existence of a Kähler-Einstein metric became known as theYau-Tian-Donaldson conjecture. In 2015,Xiuxiong Chen, Donaldson, andSong Sun, published a proof of the conjecture, receiving theOswald Veblen Prize in Geometry for their work.^[12][13][14] 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.^{[T15][15][16]}

In one of his first articles, Tian studied the space of Calabi-Yau metrics on a Kähler manifold.^[T87a] 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.^[17] As an application, Tian found a formula for theWeil-Petersson metric on the moduli space of Calabi-Yau metrics in terms of theperiod mapping.^[T87a][18]

Motivated by the Kähler-Einstein problem and a conjecture of Yau relating toBergman metrics, Tian studied the following problem. LetL be aline bundle over a Kähler manifoldM, and fix a hermitian bundle metric whose curvature form is a Kähler form onM. Suppose that for sufficiently largem, an orthonormal set of holomorphic sections of the line bundleL^⊗m defines a projective embedding ofM. One can pull back theFubini-Study metric to define a sequence of metrics onM asm increases. Tian showed that a certain rescaling of this sequence will necessarily converge in theC²topology to the original Kähler metric.^[T90a] The refined asymptotics of this sequence were taken up in a number of influential subsequent papers by other authors, and are particularly important inSimon Donaldson's program on extremal metrics.^{[19][20][21][22][23]} The approximability of a Kähler metric by Kähler metrics induced from projective embeddings is also relevant to Yau's picture of the Yau-Tian-Donaldson conjecture, as indicated above.

In a highly technical article,Xiuxiong Chen and Tian studied the regularity theory of certain complexMonge-Ampère equations, with applications to the study of the geometry of extremal Kähler metrics.^[CT08] Although their paper has been very widely cited, Julius Ross and David Witt Nyström foundcounterexamples to the regularity results of Chen and Tian in 2015.^[24] It is not clear which results of Chen and Tian's article remain valid.

Pseudoholomorphic curves were shown byMikhail Gromov in 1985 to be powerful tools insymplectic geometry.^[25] In 1991,Edward Witten conjectured a use of Gromov's theory to defineenumerative invariants.^[26] Tian andYongbin Ruan found the details of such a construction, proving that the various intersections of the images of pseudo-holomorphic curves is independent of many choices, and in particular gives an associative multilinear mapping on thehomology of certain symplectic manifolds.^[RT95] This structure is known asquantum cohomology; a contemporaneous and similarly influential approach is due toDusa McDuff andDietmar Salamon.^[27] Ruan and Tian's results are in a somewhat more general setting.

WithJun Li, Tian gave a purely algebraic adaptation of these results to the setting ofalgebraic varieties.^[LT98b] This was done at the same time asKai Behrend andBarbara Fantechi, using a different approach.^[28]

Li and Tian then adapted theiralgebro-geometric work back to the analytic setting in symplectic manifolds, extending the earlier work of Ruan and Tian.^[LT98a] Tian and Gang Liu made use of this work to prove the well-known Arnold conjecture on the number offixed points of Hamiltoniandiffeomorphisms.^[LT98c] However, these papers of Li-Tian and Liu-Tian on symplectic Gromov-Witten theory have been criticized byDusa McDuff andKatrin Wehrheim as being incomplete or incorrect, saying that Li and Tian's article^[LT98a] "lacks almost all detail" on certain points and that Liu and Tian's article^[LT98c] has "serious analytic errors."^[29]

In 1995, Tian and Weiyue Ding studied theharmonic map heat flow of a two-dimensional closedRiemannian manifold into a closed Riemannian manifoldN.^[DT95] In a seminal 1985 work, following the 1982 breakthrough of Jonathan Sacks andKaren Uhlenbeck,Michael Struwe had studied this problem and showed that there is a weak solution which exists for all positive time. Furthermore, Struwe showed that the solutionu is smooth away from finitely many spacetime points; given any sequence of spacetime points at which the solution is smooth and which converge to a given singular point(p,T), one can perform some rescalings to (subsequentially) define a finite number ofharmonic maps from the round 2-dimensional sphere intoN, called "bubbles." Ding and Tian proved a certain "energy quantization," meaning that the defect between the Dirichlet energy ofu(T) and the limit of the Dirichlet energy ofu(t) ast approachesT is exactly measured by the sum of the Dirichlet energies of the bubbles. Such results are significant in geometric analysis, following the original energy quantization result ofYum-Tong Siu andShing-Tung Yau in their proof of the Frankel conjecture.^[30] The analogous problem for harmonic maps, as opposed to Ding and Tian's consideration of the harmonic map flow, was considered by Changyou Wang around the same time.^[31]

