Approximately holomorphic techniques and monodromy invariants in symplectic topology

这个项目旨在使用近似全纯技术（由Donaldson引入并由Auroux进一步发展）和相应的单不变量来研究辛流形的拓扑。辛流形上的近似全纯线性系统产生了诸如Lefschetz铅笔和映射到复射影平面的结构，其单性由映射类群或编织群中具有值的态射描述。通过对辛流形的单不变量的研究，将对辛流形与复射影流形之间的关系：辛与复变形等价、同位素与非同位素现象、辛流形的拓扑约束等问题有新的认识。此外，将单调不变量与Gromov-Witten不变量或flower同调联系起来应该有助于理解镜像对称，而该项目的更多组合方面与映射类和辫群的算法和计算复杂性密切相关。辛流形是具有特殊结构的几何空间，最早出现在经典力学的哈密顿公式中。数学家最近对它们的几何和拓扑（内在结构）非常感兴趣，部分原因是由于理论物理（弦理论）的激励问题。本项目旨在研究辛流形的拓扑结构，使用首先由S. Donaldson和随后由Auroux提出的方法，这使得通过涉及辫群（与结密切相关的概念）的组合不变量获得完整的描述成为可能。该项目的主要目标之一是将辛流形的拓扑特征与复杂代数流形（一种更特殊、更容易理解的几何空间类别）的拓扑特征联系起来。此外，在其他领域的一些应用，如数学物理（弦理论中的“镜像对称”对偶）和密码学（涉及辫群的组合问题的计算复杂性）将被探索。

DMS-0244844Denis AurouxThis project aims to study the topology of symplectic manifolds usingapproximately holomorphic techniques (introduced by Donaldson andfurther developped by Auroux) and the corresponding monodromyinvariants. Approximately holomorphic linear systems on symplecticmanifolds give rise to structures such as Lefschetz pencils and maps tothe complex projective plane, whose monodromy is described by morphismswith values in mapping class groups or braid groups. By studying themonodromy invariants of symplectic manifolds, new insight will beobtained into the relationships between symplectic manifolds andcomplex projective manifolds: symplectic versus complex deformationequivalence, isotopy and non-isotopy phenomena, topological constraintson symplectic manifolds. In addition, relating monodromy invariantswith Gromov-Witten invariants or Floer homology should help tounderstand mirror symmetry, while the more combinatorial aspects of theproject are closely related to the algorithmics and computationalcomplexity of mapping class and braid groups.Symplectic manifolds are geometric spaces with special structures,which first arose in the Hamiltonian formulation of classicalmechanics. Mathematicians have recently become very interested in theirgeometry and topology (intrinsic structure), in part due to motivatingquestions from theoretical physics (string theory). This project aimsto study the topology of symplectic manifolds using an approachdevelopped first by S. Donaldson and subsequently by Auroux, whichmakes it possible to obtain a complete description by combinatorialinvariants involving braid groups (a concept closely related to knots).One of the main goals of the project is to relate the topologicalfeatures of symplectic manifolds with those of complex algebraicmanifolds (a more special, much better understood class of geometricspaces). In addition, some applications to other domains such asmathematical physics (the "mirror symmetry" duality in string theory)and cryptography (the computational complexity of combinatorialproblems involving braid groups) will be explored.