Rigid and Flexible Symplectic Topology

从20世纪80年代辛拓扑的第一步开始，柔性方法和刚性方法就起到了重要的补充作用，尽管M.Gromov引入了全纯曲线方法，但刚性方法在这门学科的发展中起到了主导作用。然而，随着最近的一些重要进展，将柔性侧的结果几乎推到与刚性侧的边界上是可行的。在目前的资助下，这项研究致力于辛拓扑中刚性和柔性方法的发展。在刚性方面最重要的工作包括：完成辛场理论(SFT)的代数基础，包括与Legendrian子流形不变量相关的理论，与可积系统理论的联系，以及进一步发展和推广Legendrian手术技术到所有SFT不变量，并应用于哈密顿动力学，特别是准态和拟态射理论。在柔性方面，研究方向是进一步发展柔性Weinstein流形理论，这是K.Cieliebak，E.Murphy和PI以前发现的一类新的辛流形，以及具有最少交点的拉格朗日浸入理论。另一门应该同时受益于柔性和刚性方法的新学科是辛伪同伦理论，它是理解辛群和接触变换群拓扑的必要组成部分。辛拓扑处于几个数学学科的十字路口，如低维、代数和几何拓扑、哈密顿动力学、代数几何、镜像对称数学理论和可积系统理论。它的发展，特别是辛场理论的发展，极大地改变了这些地区的面貌。在目前的资助下的研究应该会带来辛拓扑机的进一步发展，特别是辛场论的代数形式，以及它在动力学、低维拓扑和量子可积系统理论中的应用的新的视野。

From the very first steps of symplectic topology in 1980s flexible and rigid methods played important complementary roles, thoughwith introduction by M. Gromov of the method of holomorphic curves, rigid methods played the dominating role in the development of the subject. However, it became recently feasible, with a number of important recent advances, to push the results on the flexible side almost to the borderline with the rigid side. The research under the current grant pursues the development of rigid and flexible methods in symplectic topology. Most important tasks on the rigid side include: completion of the algebraic foundations of symplectic field theory (SFT), including the relative theory concerning invariants of Legendrian submanifolds, connections with the theory of integrable systems, and further development and extension of Legendrian surgery technique to all SFT invariants with applications to Hamiltonian dynamics, especially to the theory of quasi-states and quasi-morphisms. On the flexible side, the research will be directed towards further development of the theory of flexible Weinstein manifolds, a new class of symplectic manifolds previously discovered by K. Cieliebak, E. Murphy and the PI, as well as the theory of Lagrangian immersions with minimal number of intersection points. Another new subject which should benefit from both, flexible and rigid methods is symplectic pseudo-isotopy theory, a necessary ingredient towards understanding topology of groups of symplectic and contact transformations.Symplectic topology is at the crossroads of several mathematical disciplines such as low-dimensional, algebraic and geometric topology, Hamiltonian dynamics, algebraic geometry, mathematical theory of mirror symmetry and theory of integrable systems. Its development, and in particular the development of symplectic field theory, significantly changed the face of these areas. The research under the current grant should bring both, the further development of symplectic topological machinery, and in particular of the algebraic formalism of symplectic field theory, as well as new horizons for its applications to dynamics, low-dimensional topology and theory of quantum integrable systems.