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Following the tremendous reception of our first volume on topological groups called ""Topological Groups: Yesterday, Today, and Tomorrow"", we now present our second volume. Like the first volume, this collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Well-known researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.

coarse structure --- descriptive set theory --- group representation --- thick set --- free topological group --- character --- selectively sequentially pseudocompact --- separable topological group --- quotient group --- Chabauty topology --- pseudo-?-bounded --- topological group --- free precompact Boolean group --- right-angled Artin groups --- Neretin’s group --- coarse space --- ultrafilter space --- endomorphism --- separable --- absolutely closed topological group --- tree --- strongly pseudocompact --- Gromov’s compactification --- dynamical system --- semigroup compactification --- tame function --- compact topological semigroup --- vast set --- space of closed subgroups --- reflexive group --- Lie group --- matrix coefficient --- maximal ideal --- topological semigroup --- ballean --- continuous inverse algebra --- extension --- subgroup --- Thompson’s group --- scale --- isomorphic embedding --- arrow ultrafilter --- H-space --- paratopological group --- pseudocompact --- Ramsey ultrafilter --- fibre bundle --- locally compact group --- product --- large set in a group --- Vietoris topology --- topological group of compact exponent --- Bourbaki uniformity --- p-compact --- mapping cylinder --- syndetic set --- p-adic Lie group --- Boolean topological group --- non-trivial convergent sequence --- fixed point algebra --- polish group topologies --- varieties of coarse spaces --- piecewise syndetic set --- maximal space

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Following the tremendous reception of our first volume on topological groups called ""Topological Groups: Yesterday, Today, and Tomorrow"", we now present our second volume. Like the first volume, this collection contains articles by some of the best scholars in the world on topological groups. A feature of the first volume was surveys, and we continue that tradition in this volume with three new surveys. These surveys are of interest not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics. Well-known researchers who contributed to this volume include Taras Banakh, Michael Megrelishvili, Sidney A. Morris, Saharon Shelah, George A. Willis, O'lga V. Sipacheva, and Stephen Wagner.

coarse structure --- descriptive set theory --- group representation --- thick set --- free topological group --- character --- selectively sequentially pseudocompact --- separable topological group --- quotient group --- Chabauty topology --- pseudo-?-bounded --- topological group --- free precompact Boolean group --- right-angled Artin groups --- Neretin’s group --- coarse space --- ultrafilter space --- endomorphism --- separable --- absolutely closed topological group --- tree --- strongly pseudocompact --- Gromov’s compactification --- dynamical system --- semigroup compactification --- tame function --- compact topological semigroup --- vast set --- space of closed subgroups --- reflexive group --- Lie group --- matrix coefficient --- maximal ideal --- topological semigroup --- ballean --- continuous inverse algebra --- extension --- subgroup --- Thompson’s group --- scale --- isomorphic embedding --- arrow ultrafilter --- H-space --- paratopological group --- pseudocompact --- Ramsey ultrafilter --- fibre bundle --- locally compact group --- product --- large set in a group --- Vietoris topology --- topological group of compact exponent --- Bourbaki uniformity --- p-compact --- mapping cylinder --- syndetic set --- p-adic Lie group --- Boolean topological group --- non-trivial convergent sequence --- fixed point algebra --- polish group topologies --- varieties of coarse spaces --- piecewise syndetic set --- maximal space

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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 lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks-a sort of potato-stamp method-Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.

Symmetry (Mathematics) --- Symmetry (Art) --- Abstract algebra. --- Addition. --- Algorithm. --- Antisymmetry. --- Arc length. --- Boundary value problem. --- Cartesian coordinate system. --- Circular motion. --- Circumference. --- Coefficient. --- Complex analysis. --- Complex multiplication. --- Complex number. --- Complex plane. --- Computation. --- Coordinate system. --- Coset. --- Cyclic group. --- Derivative. --- Diagonal. --- Diagram (category theory). --- Dihedral group. --- Division by zero. --- Domain coloring. --- Dot product. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein integer. --- Epicycloid. --- Equation. --- Euler's formula. --- Even and odd functions. --- Exponential function. --- Fourier series. --- Frieze group. --- Function (mathematics). --- Function composition. --- Function space. --- Gaussian integer. --- Geometry. --- Glide reflection. --- Group (mathematics). --- Group theory. --- Homomorphism. --- Horocycle. --- Hyperbolic geometry. --- Ideal point. --- Integer. --- Lattice (group). --- Linear interpolation. --- Local symmetry. --- M. C. Escher. --- Main diagonal. --- Mathematical proof. --- Mathematical structure. --- Mathematics. --- Mirror symmetry (string theory). --- Mirror symmetry. --- Morphing. --- Natural number. --- Normal subgroup. --- Notation. --- Ordinary differential equation. --- Parallelogram. --- Parametric equation. --- Parametrization. --- Periodic function. --- Plane symmetry. --- Plane wave. --- Point group. --- Polynomial. --- Power series. --- Projection (linear algebra). --- Pythagorean triple. --- Quantity. --- Quotient group. --- Real number. --- Reciprocal lattice. --- Rectangle. --- Reflection symmetry. --- Right angle. --- Ring of integers. --- Rotational symmetry. --- Scientific notation. --- Special case. --- Square lattice. --- Subgroup. --- Summation. --- Symmetry group. --- Symmetry. --- Tetrahedron. --- Theorem. --- Translational symmetry. --- Trigonometric functions. --- Unique factorization domain. --- Unit circle. --- Variable (mathematics). --- Vector space. --- Wallpaper group. --- Wave packet.

