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Conformal invariants (conformally invariant tensors, conformally covariant differential operators, conformal holonomy groups etc.) are of central significance in differential geometry and physics. Well-known examples of conformally covariant operators are the Yamabe, the Paneitz, the Dirac and the twistor operator. These operators are intimely connected with the notion of Branson’s Q-curvature. The aim of these lectures is to present the basic ideas and some of the recent developments around Q -curvature and conformal holonomy. The part on Q -curvature starts with a discussion of its origins and its relevance in geometry and spectral theory. The following lectures describe the fundamental relation between Q -curvature and scattering theory on asymptotically hyperbolic manifolds. Building on this, they introduce the recent concept of Q -curvature polynomials and use these to reveal the recursive structure of Q -curvatures. The part on conformal holonomy starts with an introduction to Cartan connections and its holonomy groups. Then we define holonomy groups of conformal manifolds, discuss its relation to Einstein metrics and recent classification results in Riemannian and Lorentzian signature. In particular, we explain the connection between conformal holonomy and conformal Killing forms and spinors, and describe Fefferman metrics in CR geometry as Lorentzian manifold with conformal holonomy SU(1,m).

Conformal geometry. --- Geometry, Differential. --- Conformal geometry --- Geometry, Differential --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Differential geometry --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Mathematics. --- Differential geometry. --- Differential Geometry. --- Global differential geometry.

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Conformal geometry --- Lie groups --- Planetary theory --- Symmetry (Physics) --- Invariance principles (Physics) --- Symmetry (Chemistry) --- Conservation laws (Physics) --- Physics --- Planets, Theory of --- Celestial mechanics --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry --- Kepler, Johannes --- Kepler, Jean --- Kepler, Johann --- Keppler, Giovanni --- Kepler, Johannes, --- Kepler, Johannes, - 1571-1630

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"This is a one-stop introduction to the methods of ergodic theory applied to holomorphic iteration. The authors begin with introductory chapters presenting the necessary tools from ergodic theory thermodynamical formalism, and then focus on recent developments in the field of 1-dimensional holomorphic iterations and underlying fractal sets, from the point of view of geometric measure theory and rigidity. Detailed proofs are included. Developed from university courses taught by the authors, this book is ideal for graduate students. Researchers will also find it a valuable source of reference to a large and rapidly expanding field. It eases the reader into the subject and provides a vital springboard for those beginning their own research. Many helpful exercises are also included to aid understanding of the material presented and the authors provide links to further reading and related areas of research"--Provided by publisher.

Conformal geometry. --- Fractals. --- Ergodic theory. --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Fractal geometry --- Fractal sets --- Geometry, Fractal --- Sets, Fractal --- Sets of fractional dimension --- Dimension theory (Topology) --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry

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This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geometry can elucidate the existence and stability of the basic solutions of the theory. Introducing the differential geometric, spinorial and PDE background required to gain a deep understanding of conformal methods, this text provides an accessible account of key results in mathematical relativity over the last thirty years, including the stability of de Sitter and Minkowski spacetimes. For graduate students and researchers, this self-contained account includes useful visual models to help the reader grasp abstract concepts and a list of further reading, making this an ideal reference companion on the topic. This title, first published in 2016, has been reissued as an Open Access publication on Cambridge Core.

Conformal geometry. --- Conformal mapping. --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry --- General relativity (Physics) --- Geometry, Differential. --- Einstein field equations --- Space and time. --- Mathematics. --- Numerical solutions.

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Recent developments in biology and nanotechnology have stimulated a rapidly growing interest in the mechanics of thin, flexible ribbons and Mobius bands. This edited volume contains English translations of four seminal papers on this topic, all originally written in German; of these, Michael A. Sadowsky published the first in 1929, followed by two others in 1930, and Walter Wunderlich published the last in 1962. The volume also contains invited, peer-reviewed, original research articles on related topics. Previously published in the Journal of Elasticity, Volume 119, Issue 1-2, 2015.

Conformal geometry. --- Mechanics -- Mathematics. --- Materials Science --- Applied Mathematics --- Chemical & Materials Engineering --- Engineering & Applied Sciences --- Mechanics --- Mathematics. --- Classical mechanics --- Newtonian mechanics --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Physics --- Dynamics --- Quantum theory --- Geometry --- Mechanics. --- Mechanics, Applied. --- Computer science. --- Nanotechnology. --- Solid Mechanics. --- Computational Science and Engineering. --- Biological and Medical Physics, Biophysics. --- Molecular technology --- Nanoscale technology --- High technology --- Informatics --- Science --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Computer mathematics. --- Biophysics. --- Biological physics. --- Biological physics --- Biology --- Medical sciences --- Computer mathematics --- Electronic data processing --- Mathematics