View source for Biography:Tian Gang

{{short description|Chinese mathematician (born 1958)}}
{{Infobox scientist
| name              = Tian Gang
| image             = Gang Tian.jpeg
| image_size        = 210px
| caption           = Tian at [[Organization:Mathematical Research Institute of Oberwolfach|Oberwolfach]] in 2005
| birth_date        = {{birth date and age|1958|11|24|df=y}}
| birth_place       = Nanjing, Jiangsu, China
| death_date        = 
| death_place       = 
| nationality       = Chinese
| fields            = [[Mathematics]]
| workplaces        = [[Organization:Princeton University|Princeton University]]<br />Peking University
| alma_mater        = [[Organization:Harvard University|Harvard University]]<br />Peking University<br />Nanjing University
| doctoral_advisor  = [[Biography:Shing-Tung Yau|Shing-Tung Yau]]
| doctoral_students = [[Biography:Nataša Šešum|Nataša Šešum]]<br />Wei Dongyi
| known_for         = Yau-Tian-Donaldson conjecture<br />[[K-stability]]<br />[[K-stability of Fano varieties]]
| awards            = Veblen Prize (1996)<br />Alan T. Waterman Award (1994)
| module            = {{Chinese |child=yes |s=田刚 |t=田剛 |p=Tián Gāng}}
| thesis_title      = Kähler Metrics on Algebraic Manifolds
| thesis_year       = 1988
}}
'''Tian Gang''' ({{zh|s=田刚}}; born November 24, 1958)<ref>{{cite web |url=http://www.ams.org/notices/199603/comm-veblen.pdf |title=1996 Oswald Veblen Prize |publisher=AMS |date=1996}}</ref> is a Chinese [[Biography:Mathematician|mathematician]]. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at [[Organization:Princeton University|Princeton University]]. He is known for contributions to the mathematical fields of [[Kähler manifold|Kähler geometry]], Gromov-Witten theory, and [[Geometric analysis|geometric analysis]].

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. He qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a [[Social:Doctor of Philosophy|Ph.D.]] in [[Mathematics|mathematics]] from [[Organization:Harvard University|Harvard University]], under the supervision of [[Biography:Shing-Tung Yau|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 [[Organization:Massachusetts Institute of Technology|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);<ref>Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html</ref> from 2013 to 2017 he was the Dean of School of Mathematical Sciences at Peking University.<ref>History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html</ref> He and [[Biography:John Milnor|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 [[Place:Paris|Paris]]. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy.<ref>{{Cite web|url=https://www.ictp.it/about-ictp/governance.aspx|title=ICTP - Governance|website=www.ictp.it|language=en|access-date=2018-05-28}}</ref>

Tian has served on many committees, including for the Abel Prize and the Leroy P. Steele Prize.<ref>http://www.ams.org/notices/201304/rnoti-p480.pdf {{Bare URL PDF|date=March 2022}}</ref> 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 [[Annals of Mathematics]] and the [[Organization:Journal of the American Mathematical Society|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 manifold|Kähler geometry]], and in particular to the study of Kähler-Einstein metrics. [[Biography:Shing-Tung Yau|Shing-Tung Yau]], in his renowned resolution of the [[Calabi conjecture]], had settled the case of [[Closed manifold|closed]] Kähler manifolds with nonpositive first Chern class. His work in applying the [[Method of continuity|method of continuity]] showed that {{math|''C''<sup>0</sup>}} 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 noncompact settings, where they obtained partial results.{{ran|TY90}} They also extended their work to allow orbifold singularities.{{ran|TY91}}

Tian introduced the "{{mvar|α}}-invariant," which is essentially the optimal constant in the [[Trudinger's theorem|Moser-Trudinger inequality]] when applied to Kähler potentials with a supremal value of 0. He showed that if the {{mvar|α}}-invariant is sufficiently large (i.e. if a sufficiently strong Moser-Trudinger inequality holds), then {{math|''C''<sup>0</sup>}} control in Yau's method of continuity could be achieved.{{ran|T87b}} This was applied to demonstrate new examples of Kähler-Einstein surfaces. The case of Kähler surfaces was revisited by Tian in 1990, giving a complete resolution of the Kähler-Einstein problem in that context.{{ran|T90b}} 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.<ref>Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.</ref><ref>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.</ref><ref>Anderson, Michael T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445.</ref>