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The study of exponential sums over finite fields, begun by Gauss nearly two centuries ago, has been completely transformed in recent years by advances in algebraic geometry, culminating in Deligne's work on the Weil Conjectures. It now appears as a very attractive mixture of algebraic geometry, representation theory, and the sheaf-theoretic incarnations of such standard constructions of classical analysis as convolution and Fourier transform. The book is simultaneously an account of some of these ideas, techniques, and results, and an account of their application to concrete equidistribution questions concerning Kloosterman sums and Gauss sums.

Group theory --- Algebraic geometry --- Number theory --- 511.33 --- Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- 511.33 Analytical and multiplicative number theory. Asymptotics. Sieves etc. --- Gaussian sums --- Homology theory --- Kloosterman sums --- Monodromy groups --- Kloostermann sums --- Sums, Kloosterman --- Sums, Kloostermann --- Exponential sums --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Gauss sums --- Sums, Gaussian --- Analytical and multiplicative number theory. Asymptotics. Sieves etc --- Gaussian sums. --- Kloosterman sums. --- Homology theory. --- Monodromy groups. --- Number theory. --- Nombres, Théorie des. --- Exponential sums. --- Sommes exponentielles. --- Arithmetic --- Arithmétique --- Geometry, Algebraic. --- Géométrie algébrique --- Abelian category. --- Absolute Galois group. --- Absolute value. --- Additive group. --- Adjoint representation. --- Affine variety. --- Algebraic group. --- Automorphic form. --- Automorphism. --- Big O notation. --- Cartan subalgebra. --- Characteristic polynomial. --- Classification theorem. --- Coefficient. --- Cohomology. --- Cokernel. --- Combination. --- Commutator. --- Compactification (mathematics). --- Complex Lie group. --- Complex number. --- Conjugacy class. --- Continuous function. --- Convolution theorem. --- Convolution. --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Direct sum. --- Dual basis. --- Eigenvalues and eigenvectors. --- Empty set. --- Endomorphism. --- Equidistribution theorem. --- Estimation. --- Exactness. --- Existential quantification. --- Exponential sum. --- Exterior algebra. --- Faithful representation. --- Finite field. --- Finite group. --- Four-dimensional space. --- Frobenius endomorphism. --- Fundamental group. --- Fundamental representation. --- Galois group. --- Gauss sum. --- Homomorphism. --- Integer. --- Irreducibility (mathematics). --- Isomorphism class. --- Kloosterman sum. --- L-function. --- Leray spectral sequence. --- Lie algebra. --- Lie theory. --- Maximal compact subgroup. --- Method of moments (statistics). --- Monodromy theorem. --- Monodromy. --- Morphism. --- Multiplicative group. --- Natural number. --- Nilpotent. --- Open problem. --- P-group. --- Pairing. --- Parameter space. --- Parameter. --- Partially ordered set. --- Perfect field. --- Point at infinity. --- Polynomial ring. --- Prime number. --- Quotient group. --- Representation ring. --- Representation theory. --- Residue field. --- Riemann hypothesis. --- Root of unity. --- Sheaf (mathematics). --- Simple Lie group. --- Skew-symmetric matrix. --- Smooth morphism. --- Special case. --- Spin representation. --- Subgroup. --- Support (mathematics). --- Symmetric matrix. --- Symplectic group. --- Symplectic vector space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Trivial representation. --- Variable (mathematics). --- Weil conjectures. --- Weyl character formula. --- Zariski topology.