A study on foliations, contactstrucures, and symplectic styructures on 3 and 4 dimensional manifolds

三松和研究员Miyoshi与其他人合作，研究了与叶理和(所谓的瑟斯顿-温克肯珀)接触结构有关的欧拉切向丛，这是接触结构向叶理收敛的典型类别。特别是，他们从单行的拓扑学观点出发，研究了欧拉类的(非)消失以及对瑟斯顿不等式和本尼昆不等式的违反。因此，具有边界的曲面的某一类映射类既不能表示为仅右手Dehn扭曲的乘积，也不能仅表示为右手Dehn扭曲的乘积。这一结果被介绍在几个研讨会上，包括2006年3月的MSJ年度会议，作为Miyoshi的特别邀请演讲。本文正在提交中。研究者小野从Floer理论和Seiberg-Witten理论研究了辛同调。他对这一领域的贡献是深远的。他从接触微分同胚群的角度研究了叶化理论与接触结构理论之间的关系。研究人员松本进一步深入到层理理论，研究了Lie层理的末端。首席研究人员还在保体积微分同胚几何和无限维哈密顿系统的几何框架下研究了不可压缩流体动力学。他证明了从这种整体微分几何的观点来看，即使对有耗散的粘性流体也是有效的。这项研究在许多研讨会上发表，特别是在2005年9月的MSJ年会上作为一个特别的项目演讲。

The head Mitsumatsu and an investigator Miyoshi colaborated with others to study the euler class of tangent bundles to foliations and (so called Thurston-Winkelnkemper's) contact structures which are associate with spinnable structures, as a typical class of convergences of contact structures to foliations. Especially they studied the (non-)vanishing of the euler class and the violation of Thurston's inequality and Bennequin's inequlity, from the topological view point of monodromies. As a consequence, a certain class of mapping classes of a surface with boundary can be presented neither as a product of only right-handed Dehn twists nor as that of only right-handed ones. This result was presented in several symposiums including the annual meeting of MSJ in March 2006 as a special invited talk by Miyoshi. The paper is under submission.The investigator Ono studied the symplectec homology from Floer theory as well as from Seiberg-Witten theory. Including the solution to the Flux conjecture as well as the detemination of the symplectic filling of the link of simple singularities, his contributions to this area are profound.The investigator Tsuboi studied the relationship between foliation theory and that of contact structures from the view point of the group of contact diffeomorphisms. The investigator Matsumoto stepped further to the foliation theory and studied the ends of Lie foliations.The head investigator also studied the incompressible fluid dynamics in the framework of the geometry of volume preserving diffeomorphisms and infinite dimensional Hamiltonian systems. He proved that looking from the point of view of such global differential geometry is still valid even to viscous fluides with dissipations. This study is presented in many symposiums, especially as a special project talk in the annual meeting of MSJ in September 2005.