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Mirror symmetry has undergone dramatic progress during the last five years. Tremendous insight has been gained on a number of key issues. This volume surveys these results. Some of the contributions in this work have appeared elsewhere, while others were written specifically for this collection. The areas covered are organized into 4 sections, and each presents papers by both physicists and mathematicians.

This volume collects the most important developments that have taken place in mathematical physics since 1991. It is an essential reference tool for both mathematics and physics libraries and for students of physics and mathematics. ... Read more

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This book is somewhat dated since Mirror Symmetry III has recently appeared. However, this volume is now being reprinted and so a review is appropriate. The topic of mirror symmetry has to rank as the most fascinating one in all of modern mathematics. It has its origins in high energy physics, and as such to understand it fully, one must have a background in both quantum field theory and in some fairly esoteric mathematics, such as algebraic geometry, algebraic topology, and complex manifolds. A mathematician familiar with these areas will also have to get used to the sometimes hand-waving arguments used by physicists to discuss some of the concepts in mirror symmetry. The book is a collection of articles written by well-known experts in the field, and each article has varying degrees of physics/mathematics interplay. The literature on mirror symmetry has been dubbed "physical mathematics" by some mathematicians because of the physical constructions employed that are not based on rigorous mathematics. The powerful connections between quantum field theory and algebraic geometry are brought out in the first article of the book by B.R. Greene and H.Oogurt. Correlation functions in quantum field theory are related to the collection of holomorphic maps to rational curves on a complex compact 3-dimensional Kahler manifold. This theme of relating correlation functions in quantum field theory and superstring theory to various structures in algebraic geometry and complex manifolds is played out in the rest of the book.Most fascinating is the construction of the mirror manifolds themselves, for these look the same from a physical string-theory standpoint but are different as complex manifolds. The properties needed by physics imply the relevant manifolds are Calabi-Yau, many having been found by computer. The challenge of defining mirror symmetry rigorously is brought out in many articles in the book, with toric varieties showing the most promise: the enlarged Kahler moduli space of a given manifold is isomorphic to the complex structure moduli space of its mirror. Interestingly, some of the authors show that topology change can occur intrinsically in string theories; this is to be contrasted with "effective" changes in topology that occur in quantum field theory (and even in quantum mechanics). Again, toric varieties play the predominant role in showing this behavior. Some of the authors discuss the connection between mirror symmetry and enumerative geometry. This connection has stimulated much research, and the book serves as an introduction to the fascinating area of quantum cohomology.I think it would be fair to classify the results in this book as a kind of "quantization of mathematics". The reasonable effectiveness of physics in mathematics, to rephrase the statement by Eugene Wigner, is brought out brilliantly by the authors in this book. ... Read more

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Current Developments in Mathematics (CDM) is the proceedings of the conference jointly hosted by Harvard University and the Massachusetts Institute of Technology. The CDM Conference is an annual seminar devoted to surveying the recent developments in mathematics each year. In choosing speakers, the hosts take a broad look at the field of geometry, searching out geometers who go beyond the classical perceptions of its topics. All speakers are prominent specialists in the fields of algebraic geometry, mathematical physics, and other areas.Table of Contents:1. Mirror Symmetry: A Brief Review of the First 10 Years, by B. R. Greene2. Mirror Principle: A Survey, by Bong H. Lian, Kefeng Liu, and S.-T. Yau3. Developments in Symplectic Topology, by P. B. Kronheimer4. O-minimal Structures and Real Analytic Geometry, by Lou Van Den Dries5. Asymptotic Solutions to Dynamics of Many-Body Systems and Classical Continuum Equations, by U. Frisch6. Geometric and Ergodic Theory of Hyperbolic Dynamical Systems, by Lai-Sang Young ... Read more

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Based on lectures held at the Grand Hotel in Taipei, Taiwan from December 17-22, 2001 this volume contains a range of papers by 21st century Chinese mathematicians. The book also announces the winners of the Chern prizes and the Morningside medals. Subjects covered in the articles include: algebra and representation theory; algebraic geometry; number theory; Riemannian geometry; symplectic geometry; dynamical systems; hyperbolic, elliptic, and parabolic partial differential equations; mathematical biology; mathematical physics; applied analysis; computational mathematics; combinatorics; statistics; and mathematical education.

Table of Contents: 1. Preface 2. Committees of ICCM 2001 3. Morningside Lifetime Achievement Award in Mathematics 4. Morningside Medal of Mathematics 5. Morningside Award Medalists 6. Chern Prize Recipients 7. Photographs 8. Speakers of the Conference 9. Morningside Lectures 10. Plenary Lectures 11. Invited Lectures 12. Contributed Lectures 13. Participants ... Read more