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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 --- Dynamics --- Géométrie conforme --- Dynamique --- Conformal geometry. --- Dynamics. --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematical Sciences --- Applied Mathematics --- Complex Analysis --- General and Others

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"This memoir is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps f of a finite union of compact intervals I in R into R with non-flat critical points, such that on its maximal forward invariant set K the map f is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of f K are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pullbacks). We prove that several definitions of geometric pressure P(t), that is pressure for the map f K and the potential - t log f, give the same value (including pressure on periodic orbits, "tree" pressure, variational pressures and conformal pressure). Finally we prove that, provided all periodic orbits in K are hyperbolic repelling, the function P(t) is real analytic for t between the "condensation" and "freezing" parameters and that for each such t there exists unique equilibrium (and conformal) measure satisfying strong statistical properties"--

Lyapunov functions. --- Riemann surfaces. --- Differential equations. --- Riemann, Surfaces de. --- Liapounov, Fonctions de. --- Équations différentielles. --- Conformal geometry. --- Mappings (Mathematics) --- Géométrie conforme --- Applications (Mathématiques) --- Conformal geometry --- Riemann surfaces --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Transformations (Mathematics) --- Surfaces, Riemann --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry

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Conformal geometry --- Discrete groups --- Manifolds (Mathematics) --- 514.763.4 --- Geometry, Differential --- Topology --- Groups, Discrete --- Discrete mathematics --- Infinite groups --- Circular geometry --- Geometry of inverse radii --- Inverse radii, Geometry of --- Inversion geometry --- Möbius geometry --- Geometry --- 514.763.4 Manifolds with complex or almost-complex structure. Hermitian manifolds. Kähler manifolds --- Manifolds with complex or almost-complex structure. Hermitian manifolds. Kähler manifolds --- Differential geometry. Global analysis

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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