Tian Gang

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Note: This article places the family name before the person's first name. This is the usual order in Chinese. Tian is the family name here, Gang is the first name.

Tian Gang ( Chinese  田 剛  /  田 刚 , Pinyin Tián Gāng ; born November 24, 1958 in Nanjing ) is a Chinese mathematician who deals with differential geometry and topology.

Tian Gang in Oberwolfach 2005

Life

Tian studied at Nanjing University (pre-diploma 1982) and Peking University , where he received his diploma in mathematics in 1984. In 1988 he received his PhD under Shing-Tung Yau from Harvard University . He then went to Princeton University , the State University of New York at Stony Brook and, from 1991, at the Courant Institute of Mathematical Sciences of New York University . From 1995 he was at the Massachusetts Institute of Technology (MIT). Today he is also a math professor at Princeton University and Beijing University (“Cheung Kong Scholar Professor” since 1998). He was visiting professor at the Institut des Hautes Études Scientifiques (IHES), the Institute for Advanced Study , Stanford University (Bergmann Lecture 1994) and the Academia Sinica in Beijing.

From 1991 to 1993 he was a Sloan Research Fellow . In 1994 he received the Alan Waterman Prize of the National Science Foundation of the USA and in 1996 the Oswald Veblen Prize . In 2004 he became a member of the American Academy of Arts and Sciences . In 1990 he was invited speaker at the International Congress of Mathematicians (ICM) in Kyōto (Kähler-Einstein metrics on algebraic manifolds) and in 2002 he gave a plenary lecture at the ICM in Beijing (Geometry and Nonlinear Analysis). In 2012/13 and 2013/14 he was on the Abel Prize Committee.

Aaron Naber is one of his PhD students .

Act

Tian initially dealt with the existence of Kähler-Einstein metrics on compact complex manifolds following his teacher Yau. That is, the question of those manifolds that allow Kähler metrics as well as being Einstein manifolds (their Ricci curvature is proportional to the metric tensor , whereby the sign of the proportionality constant depends on the first Chern class ). Examples are the Calabi-Yau manifolds important in string theory (in which the first Chern class vanishes). The existence in the case of negative first Chern class was proved by Thierry Aubin in 1976 and in the case of vanishing Chern classes the existence followed from Yau's proof of the Calabi conjecture (1977). In the case of the positive Chern class, Yau found a counterexample (the complex projective plane with blow-up in two points). The question of the existence of Kähler-Einstein metrics on complex surfaces with a positive Chern class was then completely clarified by Tian. He also demonstrated the stability (in the sense of David Mumford's geometric invariant theory ) of the Kähler-Einstein metric in this case (which was suggested by Yau). In 2012 he announced a proof of the conjecture by Simon Donaldson , Yau and Tian, ​​which formulates a criterion for the existence of Kähler-Einstein metrics on compact Kähler manifolds with positive first Chern class (Fano manifolds). At the same time, Donaldson, Xiuxiong Chen, and Song Sun announced evidence, and a priority dispute ensued.

Tian found an explicit formula for the Weil-Petersson metric on modular spaces of polarized Calabi-Yau manifolds.

The name of Tian is also associated with the Bogomolov-Tian-Todorov theorem about the smoothness (absence of obstructions) of the modular space of Calabi-Yau spaces (with Bogomolov (preliminary work), Andrei Todorov , who also proved it).

He also investigated the module spaces of curves in algebraic and symplectic geometry and quantum cohomology with Ruan Yongbin (deformations of the cohomology ring of symplectic manifolds, specifically they proved the associativity of the quantum cohomology ring).

In 2006 he played an important role in the verification of the correctness of the proof of the Poincaré conjecture by Grigori Perelman . With John Morgan he published a complete version of the proof (John Morgan, Tian "Ricci Flow and the Poincare Conjecture", Clay Mathematics Institute 2007), which had previously only been published by Perelman in preprints (and not with all the necessary details).

In 2009, he and Jian Song proposed an analytical minimal model program (MMP) with Ricci flow.

Fonts

  • Canonical metrics in Kähler Geometry . Birkhäuser, 2000
  • with John Morgan: Ricci Flow and the Poincaré Conjecture . American Mathematical Society, 2007

Web links

References

  1. ↑ Tiang: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric . In: Yau: Mathematical aspects of string theory . World Scientific, 1987, pp. 629-646
  2. ↑ According to Todorov, originally by him.
  3. ^ Song, Tian: The Kahler-Ricci flow through singularities . 2009, arxiv : 0909.4898
personal data
SURNAME Tian, ​​gang
ALTERNATIVE NAMES 田 刚 (Chinese)
BRIEF DESCRIPTION Chinese mathematician
DATE OF BIRTH November 24, 1958
PLACE OF BIRTH Nanjing