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 [[Kobayashi–Hitchin correspondence|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|geometric invariant theory]]. It was generally understood, especially following work of Akito Futaki,<ref>Futaki, A. An obstruction to the existence of Einstein Kähler metrics. Invent. Math. 73 (1983), no. 3, 437–443.</ref> 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 [[Orbifold|orbifold]]s.{{ran|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.{{ran|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 of [[K-stability]] and showing that any Kähler-Einstein manifold must be [[K-stability of Fano varieties|K-stable]].{{ran|T97}}

[[Biography:Simon Donaldson|Simon Donaldson]], in 2002, modified and extended Tian's definition of K-stability.<ref>Donaldson, S.K. Scalar curvature and stability of toric varieties. J. Differential Geom. 62 (2002), no. 2, 289–349.</ref> 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, [[Biography:Xiuxiong Chen|Xiuxiong Chen]], Donaldson, and [[Biography:Song Sun|Song Sun]], published a proof of the conjecture, receiving the Oswald Veblen Prize in Geometry for their work.<ref>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.</ref><ref>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.</ref><ref>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.</ref> 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.{{ran|T15}}<ref>Xiuxiong Chen, Simon, Donaldson, and Song Sun. [http://wwwf.imperial.ac.uk/~skdona/KEDEVELOPMENTS-9-19-2013.PDF On some recent developments in Kähler geometry.]</ref><ref>Gang Tian. [http://tian.bicmr.pku.edu.cn/Documents_on_K_hler_Geometry.htm Response to CDS.]</ref>

===Kähler geometry===
In one of his first articles, Tian studied the space of Calabi-Yau metrics on a Kähler manifold.{{ran|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.<ref>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.</ref> 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|period mapping]].{{ran|T87a}}<ref>Huybrechts, Daniel. Complex geometry. An introduction. [Chapter 6.] Universitext. Springer-Verlag, Berlin, 2005. xii+309 pp. {{ISBN|3-540-21290-6}}</ref>

Motivated by the Kähler-Einstein problem and a conjecture of Yau relating to [[Bergman metric]]s, Tian studied the following problem. Let {{mvar|L}} be a line bundle over a Kähler manifold {{mvar|M}}, and fix a hermitian bundle metric whose curvature form is a Kähler form on {{mvar|M}}. Suppose that for sufficiently large {{mvar|m}}, an orthonormal set of holomorphic sections of the line bundle {{math|''L''<sup>⊗''m''</sup>}} defines a projective embedding of {{mvar|M}}. One can pull back the Fubini-Study metric to define a sequence of metrics on {{mvar|M}} as {{mvar|m}} increases. Tian showed that a certain rescaling of this sequence will necessarily converge in the {{math|''C''<sup>2</sup>}} topology to the original Kähler metric.{{ran|T90a}} The refined asymptotics of this sequence were taken up in a number of influential subsequent papers by other authors, and are particularly important in [[Biography:Simon Donaldson|Simon Donaldson]]'s program on extremal metrics.<ref>Zelditch, Steve. Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331.</ref><ref>Catlin, David. The Bergman kernel and a theorem of Tian. Analysis and geometry in several complex variables (Katata, 1997), 1–23, Trends Math., Birkhäuser Boston, Boston, MA, 1999.</ref><ref>Lu, Zhiqin. On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122 (2000), no. 2, 235–273.</ref><ref>Donaldson, S.K. Scalar curvature and projective embeddings. I. J. Differential Geom. 59 (2001), no. 3, 479–522.</ref><ref>Donaldson, S.K. Lower bounds on the Calabi functional. J. Differential Geom. 70 (2005), no. 3, 453–472.</ref> 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, [[Biography:Xiuxiong Chen|Xiuxiong Chen]] and Tian studied the regularity theory of certain complex Monge-Ampère equations, with applications to the study of the geometry of extremal Kähler metrics.{{ran|CT08}} Although their paper has been very widely cited, Julius Ross and David Witt Nyström found counterexamples to the regularity results of Chen and Tian in 2015.<ref>Ross, Julius; Nyström, David Witt. Harmonic discs of solutions to the complex homogeneous Monge-Ampère equation. Publ. Math. Inst. Hautes Études Sci. 122 (2015), 315–335.</ref> It is not clear which results of Chen and Tian's article remain valid.

