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An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a product for orbifolds and has had significant impact. The final chapter includes explicit computations for a number of interesting examples.

Orbifolds. --- Topology --- Manifolds (Mathematics) --- Orbifolds --- 512.7 --- 515.14 --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Geometry, Differential --- 515.14 Algebraic topology --- Algebraic topology --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Topology. --- Homology theory. --- Quantum theory. --- String models. --- Models, String --- String theory --- Nuclear reactions --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Cohomology theory --- Contrahomology theory

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This book seeks to be a guidebook on the journey towards the minimal supersymmetric standard model down the orbifold road. It takes the viewpoint that the chirality of matter fermions is an essential aspect that orbifold compactification allows to derive from higher-dimensional string theories in a rather straight-forward manner. Halfway between a textbook and a tutorial review, Quarks and Leptons from Orbifolded Superstring is intended for the graduate student and particle phenomenologist wishing to get acquainted with this field.

Particles (Nuclear physics) --- Supersymmetry. --- Phenomenological theory (Physics) --- Orbifolds. --- Superstring theories. --- Particules (Physique nucléaire) --- Supersymétrie --- Supercordes (Physique nucléaire) --- Electronic books. -- local. --- Particles (Nuclear physics). --- Phenomenological theory (Physics). --- Quarks --- Leptons (Nuclear physics) --- Supersymmetry --- Orbifolds --- Superstring theories --- Atomic Physics --- Nuclear Physics --- Physics --- Physical Sciences & Mathematics --- Superstrings (Nuclear physics) --- Theories, Superstring --- Phenomenology in physics --- Unified theories --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Physics. --- Quantum field theory. --- String theory. --- Nuclear physics. --- Elementary particles (Physics). --- Particle and Nuclear Physics. --- Quantum Field Theories, String Theory. --- Elementary Particles, Quantum Field Theory. --- String models --- Manifolds (Mathematics) --- Mathematical physics --- Symmetry (Physics) --- Nuclear physics --- Quantum theory. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Models, String --- String theory --- Nuclear reactions --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Atomic nuclei --- Atoms, Nuclei of --- Nucleus of the atom --- Quarks. --- Partons --- Quark-gluon interactions --- Fermions

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The present Special Issue of Symmetry is devoted to two important areas of global Riemannian geometry, namely submanifold theory and the geometry of Lie groups and homogeneous spaces. Submanifold theory originated from the classical geometry of curves and surfaces. Homogeneous spaces are manifolds that admit a transitive Lie group action, historically related to F. Klein's Erlangen Program and S. Lie's idea to use continuous symmetries in studying differential equations. In this Special Issue, we provide a collection of papers that not only reflect some of the latest advancements in both areas, but also highlight relations between them and the use of common techniques. Applications to other areas of mathematics are also considered.

warped products --- vector equilibrium problem --- Laplace operator --- cost functional --- pointwise 1-type spherical Gauss map --- inequalities --- homogeneous manifold --- finite-type --- magnetic curves --- Sasaki-Einstein --- evolution dynamics --- non-flat complex space forms --- hyperbolic space --- compact Riemannian manifolds --- maximum principle --- submanifold integral --- Clifford torus --- D’Atri space --- 3-Sasakian manifold --- links --- isoparametric hypersurface --- Einstein manifold --- real hypersurfaces --- Kähler 2 --- *-Weyl curvature tensor --- homogeneous geodesic --- optimal control --- formality --- hadamard manifolds --- Sasakian Lorentzian manifold --- generalized convexity --- isospectral manifolds --- Legendre curves --- geodesic chord property --- spherical Gauss map --- pointwise bi-slant immersions --- mean curvature --- weakly efficient pareto points --- geodesic symmetries --- homogeneous Finsler space --- orbifolds --- slant curves --- hypersphere --- ??-space --- k-D’Atri space --- *-Ricci tensor --- homogeneous space