The new results on lattice deformation of current algebra -- Baxterization, dynamical systems, and the symmetries of integrability -- A lecture on the Calogero-Sutherland models -- Quantum group and magnetic translations. Bethe-Ansatz solution for bloch electrons in a magnetic field -- Symplectic geometry of the Chern-Simons theory -- Quantization of field theories generalizing Gravity-Yang-Mills systems on the cylinder -- Wodzicki residue and anomalies of current algebras -- Spacetime locality of the antifield formalism: General theorems illustrated by means of examples -- On supersymmetric and topological quantum mechanical models -- Structures of K.Saito theory of primitive form in topological theories coupled to topological gravity -- String theory and classical integrable systems -- On background independence in string theory -- Lectures on mirror symmetry.

This is a collection of papers on a variety of topics of current interest in mathematical physics: integrable systems, quantum groups, topological quantum theory, string theory. Some of the contributions are lengthy reviews of lasting value on subjects like symplectic geometry of the Chern-Simons theory or on mirror symmetry. The book addresses graduate students as well as researchers in mathematical physics.