# Tian Gang

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Tian Gang (Chinese: 田刚; born November 24, 1958)[1] is a Chinese mathematician. He is an academician of the Chinese Academy of Sciences and of the American Academy of Arts and Sciences. He is known for his contributions to geometric analysis and quantum cohomology especially Gromov-Witten invariants, among other fields. He has been Vice President of Peking University since February 2017.[2]

Tian Gang
Tian at Oberwolfach in 2005
Born24 November 1958 (age 60)
NationalityChina
Alma materHarvard University
Peking University
Nanjing University
Known forBogomolov–Tian–Todorov theorem
AwardsVeblen Prize (1996)
Alan T. Waterman Award (1994)
Scientific career
FieldsMathematics
InstitutionsPrinceton University
Peking University
Doctoral studentsNataša Šešum
Chinese name
Simplified Chinese田刚

## 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 Ph.D. in mathematics from Harvard University, under the supervision of Shing-Tung Yau. This work was so exceptional he was invited to present it at the Geometry Festival that year. 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 awarded the Alan T. Waterman Award in 1994, and the Veblen Prize in 1996. In 2004 Tian was elected a fellow of the American Academy of Arts and Sciences.

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);[3] he has also been Dean of School of Mathematical Sciences, Peking University since 2013.[4] 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.[5]

## Mathematical contributions

Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds under the direction of Yau. In particular he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern class, and showed that hypersurfaces with a Kähler–Einstein metric are stable in the sense of geometric invariant theory. He proved that a Kähler manifold with trivial canonical bundle has trivial obstruction space, known as the Bogomolov–Tian–Todorov theorem[6]

Tian found an explicit formula for Weil-Petersson metric on moduli space of polarized Calabi-Yau manifolds.[7]

Tian made foundational contributions to Gromov-Witten theory. He (jointly with Jun Li) constructed virtual fundamental cycles of the moduli spaces of maps from curves in both algebraic geometry and symplectic geometry. He also (jointly with Y. Ruan) showed that the quantum cohomology ring of a semi-positive symplectic manifold is associative.

He introduced the Analytical Minimal Model program which is known as Tian-Song program in birational geometry. In Kähler geometry he has a new theory which is known as Cheeger-Colding-Tian's theory. Tian's alpha-invariant was introduced by him and was later given an algebraic interpretation by János Kollár and Jean-Pierre Demailly.

He (with Yau and Simon Donaldson) proposed the Yau-Tian-Donaldson conjecture. It was proven by Xiuxiong Chen, Donaldson and Song Sun in 2014.[8][9][10] Tian also gave a proof electronically published on September 16, 2015.[11][12]

In 2006, together with John Morgan and others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman.[13]

Tian served as one of the five members of the Abel Prize Committee.[14] He was also one of the five members of the Ramanujan Prize selection committee. In 2012, he became a member of Leroy P. Steele Prize Committee in AMS.[15]

## Editorial positions

Gang Tian is member of the editorial boards of a number of journals in Mathematics.

1. Annals of Mathematics[16]

2. Annali della Scuola Normale Superiore[17]

3. Journal of Symplectic Geometry[18]

4. Journal of the American Mathematical Society,1995-1998.[19]

5. Geometry & Topology[20]

6. The Journal of Geometric Analysis[21]

7. Geometric and Functional Analysis[22]

9. International Mathematics Research Notices[24]

10. Pacific Journal of Mathematics, 1994-1998.

11. Communications in Analysis and Geometry, 1994-2000.

12. Acta Mathematica Sinica,[25]

13. Mathematics Revista Matemática Complutense [26]

14. Communications in Mathematics and Statistics,[27]

15. Communication in Contemporary Mathematics,[28]

## Selected publications

Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629—646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.

Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with ${\displaystyle C_{1}(M)>0}$ . Invent. Math. 89 (1987), no. 2, 225—246.

Tian, G.; Yau, Shing-Tung. Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), no. 3, 579—609.

Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101—172.

Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32 (1990), no. 1, 99—130.

