The focus of this project is to apply methods from differential topology, geometric analysis and algebraic geometry to study symplectic four manifolds. Symplectic four manifolds can be divided into four categories according to their Kodaira dimensions, which take values -1, 0, 1 and 2. The classification of those with Kodaira dimenision -1 has been achieved, to which the investigator has made essential contribution. Tian-Jun Li proposes to classify those with Kodaira dimension 0. Fiber sum is the most powerful construction of symplectic four manifolds, and one would like manifolds constructed this way to be minimal. Tian-Jun Li has shown that a large class of fiber sums are indeed minimal. The investigator believes that he can prove it for all fiber sums using his work on the minimal genus problem for rational surfaces.
An n manifold is a space that locally looks like the Euclidean space of dimension n. For example, the space-time universe we live in is a four manifold. A symplectic four manifold is a four manifold with a symplectic structure, a very basic structure that underlies almost all the equations of classical and quantum physics. Thus symplectic four manifolds play a central role in mathematics and physics. The investigator aims to gain some understanding of the fundamental problem: classifying symplectic four manifolds.
