# Gang Tian

This is a Chinese name; the family name is Tian.
Gang Tian
Gang Tian at Oberwolfach in 2005
Born 24 November 1958 (age 57)
Nanjing, Jiangsu, China
Nationality China
Fields Mathematics
Institutions Princeton University
Peking University
Alma mater Harvard University
Peking University
Nanjing University
Doctoral students Nataša Šešum
Known for Bogomolov–Tian–Todorov theorem
Notable awards Veblen Prize (1996)
Alan T. Waterman Award (1994)

Tian Gang (simplified Chinese: 田刚; traditional Chinese: 田剛; pinyin: Tián Gāng; born November 1958)[1] is a Chinese-American mathematician and an academician 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 was born in Nanjing, and was a professor of mathematics at MIT from 1995–2006 (holding the chair of Simons Professor of Mathematics from 1996), but now divides his time between Princeton University and Peking University. His employment at Princeton started from 2003, and now he is entitled Higgins Professor of Mathematics; starting 2005, he has been the director of Beijing International Center for Mathematical Research (BICMR);[2] he has also been Dean of School of Mathematical Sciences, Peking University since 2013.[3] He with John Milnor involved as Senior Scholars of The Clay Mathematics Institute (CMI). Since 2011, Gang Tian become director of Sino-French Research Program in Mathematic in "le Centre National de la Recherche Scientifique"(CNRS) in Paris. Since 2010, He became Scientific council for International Center for Theoretical Physics in Trieste in Italy.

## Biography

Tian 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, after having studied under 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 the School of Mathematical Sciences at Peking University, under the "Cheung Kong Scholars Programme" (长江计划) of the Ministry of Education. 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 inducted into the American Academy of Arts and Sciences.

## 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.[4]

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 introduced by him and later Kollár and Demailly gave an algebraic interpretation to Tian's alpha-invariant.

He (with Yau and Donaldson) proposed the Yau-Tian-Donaldson conjecture. It was first solved by Chen, Donaldson and Sun in January 13, 2014. [5][6][7] Later, Tian also gave a proof in August, 2015.[8][9]

In 2006, together with John Morgan of Columbia University (now at Stony Brook University), amongst others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman.[10]

Gang Tian was once one of the five members of the Abel Prize Committee, which is sometimes considered to be the second most important prize in mathematics after the Fields medal.[11]

Gang Tian was also once one of the five members of the Ramanujan Prize selection committee.

Since 2012 he became member of Leroy P. Steele Prize Committee in AMS.[12]

## Editorial Positions

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

1. Annals of Mathematics[13]

2. Annali della Scuola Normale Superiore[14]

3. Journal of Symplectic Geometry[15]

4. Journal of the American Mathematical Society,1995-1998.[16]

5. Geometry & Topology[17]

6. The Journal of Geometric Analysis[18]

7. Geometric and Functional Analysis[19]

9. International Mathematics Research Notices[21]

10. Pacific Journal of Mathematics, 1994-1998.

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

12. Acta Mathematica Sinica,[22]

13. Mathematics Revista Matemática Complutense [23]

14. Communications in Mathematics and Statistics,[24]

15. Communication in Contemporary Mathematics,[25]

## 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.

## Students

Richard Bamler, Princeton, 2011[26]

Xiaodong Cao, MIT, 2002;[28]

Eaman Eftekhary,Princeton;2004[29]

Sandra Francisco, MIT, 2005;

Alessandro Ghigi, Scuola Normale Superiore, Pisa, 2003,[30]

Hans-Joachim Hein, Princeton, 2010;

Zuoliang Hou, MIT, 2004;

Hassan Jolany, University of Lille1, France, 2016 [31]

Ljudmila Kamenova, MIT, 2006;

Chi Li, Princeton, 2012;

Peng Lu, State University of New York at Stony Brook, 1996;

Zhiqin Lu, New York University, 1997;

Heather Macbeth, Princeton 2015

Aaron Naber, Princeton, 2009;

Dragos Oprea, MIT, 2005;

Yanir Rubinstein, MIT, 2008;

Giulia Saccà, Princeton 2013;

Sema Salur, Michigan State University, 2000;

Bianca Santoro, MIT, 2006;

Natasa Sesum, MIT, 2004;

Liangming Shen, Princeton 2015

Jake Solomon, MIT, 2006;

Jun S. Song, MIT, 2001;

Michael Usher, MIT, 2004;

Lijing Wang, MIT, 2003;

Hao Wu, MIT, 2004;

Zhiyu Wu, Columbia University, 1998;

Guangbo Xu, Princeton 2013;

Baozhong Yang, MIT, 2000;

Zhou Zhang, MIT, 2006;

## References

1. ^ http://www.ams.org/notices/199603/comm-veblen.pdf
2. ^ Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html
3. ^ History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html
4. ^ The history about Tian-Todorov Lemma
5. ^ 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.
6. ^ 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.
7. ^ 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.
8. ^ 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
9. ^ 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
10. ^ Morgan, John W.; Gang Tian (25 July 2006). "Ricci Flow and the Poincaré Conjecture". arXiv:math.DG/0607607.
11. ^ http://www.ams.org/notices/201304/rnoti-p480.pdf
12. ^ http://www.eusja.org/top-secret-of-abel-prize/
13. ^ http://annals.math.princeton.edu/board
14. ^ http://annaliscienze.sns.it/index.php?page=EditorialBoard
15. ^ http://intlpress.com/site/pub/pages/journals/items/jsg/_home/editorial/index.html
16. ^ http://www.ams.org/publications/journals/journalsframework/jams/editorial_history_jams
18. ^ http://www.springer.com/mathematics/geometry/journal/12220?detailsPage=editorialBoard
19. ^ http://www.springer.com/birkhauser/mathematics/journal/39?detailsPage=editorialBoard