Rigid and Flexible Symplectic Topology

从20世纪80年代辛拓扑学的第一步开始，柔性方法和刚性方法就起着重要的互补作用，尽管随着M. Gromov引入全纯曲线方法，刚性方法在该学科的发展中起着主导作用。然而，随着最近一些重要的进展，将柔性方面的结果几乎推向刚性方面的边界，这在最近变得可行。目前的研究经费旨在发展辛拓扑的刚性和柔性方法。在刚性方面最重要的任务包括：完成辛场论（SFT）的代数基础，包括关于Legendrian子流形不变量的相关理论，与可积系统理论的联系，以及将Legendrian手术技术进一步发展和推广到所有SFT不变量，并将其应用于哈密顿动力学，特别是拟态和拟态的理论。在柔性方面，研究方向将是进一步发展柔性Weinstein流形理论，这是K. Cieliebak， E. Murphy和PI先前发现的一类新的辛流形，以及具有最小交点数的Lagrangian浸入理论。另一个将受益于柔性和刚性两种方法的新学科是辛伪同位素理论，它是理解辛和接触变换群拓扑结构的必要组成部分。辛拓扑学是几个数学学科如低维、代数和几何拓扑学、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的交叉点。它的发展，特别是辛场论的发展，极大地改变了这些领域的面貌。本基金的研究将为辛拓扑力学，特别是辛场论的代数形式的进一步发展，以及辛拓扑力学在动力学、低维拓扑和量子可积系统理论方面的应用开辟新的前景。

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.