===Gromov-Witten theory===
[[Pseudoholomorphic curve]]s were shown by [[Biography:Mikhail Leonidovich Gromov|Mikhail Gromov]] in 1985 to be powerful tools in [[Symplectic geometry|symplectic geometry]].<ref>Gromov, M. Pseudo holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307–347.</ref> In 1991, [[Biography:Edward Witten|Edward Witten]] conjectured a use of Gromov's theory to define [[Enumerative geometry|enumerative invariants]].<ref>Witten, Edward. Two-dimensional gravity and intersection theory on moduli space. Surveys in differential geometry (Cambridge, MA, 1990), 243–310, Lehigh Univ., Bethlehem, PA, 1991.</ref> Tian and Yongbin 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 the [[Homology (mathematics)|homology]] of certain symplectic manifolds.{{ran|RT95}} This structure is known as [[Quantum cohomology|quantum cohomology]]; a contemporaneous and similarly influential approach is due to [[Biography:Dusa McDuff|Dusa McDuff]] and Dietmar Salamon.<ref>McDuff, Dusa; Salamon, Dietmar. J-holomorphic curves and quantum cohomology. University Lecture Series, 6. American Mathematical Society, Providence, RI, 1994. viii+207 pp. {{ISBN|0-8218-0332-8}}</ref> Ruan and Tian's results are in a somewhat more general setting.

With Jun Li, Tian gave a purely algebraic adaptation of these results to the setting of [[Algebraic variety|algebraic varieties]].{{ran|LT98b}} This was done at the same time as [[Biography:Kai Behrend|Kai Behrend]] and [[Biography:Barbara Fantechi|Barbara Fantechi]], using a different approach.<ref>Behrend, K.; Fantechi, B. The intrinsic normal cone. Invent. Math. 128 (1997), no. 1, 45–88.</ref>

Li and Tian then adapted their algebro-geometric work back to the analytic setting in symplectic manifolds, extending the earlier work of Ruan and Tian.{{ran|LT98a}} Tian and Gang Liu made use of this work to prove the well-known Arnold conjecture on the number of fixed points of Hamiltonian diffeomorphisms.{{ran|LT98c}} However, these papers of Li-Tian and Liu-Tian on symplectic Gromov-Witten theory have been criticized by [[Biography:Dusa McDuff|Dusa McDuff]] and [[Biography:Katrin Wehrheim|Katrin Wehrheim]] as being incomplete or incorrect, saying that Li and Tian's article {{ran|LT98a}} "lacks almost all detail" on certain points and that Liu and Tian's article {{ran|LT98c}} has "serious analytic errors."<ref>McDuff, Dusa; Wehrheim, Katrin. The fundamental class of smooth Kuranishi atlases with trivial isotropy. J. Topol. Anal. 10 (2018), no. 1, 71–243.</ref>

