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Books > Science & Mathematics > Mathematics > Geometry > General
This monograph on the applications of cube complexes constitutes a
breakthrough in the fields of geometric group theory and 3-manifold
topology. Many fundamental new ideas and methodologies are
presented here for the first time, including a cubical
small-cancellation theory that generalizes ideas from the 1960s, a
version of Dehn Filling that functions in the category of special
cube complexes, and a variety of results about right-angled Artin
groups. The book culminates by establishing a remarkable theorem
about the nature of hyperbolic groups that are constructible as
amalgams. The applications described here include the virtual
fibering of cusped hyperbolic 3-manifolds and the resolution of
Baumslag's conjecture on the residual finiteness of one-relator
groups with torsion. Most importantly, this work establishes a
cubical program for resolving Thurston's conjectures on hyperbolic
3-manifolds, and validates this program in significant cases.
Illustrated with more than 150 color figures, this book will
interest graduate students and researchers working in geometry,
algebra, and topology.
This collection of contributions originates from the
well-established conference series "Fractal Geometry and
Stochastics" which brings together researchers from different
fields using concepts and methods from fractal geometry. Carefully
selected papers from keynote and invited speakers are included,
both discussing exciting new trends and results and giving a gentle
introduction to some recent developments. The topics covered
include Assouad dimensions and their connection to analysis,
multifractal properties of functions and measures, renewal theorems
in dynamics, dimensions and topology of random discrete structures,
self-similar trees, p-hyperbolicity, phase transitions from
continuous to discrete scale invariance, scaling limits of
stochastic processes, stemi-stable distributions and fractional
differential equations, and diffusion limited aggregation.
Representing a rich source of ideas and a good starting point for
more advanced topics in fractal geometry, the volume will appeal to
both established experts and newcomers.
This is a unified treatment of the various algebraic approaches
to geometric spaces. The study of algebraic curves in the complex
projective plane is the natural link between linear geometry at an
undergraduate level and algebraic geometry at a graduate level, and
it is also an important topic in geometric applications, such as
cryptography.
380 years ago, the work of Fermat and Descartes led us to study
geometric problems using coordinates and equations. Today, this is
the most popular way of handling geometrical problems. Linear
algebra provides an efficient tool for studying all the first
degree (lines, planes) and second degree (ellipses, hyperboloids)
geometric figures, in the affine, the Euclidean, the Hermitian and
the projective contexts. But recent applications of mathematics,
like cryptography, need these notions not only in real or complex
cases, but also in more general settings, like in spaces
constructed on finite fields. And of course, why not also turn our
attention to geometric figures of higher degrees? Besides all the
linear aspects of geometry in their most general setting, this book
also describes useful algebraic tools for studying curves of
arbitrary degree and investigates results as advanced as the Bezout
theorem, the Cramer paradox, topological group of a cubic, rational
curves etc.
Hence the book is of interest for all those who have to teach or
study linear geometry: affine, Euclidean, Hermitian, projective; it
is also of great interest to those who do not want to restrict
themselves to the undergraduate level of geometric figures of
degree one or two.
Focusing methodologically on those historical aspects that are
relevant to supporting intuition in axiomatic approaches to
geometry, the book develops systematic and modern approaches to the
three core aspects of axiomatic geometry: Euclidean, non-Euclidean
and projective. Historically, axiomatic geometry marks the origin
of formalized mathematical activity. It is in this discipline that
most historically famous problems can be found, the solutions of
which have led to various presently very active domains of
research, especially in algebra. The recognition of the coherence
of two-by-two contradictory axiomatic systems for geometry (like
one single parallel, no parallel at all, several parallels) has led
to the emergence of mathematical theories based on an arbitrary
system of axioms, an essential feature of contemporary
mathematics.
This is a fascinating book for all those who teach or study
axiomatic geometry, and who are interested in the history of
geometry or who want to see a complete proof of one of the famous
problems encountered, but not solved, during their studies: circle
squaring, duplication of the cube, trisection of the angle,
construction of regular polygons, construction of models of
non-Euclidean geometries, etc. It also provides hundreds of figures
that support intuition.
Through 35 centuries of the history of geometry, discover the
birth and follow the evolution of those innovative ideas that
allowed humankind to develop so many aspects of contemporary
mathematics. Understand the various levels of rigor which
successively established themselves through the centuries. Be
amazed, as mathematicians of the 19th century were, when observing
that both an axiom and its contradiction can be chosen as a valid
basis for developing a mathematical theory. Pass through the door
of this incredible world of axiomatic mathematical theories
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