Approximately holomorphic techniques and monodromy invariants in symplectic topology

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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.