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Mathematical Notes, 29Originally published in 1983.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.

Differential equations, Elliptic --- Schrödinger operator. --- Eigenfunctions. --- Functions, Proper --- Proper functions --- Boundary value problems --- Differential equations --- Integral equations --- Operator, Schrödinger --- Differential operators --- Quantum theory --- Schrödinger equation --- Numerical solutions. --- Numerical solutions --- Approximation. --- Ball (mathematics). --- Bounded function. --- Center of mass. --- Coefficient. --- Compact space. --- Complex number. --- Continuous function (set theory). --- Continuous function. --- Discrete spectrum. --- Distribution (mathematics). --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Equivalence class. --- Essential spectrum. --- Estimation. --- Existential quantification. --- Exponential decay. --- Function space. --- Fundamental theorem of calculus. --- Geometry. --- Ground state. --- Infimum and supremum. --- Lebesgue measure. --- Open set. --- Pointwise. --- Quadratic form. --- Quantity. --- Restriction (mathematics). --- Riemannian manifold. --- Robert Langlands. --- Schrödinger equation. --- Self-adjoint operator. --- Self-adjoint. --- Smoothness. --- Special case. --- Subset. --- Support (mathematics). --- Theorem. --- Upper and lower bounds. --- Weak solution. --- Without loss of generality.

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The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology.In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems.Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.

Braid theory --- Braids, Theory of --- Theory of braids --- Braid theory. --- Algebraic topology --- Knot theory --- Representations of groups --- 512.54 --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Knots (Topology) --- Low-dimensional topology --- 512.54 Groups. Group theory --- Groups. Group theory --- Knot theory. --- Representations of groups. --- Addition. --- Alexander polynomial. --- Algebraic structure. --- Automorphism. --- Ball (mathematics). --- Bijection. --- Braid group. --- Branched covering. --- Burau representation. --- Calculation. --- Cartesian coordinate system. --- Characterization (mathematics). --- Coefficient. --- Combinatorial group theory. --- Commutative property. --- Commutator subgroup. --- Configuration space. --- Conjugacy class. --- Corollary. --- Covering space. --- Dehn twist. --- Determinant. --- Diagram (category theory). --- Dimension. --- Disjoint union. --- Double coset. --- Eigenvalues and eigenvectors. --- Enumeration. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Faithful representation. --- Finite set. --- Free abelian group. --- Free group. --- Fundamental group. --- Geometry. --- Group (mathematics). --- Group ring. --- Groupoid. --- Handlebody. --- Heegaard splitting. --- Homeomorphism. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Identity element. --- Identity matrix. --- Inclusion map. --- Initial point. --- Integer matrix. --- Integer. --- Knot polynomial. --- Lens space. --- Line segment. --- Line–line intersection. --- Link group. --- Low-dimensional topology. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix group. --- Matrix representation. --- Monograph. --- Morphism. --- Natural transformation. --- Normal matrix. --- Notation. --- Orientability. --- Parity (mathematics). --- Permutation. --- Piecewise linear. --- Pointwise. --- Polynomial. --- Prime knot. --- Projection (mathematics). --- Proportionality (mathematics). --- Quotient group. --- Requirement. --- Rewriting. --- Riemann surface. --- Semigroup. --- Sequence. --- Special case. --- Subgroup. --- Submanifold. --- Subset. --- Symmetric group. --- Theorem. --- Theory. --- Topology. --- Trefoil knot. --- Two-dimensional space. --- Unimodular matrix. --- Unit vector. --- Variable (mathematics). --- Word problem (mathematics). --- Topologie algébrique

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This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms. The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups. The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.

Continuous groups. --- Additive group. --- Adjoint representation. --- Algebra over a field. --- Algebraic extension. --- Algebraic variety. --- Algebraically closed field. --- Analytic function. --- Analytic manifold. --- Automorphism. --- Axiom of countability. --- Ball (mathematics). --- Cardinal number. --- Characteristic polynomial. --- Coefficient. --- Commutator subgroup. --- Complex number. --- Connected component (graph theory). --- Continuous function (set theory). --- Continuous function. --- Coordinate system. --- Coset. --- Countable set. --- Covering group. --- Covering space. --- Differential algebra. --- Differential calculus. --- Differential form. --- Differential of a function. --- Dual space. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equivalence class. --- Existential quantification. --- Exponential function. --- Exterior algebra. --- Fundamental group. --- Galois group. --- General topology. --- Geometry. --- Group (mathematics). --- Group theory. --- Hermitian matrix. --- Homeomorphism. --- Homogeneous space. --- Homomorphism. --- Homotopy group. --- Identity element. --- Identity matrix. --- Infinitesimal transformation. --- Integer. --- Invariant subspace. --- Irreducible representation. --- Kronecker product. --- Lie algebra. --- Lie group. --- Linear function. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Linearization. --- Locally connected space. --- Manifold. --- Mathematical induction. --- Matrix exponential. --- Modular arithmetic. --- Module (mathematics). --- Monodromy. --- Morphism. --- Open set. --- Orthogonal group. --- Parametric equation. --- Permutation. --- Power series. --- Projective plane. --- Real number. --- Regular matrix. --- Representation theory. --- Riemann surface. --- Simply connected space. --- Skew-symmetric matrix. --- Special case. --- Subalgebra. --- Subgroup. --- Submanifold. --- Subset. --- Summation. --- Symplectic geometry. --- Symplectic group. --- Tangent space. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Trigonometric polynomial. --- Union (set theory). --- Uniqueness theorem. --- Unitary group. --- Unitary matrix. --- Variable (mathematics). --- Vector space.