The focus of this project is to apply methods from differential topology, geometric analysis and algebraic geometry to study symplectic four manifolds. For symplectic four manifolds, the equivalence between Seiberg-Witten invariants and the symplectic Gromov-Taubes invariants has led to many striking results in symplectic topology. In collaboration with Liu, Li has set up the parametrized Seiberg-Witten theory and derived a wall-crossing formula. The parametrized Seiberg-Witten theory is particularly interesting for families of symplectic manifolds. Li proposes to develop further the parametrized Seiberg-Witten and the parametrized Gromov-Taubes theories and show that these two theories are equivalent for symplectic families. The parametrized theories should be useful for studying the isotopies of diffeomorphisms as well as symplectomorphisms. And the equivalence between the two theories is expected to play an important role in the classification of symplectic four manifolds, especially those with torsion canonical classes. Recently, it has been shown that smooth Lefschetz fibrations provide a link between the topology of symplectic four manifolds, the geometry of the moduli space of curves and the algebra of the mapping class groups. The investigator plans to explore the beautiful and rich interplay to advance understanding of all these objects.
An n manifold is a space that locally looks like Euclidean space of dimension n. For example, the space-time universe we live in is a four manifold. A symplectic structure is a very basic structure that underlies almost all the equations of classical and quantum physics. A symplectic four manifold is a four manifold with a symplectic structure. Thus symplectic four manifolds play a central role in mathematics and physics. The fundamental problem is to classify all symplectic four manifolds. The investigator aims to gain some understanding of the general shape of symplectic four manifolds.
