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
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