This book presents surveys from a workshop held during the theme year in geometry and topology at the Centre de recherches mathematiques (CRM, University of Montreal). The volume is in some sense a sequel to ""Mirror Symmetry I"" (1998) and ""Mirror Symmetry II"" (1996), copublished by the AMS and International Press. Included are recent developments in the theory of mirror manifolds and the related areas of complex and symplectic geometry. The long introductory articles explain the key physical ideas and motivation, namely conformal field theory, supersymmetry, and string theory. Open problems are emphasized.Thus the book provides an efficient way for a very broad audience of mathematicians and physicists to reach the frontier of research in this fast expanding area. It features: crucial research pertaining to future developments in algebraic and symplectic geometry and to the physics of unified string theories; well-known authors who are leaders in the field; introductory article by Greene and Yau; and a solid and even blend of ideas and techniques from both mathematics and physics. Also available from the AMS is the related volume, ""Mirror Symmetry IV"" (2002).
Aspects of quantum geometry by B. R. Greene Introduction to enumerative invariants by S.-T. Yau Compactified moduli spaces of pseudo-holomorphic curves by T. H. Parker Mirror symmetry for hyper-Kahler manifolds by M. Verbitsky Connecting the web: A prognosis by M. Gross Strong coupling singularities and non-abelian gauge symmetries in $N = 2$ string theory by A. Klemm and P. Mayr Remarks on (0,2) Calabi-Yau models by S. Kachru Relations among fixed point by K. Liu An analytic discriminant for polarized algebraic $K3$ surfaces by J. Jorgenson and A. Todorov Through the looking glass by D. R. Morrison An update on (small) quantum cohomology by B. Siebert.