===Geometric analysis===
In 1995, Tian and Weiyue Ding studied the [[Harmonic map|harmonic map heat flow]] of a two-dimensional [[Closed manifold|closed]] [[Riemannian manifold]] into a closed Riemannian manifold {{mvar|N}}.{{ran|DT95}} In a seminal 1985 work, following the 1982 breakthrough of Jonathan Sacks and Karen 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 solution {{mvar|u}} 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 {{math|(''p'', ''T'')}}, one can perform some rescalings to (subsequentially) define a finite number of [[Harmonic map|harmonic map]]s from the round 2-dimensional sphere into {{mvar|N}}, called "bubbles." Ding and Tian proved a certain "energy quantization," meaning that the defect between the Dirichlet energy of {{math|''u''(''T'')}} and the limit of the Dirichlet energy of {{math|''u''(''t'')}} as {{mvar|t}} approaches {{mvar|T}} 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 of [[Biography:Yum-Tong Siu|Yum-Tong Siu]] and [[Biography:Shing-Tung Yau|Shing-Tung Yau]] in their proof of the Frankel conjecture.<ref>Siu, Yum Tong; Yau, Shing Tung. Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay. Ann. of Math. (2) 105 (1977), no. 2, 225–264.</ref> The analogous problem for [[Harmonic map|harmonic map]]s, as opposed to Ding and Tian's consideration of the harmonic map flow, was considered by Changyou Wang around the same time.<ref>Wang, Changyou. Bubble phenomena of certain Palais-Smale sequences from surfaces to general targets. Houston J. Math. 22 (1996), no. 3, 559–590.</ref>

A major paper of Tian's dealt with the [[Yang–Mills equations]].{{ran|T00a}} In addition to extending much of Karen Uhlenbeck's analysis to higher dimensions, he studied the interaction of Yang-Mills theory with [[Calibrated geometry|calibrated 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". Tian showed that the singular set is a [[Rectifiable set|rectifiable 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 showed that the singular set is calibrated. For instance, the singular set of a sequence of [[Hermitian Yang–Mills connection|hermitian Yang-Mills connections]] of uniformly bounded energy will be a holomorphic cycle. This is a significant geometric feature of the analysis of Yang-Mills connections.

===Ricci flow===
In 2006, Tian and Zhou Zhang studied the [[Ricci flow]] in the special setting of [[Closed manifold|closed]] [[Kähler manifold]]s.{{ran|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 scalar [[Maximum principle|maximum 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-dimensional complex manifolds.{{ran|ST07}}

In 2002 and 2003, [[Biography:Grigori Perelman|Grigori Perelman]] posted three papers on the [[ArXiv|arXiv]] which purported to prove the [[Poincaré conjecture]] and [[Geometrization conjecture]] in the field of three-dimensional [[Geometric topology|geometric topology]].<ref>Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. {{arxiv|math/0211159}}</ref><ref>Grisha Perelman. Ricci flow with surgery on three-manifolds. {{arxiv|math/0303109}}</ref><ref name="Per3">Grisha Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. {{arxiv|math/0307245}}</ref> 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 with [[Biography:John Morgan (mathematician)|John Morgan]], Tian published an exposition of Perelman's papers in 2007, filling in many of the details.{{ran|MT07}} Other expositions, which have also been widely studied, were written by [[Biography:Huai-Dong Cao|Huai-Dong Cao]] and Xi-Ping Zhu, and by [[Biography:Bruce Kleiner|Bruce Kleiner]] and [[Biography:John Lott (mathematician)|John Lott]].<ref>Cao, Huai-Dong; Zhu, Xi-Ping. A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006), no. 2, 165–492.</ref><ref>Kleiner, Bruce; Lott, John. Notes on Perelman's papers. Geom. Topol. 12 (2008), no. 5, 2587–2855.</ref> Morgan and Tian's exposition is the only of the three to deal with Perelman's third paper,<ref name="Per3" /> which is irrelevant for analysis of the geometrization conjecture but uses curve-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.<ref>Bahri, Abbas. Five gaps in mathematics. Adv. Nonlinear Stud. 15 (2015), no. 2, 289–319.</ref> The error, which dealt with details not present in Perelman's paper, was soon after amended by Morgan and Tian.<ref>John Morgan and Gang Tian. Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture. {{arxiv|1512.00699}} (2015)</ref>

