Mathematical Sciences: Symplectic Topology and Its Applications

所研究的问题属于辛拓扑的核问题。主题包括：接触结构的分类，特别是3流形上的可填充接触结构；伪凸（或接触型）边界拓扑对复（辛）流形拓扑的影响Legendrian节的分类；接触流形类容辛不变量的定义与研究多次谐波函数的小型理论复形态群的格律性质全纯圆盘填充技术的发展及其应用。重要的是，这些问题处于几个复杂变量和辛几何的前沿，因此这两个领域都将从这个方向的进展中受益。与数学物理的深刻联系也存在于这些分析和几何领域。

The problems to be studied are among the kernel problems of symplectic topology. The topics include: Classification of contact structures and, especially, fillable contact structures on 3-manifolds; influence of the topology of a pseudoconvex (or contact type) boundary on the topology of a complex (or symplectic) manifold; classification of Legendrian knots; definition and study of capacity-like symplectic invariants for contact manifolds; Smale-type theory for plurisubharmonic functions; metrical properties of the symplectomorphism group; developing the technique of filling by holomorphic discs and its applications. It is important that these problems are at the frontier of several complex variables and symplectic geometry, so both fields will benefit from progress in this direction. Profound connections to mathematical physics also lie in these areas of analysis and geometry.