Monodromy invariants of the fibrations over projective line

One of the main problems of geometry is to find discrete invariants distinguishing geometric objects up to some equivalence. In algebraic geometry, the classical approach, based on ideas of Riemann, Hurwitz, Lefschetz, consists of representations of complex algebraic manifolds either as finite coverings of the projective spaces (generic coverings) or as codimension one fibrations over the projective line (Lefschetz pencils). The monodromy, defined by circuits around the locus of critical values of such fibrations, defines completely these manifolds (as differentiable manifolds) and allows to hope that the invariants, connected with monodromy, defines completely these manifolds up to deformation of complex structures. Recently, Donaldson, Auroux, and Katzarkov generalized this approach to the case of four-dimensional symplectic manifolds in order to find invariants of symplectic structures on these manifolds. In the talk, the basic directions of the development and results based on this approach to classification of algebraic and symplectic manifolds will be described.