In collaboration with [[Biography:Nataša Šešum|Nataša 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.<ref>Sesum, Natasa; Tian, Gang. Bounding scalar curvature and diameter along the Kähler Ricci flow (after Perelman). J. Inst. Math. Jussieu 7 (2008), no. 3, 575–587.</ref>

==Selected publications==
{{refbegin}}
'''Research articles.'''
{{Refbegin|30em}}
{{longitem|{{rma|T87a|tw=4em|{{cite conference|last1=Tian|first1=Gang|title=Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric|book-title=Mathematical aspects of string theory|pages=629–646|series=Advanced Series in Mathematical Physics|volume=1|publisher=[[Company:World Scientific|World Scientific Publishing Co.]]|location=Singapore|isbn=9971-50-273-9|editor1-last=Yau|editor1-first=S.-T.|year=1987|conference=Conference held at the University of California, San Diego (July 21–August 1, 1986)|doi=10.1142/9789812798411_0029|mr=0915841}}}}}}
{{longitem|{{rma|T87b|tw=4em|{{cite journal|last1=Tian|first1=Gang|title=On Kähler–Einstein metrics on certain Kähler manifolds with {{math|''c''<sub>1</sub>(''M'') > 0}}|journal=[[Inventiones Mathematicae]]|volume=89|year=1987|issue=2|pages=225–246|mr=0894378|doi=10.1007/BF01389077|s2cid=122352133|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002104229}}}}}}
{{longitem|{{rma|TY87|tw=4em|{{cite journal|last1=Tian|first1=Gang|last2=Yau|first2=Shing-Tung|title=Kähler–Einstein metrics on complex surfaces with {{math|''C''<sub>1</sub> > 0}}|journal=[[Physics:Communications in Mathematical Physics|Communications in Mathematical Physics]]|volume=112|year=1987|issue=1|pages=175–203|mr=0904143|doi=10.1007/BF01217685|s2cid=121216755|url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-112/issue-1/Kåhler-Einstein-metrics-on-complex-surfaces-with-C_10/cmp/1104159814.full}}}}}}
{{longitem|{{rma|T90a|tw=4em|{{cite journal|last1=Tian|first1=Gang|title=On a set of polarized Kähler metrics on algebraic manifolds|journal=[[Journal of Differential Geometry]]|volume=32|year=1990|issue=1|pages=99–130|mr=1064867|doi=10.4310/jdg/1214445039|doi-access=free}}}}}}
{{longitem|{{rma|T90b|tw=4em|{{cite journal|last1=Tian|first1=G.|title=On Calabi's conjecture for complex surfaces with positive first Chern class|journal=[[Inventiones Mathematicae]]|volume=101|year=1990|issue=1|pages=101–172}|mr=1055713|doi=10.1007/BF01231499|bibcode=1990InMat.101..101T |s2cid=59419559|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002107589}}}}}}
{{longitem|{{rma|TY90|tw=4em|{{cite journal|last1=Tian|first1=G.|last2=Yau|first2=Shing-Tung|title=Complete Kähler manifolds with zero Ricci curvature. I|journal=[[Organization:Journal of the American Mathematical Society|Journal of the American Mathematical Society]]|volume=3|year=1990|issue=3|pages=579–609|mr=1040196|doi=10.1090/S0894-0347-1990-1040196-6|doi-access=free}}}}}}
{{longitem|{{rma|TY91|tw=4em|{{cite journal|last1=Tian|first1=Gang|last2=Yau|first2=Shing-Tung|title=Complete Kähler manifolds with zero Ricci curvature. II|journal=[[Inventiones Mathematicae]]|volume=106|year=1991|issue=1|pages=27–60|mr=1123371|doi=10.1007/BF01243902|bibcode=1991InMat.106...27T |s2cid=122638262|url=http://eudml.org/doc/143931}}}}}}
