Different Faces of Geometry - edited by the world renowned geometers S. Donaldson, Ya. Eliashberg, and M. Gromov - presents the current state, new results, original ideas and open questions from the following important topics in modern geometry: Amoebas and Tropical Geometry Convex Geometry and Asymptotic Geometric Analysis Differential Topology of 4-Manifolds 3-Dimensional Contact Geometry Floer Homology and Low-Dimensional Topology Kahler Geometry Lagrangian and Special Lagrangian Submanifolds Refined Seiberg-Witten Invariants. These apparently diverse topics have a common feature in that they are all areas of exciting current activity. The Editors have attracted an impressive array of leading specialists to author chapters for this volume: G. Mikhalkin (USA-Canada-Russia), V.D. Milman (Israel) and A.A. Giannopoulos (Greece), C. LeBrun (USA), Ko Honda (USA), P. Ozsvath (USA) and Z. Szabo (USA), C. Simpson (France), D. Joyce (UK) and P. Seidel (USA), and S. Bauer (Germany). "One can distinguish various themes running through the different contributions. There is some emphasis on invariants defined by elliptic equations and their applications in low-dimensional topology, symplectic and contact geometry (Bauer, Seidel, Ozsvath and Szabo). These ideas enter, more tangentially, in the articles of Joyce, Honda and LeBrun. Here and elsewhere, as well as explaining the rapid advances that have been made, the articles convey a wonderful sense of the vast areas lying beyond our current understanding. Simpson's article emphasizes the need for interesting new constructions (in that case of Kahler and algebraic manifolds), a point which is also made by Bauer in the context of 4-manifolds and the "11/8 conjecture". LeBrun's article gives another perspective on 4-manifold theory, via Riemannian geometry, and the challenging open questions involving the geometry of even "well-known" 4-manifolds. There are also striking contrasts between the articles. The authors have taken different approaches: for example, the thoughtful essay of Simpson, the new research results of LeBrun and the thorough expositions with homework problems of Honda. One can also ponder the differences in the style of mathematics. In the articles of Honda, Giannopoulos and Milman, and Mikhalkin, the "geometry" is present in a very vivid and tangible way; combining respectively with topology, analysis and algebra. The papers of Bauer and Seidel, on the other hand, makes the point that algebraic and algebro-topological abstraction (triangulated categories, spectra) can play an important role in very unexpected ways in concrete geometric problems." - From the Preface by the Editors

Different Faces of Geometry - edited by the world renowned geometers S. Donaldson, Ya. Eliashberg, and M. Gromov - presents the current state, new results, original ideas and open questions from the following important topics in modern geometry:

Amoebas and Tropical Geometry
Convex Geometry and Asymptotic Geometric Analysis
Differential Topology of 4-Manifolds
3-Dimensional Contact Geometry
Floer Homology and Low-Dimensional Topology
Kahler Geometry
Lagrangian and Special Lagrangian Submanifolds
Refined Seiberg-Witten Invariants.

These apparently diverse topics have a common feature in that they are all areas of exciting current activity. The Editors have attracted an impressive array of leading specialists to author chapters for this volume: G. Mikhalkin (USA-Canada-Russia), V.D. Milman (Israel) and A.A. Giannopoulos (Greece), C. LeBrun (USA), Ko Honda (USA), P. Ozsvath (USA) and Z. Szabo (USA), C. Simpson (France), D. Joyce (UK) and P. Seidel (USA), and S. Bauer (Germany).

"One can distinguish various themes running through the different contributions. There is some emphasis on invariants defined by elliptic equations and their applications in low-dimensional topology, symplectic and contact geometry (Bauer, Seidel, Ozsvath and Szabo). These ideas enter, more tangentially, in the articles of Joyce, Honda and LeBrun. Here and elsewhere, as well as explaining the rapid advances that have been made, the articles convey a wonderful sense of the vast areas lying beyond our current understanding.
Simpson's article emphasizes the need for interesting new constructions (in that case of Kahler and algebraic manifolds), a point which is also made by Bauer in the context of 4-manifolds and the "11/8 conjecture."
LeBrun's article gives another perspective on 4-manifold theory, via Riemannian geometry, and the challenging open questions involving the geometry of even "well-known" 4-manifolds.
There are also striking contrasts between the articles. The authors have taken different approaches: for example, the thoughtful essay of Simpson, the new research results of LeBrun and the thorough expositions with homework problems of Honda.
One can also ponder the differences in the style of mathematics. In the articles of Honda, Giannopoulos and Milman, and Mikhalkin, the "geometry" is present in a very vivid and tangible way; combining respectively with topology, analysis and algebra. The papers of Bauer and Seidel, on the other hand, makes the point that algebraic and algebro-topological abstraction (triangulated categories, spectra) can play an important role in very unexpected ways in concrete geometric problems." - From the Preface by the Editors "

While Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer's fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi's important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kahler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi's mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi's visionary contributions will certainly continue to shape the course of this subject far into the future.

This volume presents twelve important papers on algebraic and complex geometry-papers that have played important roles in the development of these subjects. The subject matter ranges from the birational geometry of varieties, to geometric structures of moduli spaces in algebraic and complex geometry, to the geometry of compact and non-compact Kahler manifolds. Among the authors and topics are: Raoul Bott on a residue formula for holomorphic vector-fields; Armand Borel on metric properties of arithmetic quotients of symmetric spaces and an extension theorem; P. B. A. Kronheimer on Torelli-type theorem for gravitational instantons; Victor Guillemin on Kaehler structures on toric varieties; S. K. Donaldson on scalar curvature and stability of toric varieties; and Gang Liu on the volume growth of Kahler manifolds with nonnegative bisectional curvature. With a preface by Jun Li.

This volume presents twelve outstanding papers which have had a deep influence upon the development of geometry during the last fifty years. Among the authors and topics are: H. P. McKean, Jr. and I. M. Singer on curvature and the eigenvalues of the Laplacian; Eugenio Calabi on minimal immersions of surfaces in Euclidean spheres; H. Blaine Lawson, Jr. and Shing-Tung Yau on compact manifolds of nonpositive curvature; Mikhael Gromov on filling Riemannian manifolds; G. Perelman on a proof of the soul conjecture of Cheeger and Gromoll; and Jeff Cheeger and Tobias H. Colding on the structure of spaces with Ricci curvature bounded below. With a preface by Shing-Tung Yau.

This volume presents twelve important papers which have formed some the cornerstones of geometric analysis. Their subject matter ranges over the areas of conformal geometry, gauge theory, symplectic geometry, mean curvature, spectral theory, and general relativity. Among the authors and topics are: Morio Obata on the conjectures on conformal transformations of Riemannian manifolds; Richard Schoen on conformal deformation of a Riemannian metric to constant scalar curvature; Clifford Henry Taubes on the stable topology of self-dual moduli spaces; Karen Uhlenbeck on harmonic maps and classical solutions of the chiral model; Z. I. Szabó on the Lichnerowicz conjecture on harmonic manifolds (1990); and Ben Andrews and Haizhong Li on embedded constant mean curvature tori in the three-sphere. With a preface by Richard Schoen.

These papers have been organized into five volumes by subject matter. The first volume deals with topology, the second with algebraic geometry, the third with geometric ideas, the fourth with geometric analysis, and the fifth with geometric flows. These five volumes provide a condensed version of the Journal of Differential Geometry, helping readers to understand the development of the field of geometry over the past fifty years.