Ruan, Yongbin; Tian, Gang. A mathematical theory of quantum cohomology. J. Differential Geom. 42 (1995), no. 2, 259—367.

Tian, Gang. Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), no. 1, 1--37.

Ruan, Yongbin; Tian, Gang. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130 (1997), no. 3, 455—516.

Li, Jun; Tian, Gang. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc. 11 (1998), no. 1, 119—174.

Liu, Gang; Tian, Gang. Floer homology and Arnold conjecture. J. Differential Geom. 49 (1998), no. 1, 1--74.

Liu, Xiaobo; Tian, Gang. Virasoro constraints for quantum cohomology. J. Differential Geom. 50 (1998), no. 3, 537—590.

Tian, Gang. Gauge theory and calibrated geometry. I. Ann. of Math. (2) 151 (2000), no. 1, 193—268.

Tian, Gang; Zhu, Xiaohua. Uniqueness of Kähler-Ricci solitons. Acta Math. 184 (2000), no. 2, 271—305.

Cheeger, J.; Colding, T. H.; Tian, G. On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12 (2002), no. 5, 873—914.

Tao, Terence; Tian, Gang. A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Amer. Math. Soc. 17 (2004), no. 3, 557—593.

Tian, Gang; Viaclovsky, Jeff. Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160 (2005), no. 2, 357—415.

Cheeger, Jeff; Tian, Gang. Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. Vol. 19, No. 2 (2006), 487—525.

Morgan, John; Tian, Gang. Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007, 525pp.

Song, Jian; Tian, Gang. The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170 (2007), no. 3, 609—653.

Chen, X. X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1--107.

Kołodziej, Sławomir; Tian, Gang A uniform ${\displaystyle L^{\infty }}$  estimate for complex Monge-Ampère equations. Math. Ann. 342 (2008), no. 4, 773–787.

Mundet i Riera, I.; Tian, G. A compactification of the moduli space of twisted holomorphic maps. Adv. Math. 222 (2009), no. 4, 1117–1196.

Rivière, Tristan; Tian, Gang The singular set of 1-1 integral currents. Ann. of Math. (2) 169 (2009), no. 3, 741–794.

Tian, Gang Finite-time singularity of Kähler-Ricci flow. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1137–1150.

## References

1. ^ "1996 Oswald Veblen Prize" (PDF). AMS. 1996.
2. ^ "民盟中央副主席田刚出任北京大学副校长". The Paper. 2017-03-01. Retrieved 2018-10-31.
3. ^ Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html
4. ^ History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html
5. ^ "ICTP - Governance". www.ictp.it. Retrieved 2018-05-28.
6. ^ [Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629—646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.]
7. ^ [Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629—646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987. ]
8. ^ Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (January 2015), no. 1, 183–197.
9. ^ 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 (January 2015), no. 1, 199–234.
10. ^ 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 (January 2015), no. 1, 235–278.
11. ^ Gang Tian: K-Stability and Kähler-Einstein Metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 7, pages 1085–1156, July 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21578/abstract
12. ^ Gang Tian: Corrigendum: K-stability and Kähler-Einstein metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 11, pages 2082–2083, September 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21612/full
13. ^ Morgan, John W.; Gang Tian (25 July 2006). "Ricci Flow and the Poincaré Conjecture". arXiv:math.DG/0607607.
14. ^ http://www.ams.org/notices/201304/rnoti-p480.pdf
15. ^ "Top Secret of Abel Prize". www.eusja.org.
16. ^ "Editorial Board - Annals of Mathematics". annals.math.princeton.edu.
17. ^ "Editorial board". annaliscienze.sns.it.
18. ^ "Journal of Symplectic Geometry". intlpress.com.
19. ^
20. ^ "Geometry & Topology". www.msp.warwick.ac.uk.
21. ^
22. ^
23. ^
24. ^ "Editorial_Board - International Mathematics Research Papers - Oxford Academic". www.oxfordjournals.org.
25. ^ "Acta Mathematica Sinica, English Series". springer.com.
26. ^
27. ^
28. ^ "CCM Editorial Board". www.worldscientific.com.