{{longitem|{{rma|DT92|tw=4em|{{cite journal|last1=Ding|first1=Wei Yue|last2=Tian|first2=Gang|title=Kähler–Einstein metrics and the generalized Futaki invariant|journal=[[Inventiones Mathematicae]]|volume=110|year=1992|issue=|pages=315–335|mr=1185586|doi=10.1007/BF01231335|bibcode=1992InMat.110..315D |s2cid=59332400|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00211030X}}}}}}
{{longitem|{{rma|DT95|tw=4em|{{cite journal|last1=Ding|first1=Weiyue|last2=Tian|first2=Gang|title=Energy identity for a class of approximate harmonic maps from surfaces|journal=Communications in Analysis and Geometry|volume=3|year=1995|issue=3–4|pages=543–554|mr=1371209|doi=10.4310/CAG.1995.v3.n4.a1|doi-access=free}}}}}}
{{longitem|{{rma|RT95|tw=4em|{{cite journal|last1=Ruan|first1=Yongbin|last2=Tian|first2=Gang|title=A mathematical theory of quantum cohomology|journal=[[Journal of Differential Geometry]]|volume=42|year=1995|issue=2|pages=259–367|mr=1366548|doi=10.4310/jdg/1214457234|doi-access=free}}}}}}
{{longitem|{{rma|ST97|tw=4em|{{cite journal|last1=Siebert|first1=Bernd|last2=Tian|first2=Gang|title=On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator|journal=[[Asian Journal of Mathematics]]|volume=1|year=1997|issue=4|pages=679–695|mr=1621570|doi=10.4310/AJM.1997.v1.n4.a2|s2cid=14494725|url=https://doi.org/10.4310/AJM.1997.v1.n4.a2}}}}}}
{{longitem|{{rma|T97|tw=4em|{{cite journal|last1=Tian|first1=Gang|title=Kähler–Einstein metrics with positive scalar curvature|journal=[[Inventiones Mathematicae]]|volume=130|year=1997|issue=1|pages=1–37|mr=1471884|doi=10.1007/s002220050176|bibcode=1997InMat.130....1T |s2cid=122529381}}}}}}
{{longitem|{{rma|LT98a|tw=4em|{{cite conference|last1=Li|first1=Jun|last2=Tian|first2=Gang|title=Virtual moduli cycles and Gromov–Witten invariants of general symplectic manifolds|book-title=Topics in symplectic 4-manifolds|conference=1st International Press Lectures presented at the University of California, Irvine (March 28–30, 1996)|pages=47–83|series=First International Press Lecture Series|volume=I|publisher=International Press|location=Cambridge, MA|year=1998|mr=1635695|editor1-last=Stern|editor1-first=Ronald J.|isbn=1-57146-019-5|arxiv=alg-geom/9608032}}}}}}
{{longitem|{{rma|LT98b|tw=4em|{{cite journal|last1=Li|first1=Jun|last2=Tian|first2=Gang|title=Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties|journal=[[Organization:Journal of the American Mathematical Society|Journal of the American Mathematical Society]]|volume=11|year=1998|issue=1|pages=119–174|mr=1467172|doi=10.1090/S0894-0347-98-00250-1|s2cid=15201721|doi-access=free}}}}}}
{{longitem|{{rma|LT98c|tw=4em|{{cite journal|last1=Liu|first1=Gang|last2=Tian|first2=Gang|title=Floer homology and Arnold conjecture|journal=[[Journal of Differential Geometry]]|volume=49|year=1998|issue=1|pages=1–74|mr=1642105|doi=10.4310/jdg/1214460936|doi-access=free}}}}}}
{{longitem|{{rma|T00a|tw=4em|{{cite journal|last1=Tian|first1=Gang|title=Gauge theory and calibrated geometry. I|journal=[[Annals of Mathematics]] |series=Second Series|volume=151|year=2000|issue=1|pages=193–268|mr=1745014|doi=10.2307/121116|jstor=121116|arxiv=math/0010015}}}}}}