A major paper of Tian's dealt with theYang–Mills equations.^[T00a] In addition to extending much ofKaren Uhlenbeck's analysis to higher dimensions, he studied the interaction of Yang-Mills theory withcalibrated geometry. Uhlenbeck had shown in the 1980s that, when given a sequence of Yang-Mills connections of uniformly bounded energy, they will converge smoothly on the complement of a subset of codimension at least four, known as the complement of the "singular set". Using techniques developed by Fanghua Lin in the study of harmonic maps,^[32] Tian showed that the singular set is arectifiable set. In the case that the manifold is equipped with a calibration, one can restrict interest to the Yang-Mills connections which are self-dual relative to the calibration. In this case, Tian claimed that the singular set is calibrated. For instance, the singular set of a sequence ofhermitian Yang-Mills connections of uniformly bounded energy will be a holomorphic cycle. This was viewed as a significant geometric feature of the analysis of Yang-Mills connections. However, it was later discovered that there are significant gaps in Tian’s article^[T00a]. In a later paper in 2004,Terence Tao and Tian^[33] addressed these issues, providing a new proof to fill the gap in the epsilon-regularity theorem originally claimed in^[T00a] (which had implicitly assumed the existence of a good gauge). Furthermore, the proof of the calibrated property of the singular set presented in^[T00a] also contains serious flaws. Specifically, the proof of Proposition 2.3.1 in^[T00a] is incorrect, as it fails to utilise the self-duality assumption, despite the fact that the statement is known to be false without this assumption.^[34] The core issue lies in the fact that it is not ruled out the possibility that the limit connection may encounter topological singularities. There is also an issue concerning the compactness of the space of generalised instantons claimed in^[T00a], as noted in the paper^[35] (see Remark 1.15).

In 2006, Tian and Zhou Zhang studied theRicci flow in the special setting of closedKähler manifolds.^[TZ06] Their principal achievement was to show that the maximal time of existence can be characterized in purely cohomological terms. This represents one sense in which the Kähler-Ricci flow is significantly simpler than the usual Ricci flow, where there is no (known) computation of the maximal time of existence from a given geometric context. Tian and Zhang's proof consists of a use of the scalarmaximum principle as applied to various geometric evolution equations, in terms of a Kähler potential as parametrized by a linear deformation of forms which is cohomologous to the Kähler-Ricci flow itself. In a notable work with Jian Song, Tian analyzed the Kähler Ricci flow on certain two-dimensionalcomplex manifolds.^[ST07]

In 2002 and 2003,Grigori Perelman posted three papers on thearXiv which purported to prove thePoincaré conjecture andGeometrization conjecture in the field of three-dimensionalgeometric topology.^[36][37][38] Perelman's papers were immediately acclaimed for many of their novel ideas and results, although the technical details of many of his arguments were seen as hard to verify. In collaboration withJohn Morgan, Tian published an exposition of Perelman's papers in 2007, filling in many of the details.^[MT07] Other expositions, which have also been widely studied, were written byHuai-Dong Cao andXi-Ping Zhu, and byBruce Kleiner andJohn Lott.^[39][40] Morgan and Tian's exposition is the only of the three to deal with Perelman's third paper,^[38] which is irrelevant for analysis of the geometrization conjecture but usescurve-shortening flow to provide a simpler argument for the special case of the Poincaré conjecture. Eight years after the publication of Morgan and Tian's book,Abbas Bahri pointed to part of their exposition of this paper to be in error, having relied upon incorrect computations of evolution equations.^[41] The error, which dealt with details not present in Perelman's paper, was soon after amended by Morgan and Tian.^[42]

In collaboration withNataša Šešum, Tian also published an exposition of Perelman's work on the Ricci flow of Kähler manifolds, which Perelman did not publish in any form.^[43]

• Tian Gang at theMathematics Genealogy Project

• 1958 births
• Living people
• 20th-century Chinese mathematicians
• 21st-century Chinese mathematicians
• Chinese expatriate academics in the United States
• Differential geometers
• Educators from Nanjing
• Fellows of the American Academy of Arts and Sciences
• Harvard University alumni
• Mathematicians from Jiangsu
• Members of the Chinese Academy of Sciences
• Peking University alumni
• Academic staff of Peking University
• Princeton University faculty
• Scientists from Nanjing
• Presidents of the Chinese Mathematical Society

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