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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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The description for this book, Lectures on Fourier Integrals. (AM-42), Volume 42, will be forthcoming.

Fourier series. --- Integrals. --- Harmonic analysis. --- Abscissa. --- Absolute value. --- Absolutely integrable function. --- Acta Mathematica. --- Addition. --- Additive function. --- Affine transformation. --- Almost periodic function. --- Analytic function. --- Antiderivative. --- Arbitrarily large. --- Arithmetic mean. --- Augustin-Louis Cauchy. --- Bernhard Riemann. --- Bessel function. --- Big O notation. --- Borel set. --- Boundary layer. --- Boundary value problem. --- Bounded function. --- Bounded variation. --- Calculation. --- Cauchy principal value. --- Characteristic function (probability theory). --- Coefficient. --- Compact space. --- Compactness theorem. --- Complex number. --- Continuous function. --- Dense set. --- Derivative. --- Differentiable function. --- Dirichlet series. --- Distribution function. --- Division by zero. --- E. W. Hobson. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Empty set. --- Equation. --- Existential quantification. --- Exponential polynomial. --- Exterior (topology). --- Fourier transform. --- Function (mathematics). --- Functional equation. --- Gamma function. --- Generating function. --- Harmonic function. --- Initial point. --- Integer. --- Integral equation. --- Interval (mathematics). --- Limit of a sequence. --- Line (geometry). --- Linear combination. --- Linear differential equation. --- Mathematische Annalen. --- Mean value theorem. --- Monotonic function. --- Null set. --- Order of integration (calculus). --- Order of integration. --- Order of magnitude. --- Parameter. --- Partial derivative. --- Partial fraction decomposition. --- Poisson formula. --- Poisson summation formula. --- Polar coordinate system. --- Polynomial. --- Power series. --- Principal part. --- Rapidity. --- Rational function. --- Rational number. --- Real variable. --- Remainder. --- Requirement. --- Set function. --- Sign (mathematics). --- Smoothness. --- Special case. --- State function. --- Step function. --- Subsequence. --- Summation. --- Theorem. --- Total variation. --- Trigonometric integral. --- Uniform convergence. --- Uniqueness theorem. --- Variable (mathematics).