{{longitem|{{rma|TZ06|tw=4em|{{cite journal|last1=Tian|first1=Gang|last2=Zhang|first2=Zhou|title=On the Kähler–Ricci flow on projective manifolds of general type|journal=[[Chinese Annals of Mathematics, Series B]]|volume=27|year=2006|issue=2|pages=179–192|mr=2243679|doi=10.1007/s11401-005-0533-x|citeseerx=10.1.1.116.5906|s2cid=16476473}}}}}} 
{{longitem|{{rma|ST07|tw=4em|{{cite journal|last1=Song|first1=Jian|last2=Tian|first2=Gang|title=The Kähler–Ricci flow on surfaces of positive Kodaira dimension|journal=[[Inventiones Mathematicae]]|volume=17|year=2007|issue=3|pages=609–653|mr=2357504|doi=10.1007/s00222-007-0076-8|arxiv=math/0602150|bibcode=2007InMat.170..609S |s2cid=735225}}}}}}
{{longitem|{{rma|CT08|tw=4em|{{cite journal|last1=Chen|first1=X. X.|last2=Tian|first2=G.|title=Geometry of Kähler metrics and foliations by holomorphic discs|journal=[[Publications Mathématiques de l'IHÉS|Publications Mathématiques de l'Institut des Hautes Études Scientifiques]]|volume=107|year=2008|pages=1–107|mr=2434691|doi=10.1007/s10240-008-0013-4|arxiv=math/0507148|s2cid=119699845|url=http://www.numdam.org/item/PMIHES_2008__107__1_0/}}}}}}
{{longitem|{{rma|T15|tw=4em|{{cite journal|last1=Tian|first1=Gang|title=K-stability and Kähler–Einstein metrics|journal=[[Communications on Pure and Applied Mathematics]]|volume=68|year=2015|issue=7|pages=1085–1156|mr=3352459|doi=10.1002/cpa.21578|arxiv=1211.4669|s2cid=119303358}} {{erratum|doi=10.1002/cpa.21612|checked=yes}}}}}}
{{Refend}}
'''Books.'''
{{longitem|{{rma|T00b|tw=4em|{{cite book|last1=Tian|first1=Gang|title=Canonical metrics in Kähler geometry|others=Notes taken by [[Biography:Meike Akveld|Meike Akveld]]|series=Lectures in Mathematics ETH Zürich|publisher=Birkhäuser Verlag|location=Basel|year=2000|isbn=3-7643-6194-8|doi=10.1007/978-3-0348-8389-4|mr=1787650}}}}}}
{{longitem|{{rma|MT07|tw=4em|{{cite book|last1=Morgan|first1=John|last2=Tian|first2=Gang|title=Ricci flow and the Poincaré conjecture|series=Clay Mathematics Monographs|volume=3|publisher=Clay Mathematics Institute|location=Cambridge, MA|year=2007|isbn=978-0-8218-4328-4|arxiv=math/0607607|mr=2334563}}<br /><br />{{cite arXiv|author-link1=John Morgan (mathematician)|last1=Morgan|first1=John|last2=Tian|first2=Gang|title=Correction to Section 19.2 of Ricci Flow and the Poincare Conjecture|eprint=1512.00699|year=2015|class=math.DG}}}}}}
{{longitem|{{rma|MT14|tw=4em|{{cite book|last1=Morgan|first1=John|last2=Tian|first2=Gang|title=The geometrization conjecture|series=Clay Mathematics Monographs|volume=5|publisher=Clay Mathematics Institute|location=Cambridge, MA|year=2014|isbn=978-0-8218-5201-9|mr=3186136}}}}}}
{{refend}}

== References ==
<!--See http://en.wikipedia.org/wiki/Wikipedia:Footnotes
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{{reflist}}

== External links ==
* {{MathGenealogy|id=37749}}

{{Veblen Prize recipients}}

{{DEFAULTSORT:Tian, Gang}}
[[Category:Differential geometers]]

{{Sourceattribution|Tian Gang}}