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51 <082.1> --- Mathematics--Series --- Vector fields. --- Differential inequalities. --- Heat equation. --- Partial differential operators. --- Champs vectoriels --- Inégalités différentielles --- Equation de la chaleur --- Opérateurs différentiels partiels --- Inégalités différentielles --- Opérateurs différentiels partiels --- Algebraic topology --- Differential inequalities --- Heat equation --- Partial differential operators --- Vector fields --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Differential operators --- Diffusion equation --- Heat flow equation --- Differential equations, Parabolic --- Inequalities (Mathematics) --- Équations aux dérivées partielles
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Vector fields on manifolds play a major role in mathematics and other sciences. In particular, the Poincaré-Hopf index theorem gives rise to the theory of Chern classes, key manifold-invariants in geometry and topology. It is natural to ask what is the ‘good’ notion of the index of a vector field, and of Chern classes, if the underlying space becomes singular. The question has been explored by several authors resulting in various answers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph.
Singularities (Mathematics) --- Vector fields. --- Direction fields (Mathematics) --- Fields, Direction (Mathematics) --- Fields, Slope (Mathematics) --- Fields, Vector --- Slope fields (Mathematics) --- Vector analysis --- Geometry, Algebraic --- Vector fields --- Differential equations, partial. --- Differentiable dynamical systems. --- Cell aggregation --- Global analysis. --- Geometry, algebraic. --- Several Complex Variables and Analytic Spaces. --- Dynamical Systems and Ergodic Theory. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Global Analysis and Analysis on Manifolds. --- Algebraic Geometry. --- Mathematics. --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Partial differential equations --- Algebraic geometry --- Geometry --- Functions of complex variables. --- Dynamics. --- Ergodic theory. --- Manifolds (Mathematics). --- Complex manifolds. --- Global analysis (Mathematics). --- Algebraic geometry. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Analytic spaces --- Manifolds (Mathematics) --- Geometry, Differential --- Topology --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Complex variables --- Elliptic functions --- Functions of real variables
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