Mathematical Contributions
Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds 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 spaces 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.
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