Gang Tian
Encyclopedia
Tian Gang is a Chinese
mathematician
and an academician
of the Chinese Academy of Sciences
. He is known for his contributions to geometric analysis and quantum cohomology, among other fields. He was born in Nanjing
, China
, 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).
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
.
s under the direct 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.
He (jointly with Jun Li) constructed the moduli space
s of maps from curves in both algebraic geometry
and symplectic geometry and studied the obstruction theory
on these moduli spaces. He also (jointly with Y. Ruan) showed that the quantum cohomology
ring of a symplectic manifold is associative.
In 2006, together with John Morgan
of Columbia University
, amongst others, Tian helped verify the proof of the Poincaré conjecture
given by Grigori Perelman
.
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 $C\sb 1(M)>0$. 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 $L^\infty$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.
Xiaodong Cao, MIT, 2002;
Sandra Francisco, MIT, 2005;
Zuoliang Hou, MIT, 2004;
Ljudmila Kamenova, MIT, 2006;
Peng Lu State, University of New York at Stony Brook, 1996;
Zhiqin Lu, New York University, 1997;
Dragos Oprea, MIT, 2005;
Yanir Rubinstein, MIT, 2008;
Sema Salur, Michigan State University, 2000;
Bianca Santoro, MIT, 2006;
Natasa Sesum, MIT, 2004;
Jake Solomon, MIT, 2006;
Michael Usher, MIT, 2004;
Lijing Wang, MIT, 2003;
Hao Wu, MIT, 2004;
Zhiyu Wu, Columbia University, 1998;
Baozhong Yang, MIT, 2000;
Zhou Zhang, MIT, 2006;
Hans-Joachim Hein, Princeton, 2010;
Richard Bamler, Princeton, 2011;
Chi Li, Princeton, 2011;
Mohammad Farajzadeh Tehrani, Princeton 2011;
Giulia Saccà, Princeton, 2012;
Guangbo Xu, Princeton 2012;
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and an academician
Academician
The title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...
of the Chinese Academy of Sciences
Chinese Academy of Sciences
The Chinese Academy of Sciences , formerly known as Academia Sinica, is the national academy for the natural sciences of the People's Republic of China. It is an institution of the State Council of China. It is headquartered in Beijing, with institutes all over the People's Republic of China...
. He is known for his contributions to geometric analysis and quantum cohomology, among other fields. He was born in Nanjing
Nanjing
' is the capital of Jiangsu province in China and has a prominent place in Chinese history and culture, having been the capital of China on several occasions...
, China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, 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
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
and Peking University
Peking University
Peking University , colloquially known in Chinese as Beida , is a major research university located in Beijing, China, and a member of the C9 League. It is the first established modern national university of China. It was founded as Imperial University of Peking in 1898 as a replacement of the...
. 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).
Biography
Tian graduated from Nanjing UniversityNanjing University
Nanjing University , or Nanking University, is one of the oldest and most prestigious institutions of higher learning in China...
in 1982, and received a master's degree
Master's degree
A master's is an academic degree granted to individuals who have undergone study demonstrating a mastery or high-order overview of a specific field of study or area of professional practice...
from Peking University in 1984. In 1988, he received a Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
from Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
, after having studied under Shing-Tung Yau
Shing-Tung Yau
Shing-Tung Yau is a Chinese American mathematician working in differential geometry. He was born in Shantou, Guangdong Province, China into a family of scholars from Jiaoling, Guangdong Province....
. This work was so exceptional he was invited to present it at the Geometry Festival
Geometry Festival
-1985 at Penn:* Marcel Berger* Pat Eberlein* Jost Eschenburg* Friedrich Hirzebruch* Blaine Lawson* Leon Simon* Scott Wolpert* Deane Yang-1986 at Maryland:* Uwe Abresch, Explicit constant mean curvature tori...
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
Ministry of Education of the People's Republic of China
The Ministry of Education of the People's Republic of China , formerly Ministry of Education, Central People's Government from 1949 to 1954, State Education Commission from 1985 to 1998, is headquartered in Beijing. It is the agency of the State Council which regulates all aspects of the...
. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award
Alan T. Waterman Award
The Alan T. Waterman Award is the United States's highest honorary award for scientists no older than 35. It is awarded on a yearly basis by the National Science Foundation. In addition to the medal, the awardee receives a grant of $500,000 to be used for advanced scientific research at the...
in 1994, and the Veblen Prize
Oswald Veblen Prize in Geometry
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen...
in 1996. In 2004 Tian was inducted into the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
.
Mathematical contributions
Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifoldComplex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s under the direct 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
Geometric invariant theory
In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
. He proved that a Kähler manifold with trivial canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
has trivial obstruction space, known as the Bogomolov–Tian–Todorov theorem.
He (jointly with Jun Li) constructed the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
s of maps from curves in both algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
and symplectic geometry and studied the obstruction theory
Obstruction theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants.-In homotopy theory:...
on these moduli spaces. He also (jointly with Y. Ruan) showed that the quantum cohomology
Quantum cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more...
ring of a symplectic manifold is associative.
In 2006, together with John Morgan
John Morgan (mathematician)
John Willard Morgan is an American mathematician, well known for his contributions to topology and geometry. He is currently the director of the Simons Center for Geometry and Physics at Stony Brook University.-Life:...
of Columbia University
Columbia University
Columbia University in the City of New York is a private, Ivy League university in Manhattan, New York City. Columbia is the oldest institution of higher learning in the state of New York, the fifth oldest in the United States, and one of the country's nine Colonial Colleges founded before the...
, amongst others, Tian helped verify the proof of the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
given by Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
.
Publications
(Selected)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 $C\sb 1(M)>0$. 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 $L^\infty$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.
Students
Vladimir Bozin, MIT, 2004;Xiaodong Cao, MIT, 2002;
Sandra Francisco, MIT, 2005;
Zuoliang Hou, MIT, 2004;
Ljudmila Kamenova, MIT, 2006;
Peng Lu State, University of New York at Stony Brook, 1996;
Zhiqin Lu, New York University, 1997;
Dragos Oprea, MIT, 2005;
Yanir Rubinstein, MIT, 2008;
Sema Salur, Michigan State University, 2000;
Bianca Santoro, MIT, 2006;
Natasa Sesum, MIT, 2004;
Jake Solomon, MIT, 2006;
Michael Usher, MIT, 2004;
Lijing Wang, MIT, 2003;
Hao Wu, MIT, 2004;
Zhiyu Wu, Columbia University, 1998;
Baozhong Yang, MIT, 2000;
Zhou Zhang, MIT, 2006;
Hans-Joachim Hein, Princeton, 2010;
Richard Bamler, Princeton, 2011;
Chi Li, Princeton, 2011;
Mohammad Farajzadeh Tehrani, Princeton 2011;
Giulia Saccà, Princeton, 2012;
Guangbo Xu, Princeton 2012;
External links
- Veblen prize citation
- M.I.T. home page for Gang Tian (probably out of date).