Approximately holomorphic techniques and monodromy invariants in symplectic topology

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

DMS-0244844 Denis Auroux This project aims to study the topology of symplectic manifolds using approximately holomorphic techniques (introduced by Donaldson and further developped by Auroux) and the corresponding monodromy invariants. Approximately holomorphic linear systems on symplectic manifolds give rise to structures such as Lefschetz pencils and maps to the complex projective plane, whose monodromy is described by morphisms with values in mapping class groups or braid groups. By studying the monodromy invariants of symplectic manifolds, new insight will be obtained into the relationships between symplectic manifolds and complex projective manifolds: symplectic versus complex deformation equivalence, isotopy and non-isotopy phenomena, topological constraints on symplectic manifolds. In addition, relating monodromy invariants with Gromov-Witten invariants or Floer homology should help to understand mirror symmetry, while the more combinatorial aspects of the project are closely related to the algorithmics and computational complexity of mapping class and braid groups. Symplectic manifolds are geometric spaces with special structures, which first arose in the Hamiltonian formulation of classical mechanics. Mathematicians have recently become very interested in their geometry and topology (intrinsic structure), in part due to motivating questions from theoretical physics (string theory). This project aims to study the topology of symplectic manifolds using an approach developped first by S. Donaldson and subsequently by Auroux, which makes it possible to obtain a complete description by combinatorial invariants involving braid groups (a concept closely related to knots). One of the main goals of the project is to relate the topological features of symplectic manifolds with those of complex algebraic manifolds (a more special, much better understood class of geometric spaces). In addition, some applications to other domains such as mathematical physics (the "mirror symmetry" duality in string theory) and cryptography (the computational complexity of combinatorial problems involving braid groups) will be explored.