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Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.

Mathematics --- Mathématiques --- Congresses --- Congrès --- 51 --- -Math --- Science --- Congresses. --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Math --- Mathématiques --- Congrès --- A priori estimate. --- Addition. --- Additive group. --- Affine space. --- Algebraic geometry. --- Algebraic group. --- Atiyah–Singer index theorem. --- Bernoulli number. --- Boundary value problem. --- Bounded operator. --- C*-algebra. --- Canonical transformation. --- Cauchy problem. --- Characteristic class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Commutative property. --- Commutative ring. --- Complex manifold. --- Complex number. --- Complex vector bundle. --- Dedekind sum. --- Degenerate bilinear form. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Ellipse. --- Elliptic operator. --- Equation. --- Euler characteristic. --- Euler number. --- Existence theorem. --- Exotic sphere. --- Finite difference. --- Finite group. --- Fourier integral operator. --- Fourier transform. --- Fourier. --- Fredholm operator. --- Hardy space. --- Hilbert space. --- Holomorphic vector bundle. --- Homogeneous coordinates. --- Homomorphism. --- Homotopy. --- Hyperbolic partial differential equation. --- Identity component. --- Integer. --- Integral transform. --- Isomorphism class. --- John Milnor. --- K-theory. --- Lebesgue measure. --- Line bundle. --- Local ring. --- Mathematics. --- Maximal ideal. --- Modular form. --- Module (mathematics). --- Monoid. --- Normal bundle. --- Number theory. --- Open set. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Piecewise linear manifold. --- Poisson bracket. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal part. --- Projective space. --- Pseudo-differential operator. --- Quadratic form. --- Rational variety. --- Real number. --- Reciprocity law. --- Resolution of singularities. --- Riemann–Roch theorem. --- Shift operator. --- Simply connected space. --- Special case. --- Square-integrable function. --- Subalgebra. --- Submanifold. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Winding number. --- Mathematics - Congresses

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This book builds upon the revolutionary discovery made in 1974 that when one passes from function f to a function J of paths joining two points A1≠A1 the connectivities R1 of the domain of f can be replaced by connectivities R1 over Q, common to the pathwise components of a basic Frechet space of classes of equivalent curves joining A1 to A1. The connectivities R1, termed "Frechet numbers," are proved independent of the choice of A1 ≠ A1, and of a replacement of Mn by any differential manifold homeomorphic to Mn.Originally published in 1976.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differentiable manifolds. --- Global analysis (Mathematics) --- Calculus of variations. --- Algebraic topology. --- Analytic function. --- Arc length. --- Axiom. --- Bernhard Riemann. --- Boundary value problem. --- Cartesian coordinate system. --- Coefficient. --- Compact space. --- Computation. --- Conjugate points. --- Connectivity (graph theory). --- Continuous function. --- Corollary. --- Countable set. --- Counting. --- Cramer's rule. --- Curve. --- Deformation theory. --- Degeneracy (mathematics). --- Derivative. --- Diffeomorphism. --- Differentiable manifold. --- Differential equation. --- Differential geometry. --- Differential structure. --- Dimension. --- Domain of a function. --- Eilenberg. --- Einstein notation. --- Equation. --- Euclidean space. --- Euler characteristic. --- Euler equations (fluid dynamics). --- Euler integral. --- Existence theorem. --- Existential quantification. --- Exotic sphere. --- Family of curves. --- Finite set. --- First variation. --- Geometry. --- Global analysis. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Implicit function theorem. --- Inference. --- Integer. --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Jacobian matrix and determinant. --- Lagrange multiplier. --- Linear combination. --- Linear map. --- Line–line intersection. --- Mathematical proof. --- Maximal set. --- Metric space. --- N-sphere. --- Neighbourhood (mathematics). --- Null vector. --- Open set. --- Pairwise. --- Parameter. --- Parametric equation. --- Parametrization. --- Partial derivative. --- Partial function. --- Phase space. --- Positive definiteness. --- Projective plane. --- Quadratic form. --- Quadratic. --- Rate of convergence. --- Rational number. --- Real variable. --- Resultant. --- Riemannian manifold. --- Scientific notation. --- Sign (mathematics). --- Special case. --- Sturm separation theorem. --- Submanifold. --- Subsequence. --- Subset. --- Taylor's theorem. --- Tensor algebra. --- Theorem. --- Theory. --- Topological manifold. --- Topological space. --- Topology. --- Tuple. --- Unit vector. --- Variable (mathematics). --- Variational analysis. --- Weierstrass function. --- Without loss of generality.