Approximately holomorphic techniques and monodromy invariants in symplectic topology

DMS-0244844 Denis Auroux该项目旨在使用近似全纯技术（由唐纳森引入并由Auroux进一步发展）和相应的monodromy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束和映射到复射影平面的结构，其单值性由映射类群或辫子群中的值的态射描述。通过研究辛流形的单值不变量，可以对辛流形与复射影流形之间的关系：辛与复变形等价、合痕与非合痕现象、辛流形上的拓扑约束等问题有新的认识。此外，将单值不变量与Gromov-Witten不变量或Floer同调相联系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子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，它首先出现在经典力学的Hamilton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，部分原因是来自理论物理（弦理论）的激发性问题。本项目旨在利用S.唐纳森和随后的Auroux，这使得有可能获得一个完整的描述组合不变量涉及辫群（一个概念密切相关的结）。该项目的主要目标之一是将拓扑特征的辛流形与复代数流形（一类更特殊，更好地理解的几何空间）。此外，一些应用到其他领域，如数学物理（“镜像对称”的对偶性在弦理论）和密码学（计算复杂性的组合问题，涉及辫子群）将进行探讨。

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.