Gang Tian

This is a Chinese name; the family name is Tian.

Gang Tian

Gang Tian at Oberwolfach in 2005

Born|  (1958-11-24) 24 November 1958 (age 67)

Nanjing, Jiangsu, China

Nationality| China
Alma mater| Harvard University
Peking University
Nanjing University
Known for| Bogomolov–Tian–Todorov theorem
Awards| Veblen Prize (1996)
Alan T. Waterman Award (1994)
Scientific career
Fields| Mathematics
Institutions| Princeton University
Peking University
Doctoral advisor| Shing-Tung Yau
Doctoral students| Nataša Šešum

Tian Gang (simplified Chinese: 田刚; traditional Chinese: 田剛; pinyin: Tián Gāng; born November 1958)[1] is a Chinese 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 the Massachusetts Institute of Technology 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 his title there is Higgins Professor of Mathematics; starting 2005, he has been the director of the Beijing International Center for Mathematical Research (BICMR);[2] he has also been Dean of School of Mathematical Sciences, Peking University since 2013.[3] 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.[4]

## Contents

* 1 Biography
* 2 Mathematical contributions
* 3 Editorial Positions
* 4 Selected publications
* 5 References
* 6 External links

## Biography

Tian 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, 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.[5]

Tian found an explicit formula for Weil-Petersson metric on moduli space of polarized Calabi-Yau manifolds.[6]

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 solved by Chen, Donaldson and Sun which was electronically published on March 28, 2014.[7][8][9] Tian also gave a proof electronically published on September 16, 2015.[10][11]

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.[12]

Gang Tian was once one of the five members of the Abel Prize Committee.[13]

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

In 2012, he became member of Leroy P. Steele Prize Committee in AMS.[14]

## Editorial Positions

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

1\. Annals of Mathematics[15]

2\. Annali della Scuola Normale Superiore[16]

3\. Journal of Symplectic Geometry[17]

4\. Journal of the American Mathematical Society,1995-1998.[18]

5\. Geometry & Topology[19]

6\. The Journal of Geometric Analysis[20]

7\. Geometric and Functional Analysis[21]

8\. Advances in Mathematics [22]

9\. International Mathematics Research Notices[23]

10\. Pacific Journal of Mathematics, 1994-1998.

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

12\. Acta Mathematica Sinica,[24]

13\. Mathematics Revista Matemática Complutense [25]

14\. Communications in Mathematics and Statistics,[26]

15\. Communication in Contemporary Mathematics,[27]

## Selected publications

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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 C1(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 L∞ 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. ↑ 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. ↑ "ICTP - Governance" (in en). https://www.ictp.it/about-ictp/governance.aspx.
5. ↑ The history about Tian-Todorov Lemma
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. ↑ 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.
8. ↑ 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.
9. ↑ 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.
10. ↑ 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
11. ↑ 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
12. ↑ Morgan, John W.; Gang Tian (25 July 2006). "Ricci Flow and the Poincaré Conjecture". arXiv:math.DG/0607607.
13. ↑ http://www.ams.org/notices/201304/rnoti-p480.pdf
14. ↑ http://www.eusja.org/top-secret-of-abel-prize/
15. ↑ http://annals.math.princeton.edu/board
16. ↑ http://annaliscienze.sns.it/index.php?page=EditorialBoard
17. ↑ http://intlpress.com/site/pub/pages/journals/items/jsg/_home/editorial/index.html
18. ↑ http://www.ams.org/publications/journals/journalsframework/jams/editorial_history_jams
19. ↑ http://www.msp.warwick.ac.uk/gt/about/journal/editorial.html
20. ↑ https://www.springer.com/mathematics/geometry/journal/12220?detailsPage=editorialBoard
21. ↑ https://www.springer.com/birkhauser/mathematics/journal/39?detailsPage=editorialBoard
22. ↑ http://www.journals.elsevier.com/advances-in-mathematics/editorial-board/
23. ↑ http://www.oxfordjournals.org/our_journals/imrp/editorial_board.html
24. ↑ https://www.springer.com/mathematics/journal/10114
25. ↑ https://www.springer.com/mathematics/journal/13163
26. ↑ https://www.springer.com/mathematics/journal/40304
27. ↑ http://www.worldscientific.com/page/ccm/editorial-board

## External links

* Veblen prize citation
* Gang Tian at the Mathematics Genealogy Project
* M.I.T. home page for Gang Tian (probably out of date).

* v
* t
* e

Recipients of the Oswald Veblen Prize in Geometry

* 1964 Christos Papakyriakopoulos
* 1964 Raoul Bott
* 1966 Stephen Smale
* 1966 Morton Brown and Barry Mazur
* 1971 Robion Kirby
* 1971 Dennis Sullivan
* 1976 William Thurston
* 1976 James Harris Simons
* 1981 Mikhail Gromov
* 1981 Shing-Tung Yau
* 1986 Michael Freedman
* 1991 Andrew Casson and Clifford Taubes
* 1996 Richard S. Hamilton and Gang Tian
* 2001 Jeff Cheeger, Yakov Eliashberg and Michael J. Hopkins
* 2004 David Gabai
* 2007 Peter Kronheimer and Tomasz Mrowka; Peter Ozsváth and Zoltán Szabó
* 2010 Tobias Colding and William Minicozzi II; Paul Seidel
* 2013 Ian Agol and Daniel Wise
* 2016 Fernando Codá Marques and André Neves
* 2019 Xiuxiong Chen, Simon Donaldson and Song Sun

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