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The Taniguchi Symposium on global analysis on manifolds focused
mainly on the relationships between some geometric structures of
manifolds and analysis, especially spectral analysis on noncompact
manifolds. Included in the present volume are expanded versions of
most of the invited lectures. In these original research articles,
the reader will find up-to date accounts of the subject.
Geometry in ancient Greece is said to have originated in the
curiosity of mathematicians about the shapes of crystals, with that
curiosity culminating in the classification of regular convex
polyhedra addressed in the final volume of Euclid's Elements. Since
then, geometry has taken its own path and the study of crystals has
not been a central theme in mathematics, with the exception of
Kepler's work on snowflakes. Only in the nineteenth century did
mathematics begin to play a role in crystallography as group theory
came to be applied to the morphology of crystals. This monograph
follows the Greek tradition in seeking beautiful shapes such as
regular convex polyhedra. The primary aim is to convey to the
reader how algebraic topology is effectively used to explore the
rich world of crystal structures. Graph theory, homology theory,
and the theory of covering maps are employed to introduce the
notion of the topological crystal which retains, in the abstract,
all the information on the connectivity of atoms in the crystal.
For that reason the title Topological Crystallography has been
chosen. Topological crystals can be described as "living in the
logical world, not in space," leading to the question of how to
place or realize them "canonically" in space. Proposed here is the
notion of standard realizations of topological crystals in space,
including as typical examples the crystal structures of diamond and
lonsdaleite. A mathematical view of the standard realizations is
also provided by relating them to asymptotic behaviors of random
walks and harmonic maps. Furthermore, it can be seen that a
discrete analogue of algebraic geometry is linked to the standard
realizations. Applications of the discussions in this volume
include not only a systematic enumeration of crystal structures, an
area of considerable scientific interest for many years, but also
the architectural design of lightweight rigid structures. The
reader therefore can see the agreement of theory and practice.
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