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Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers. Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on "ients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

Mumford-Tate groups. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Hodge theory --- Mumford-Tate groups --- Deligne torus integer. --- Galois group. --- Grothendieck conjecture. --- Hodge decomposition. --- Hodge domain. --- Hodge filtration. --- Hodge orientation. --- Hodge representation. --- Hodge structure. --- Hodge tensor. --- Hodge theory. --- Kubota rank. --- Lie algebra representation. --- Lie group. --- Mumford-Tate domain. --- Mumford-Tate group. --- Mumford-Tate subdomain. --- Noether-Lefschetz locus. --- Vogan diagram method. --- Weyl group. --- abelian variety. --- absolute Hodge class. --- algebraic geometry. --- arithmetic group. --- automorphic cohomology. --- classical group. --- compact dual. --- complex manifold. --- complex multiplication Hodge structure. --- complex multiplication. --- endomorphism algebra. --- exceptional group. --- holomorphic mapping. --- homogeneous complex manifold. --- homomorphism. --- mixed Hodge structure. --- moduli space. --- monodromy group. --- natural symmetry group. --- oriented imaginary number fields. --- period domain. --- period map. --- polarization. --- polarized Hodge structure. --- pure Hodge structure. --- reflex field. --- semisimple Lie algebra. --- semisimple Lie group. --- Γ-equivalence classes.

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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. --- 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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Outer billiards is a basic dynamical system defined relative to a convex shape in the plane. B. H. Neumann introduced this system in the 1950's, and J. Moser popularized it as a toy model for celestial mechanics. All along, the so-called Moser-Neumann question has been one of the central problems in the field. This question asks whether or not one can have an outer billiards system with an unbounded orbit. The Moser-Neumann question is an idealized version of the question of whether, because of small disturbances in its orbit, the Earth can break out of its orbit and fly away from the Sun. In Outer Billiards on Kites, Richard Schwartz presents his affirmative solution to the Moser-Neumann problem. He shows that an outer billiards system can have an unbounded orbit when defined relative to any irrational kite. A kite is a quadrilateral having a diagonal that is a line of bilateral symmetry. The kite is irrational if the other diagonal divides the quadrilateral into two triangles whose areas are not rationally related. In addition to solving the basic problem, Schwartz relates outer billiards on kites to such topics as Diophantine approximation, the modular group, self-similar sets, polytope exchange maps, profinite completions of the integers, and solenoids--connections that together allow for a fairly complete analysis of the dynamical system.

Hyperbolic spaces. --- Singularities (Mathematics) --- Transformations (Mathematics) --- Geometry, Plane. --- Plane geometry --- Algorithms --- Differential invariants --- Geometry, Differential --- Geometry, Algebraic --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Abelian group. --- Automorphism. --- Big O notation. --- Bijection. --- Binary number. --- Bisection. --- Borel set. --- C0. --- Calculation. --- Cantor set. --- Cartesian coordinate system. --- Combination. --- Compass-and-straightedge construction. --- Congruence subgroup. --- Conjecture. --- Conjugacy class. --- Continuity equation. --- Convex lattice polytope. --- Convex polytope. --- Coprime integers. --- Counterexample. --- Cyclic group. --- Diameter. --- Diophantine approximation. --- Diophantine equation. --- Disjoint sets. --- Disjoint union. --- Division by zero. --- Embedding. --- Equation. --- Equivalence class. --- Ergodic theory. --- Ergodicity. --- Factorial. --- Fiber bundle. --- Fibonacci number. --- Fundamental domain. --- Gauss map. --- Geometry. --- Half-integer. --- Homeomorphism. --- Hyperbolic geometry. --- Hyperplane. --- Ideal triangle. --- Intersection (set theory). --- Interval exchange transformation. --- Inverse function. --- Inverse limit. --- Isometry group. --- Lattice (group). --- Limit set. --- Line segment. --- Linear algebra. --- Linear function. --- Line–line intersection. --- Main diagonal. --- Modular group. --- Monotonic function. --- Multiple (mathematics). --- Orthant. --- Outer billiard. --- Parallelogram. --- Parameter. --- Partial derivative. --- Penrose tiling. --- Permutation. --- Piecewise. --- Polygon. --- Polyhedron. --- Polytope. --- Product topology. --- Projective geometry. --- Rectangle. --- Renormalization. --- Rhombus. --- Right angle. --- Rotational symmetry. --- Sanity check. --- Scientific notation. --- Semicircle. --- Sign (mathematics). --- Special case. --- Square root of 2. --- Subsequence. --- Summation. --- Symbolic dynamics. --- Symmetry group. --- Tangent. --- Tetrahedron. --- Theorem. --- Toy model. --- Translational symmetry. --- Trapezoid. --- Triangle group. --- Triangle inequality. --- Two-dimensional space. --- Upper and lower bounds. --- Upper half-plane. --- Without loss of generality. --- Yair Minsky.

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This Special Issue presents research papers on various topics within many different branches of mathematics, applied mathematics, and mathematical physics. Each paper presents mathematical theories, methods, and their application based on current and recently developed symmetric polynomials. Also, each one aims to provide the full understanding of current research problems, theories, and applications on the chosen topics and includes the most recent advances made in the area of symmetric functions and polynomials.

generalized Laguerre --- central complete Bell numbers --- rational polynomials --- Changhee polynomials of type two --- Euler polynomials --- generalized Laguerre polynomials --- Hermite --- conjecture --- Legendre --- the degenerate gamma function --- trivariate Lucas polynomials --- perfectly matched layer --- third-order character --- Euler numbers --- two variable q-Berstein operator --- entropy production --- hypergeometric function --- q-Bernoulli numbers --- q-Bernoulli polynomials --- symmetry group --- Bernoulli polynomials --- Fibonacci polynomials --- central incomplete Bell polynomials --- Chebyshev polynomials --- convolution sums --- Lucas polynomials --- Jacobi --- the modified degenerate Laplace transform --- q-Volkenborn integral on ?p --- and fourth kinds --- two variable q-Berstein polynomial --- the modified degenerate gamma function --- two variable q-Bernstein operators --- reduction method --- identity --- elementary and combinatorial methods --- generalized Bernoulli polynomials and numbers attached to a Dirichlet character ? --- explicit relations --- recursive sequence --- Fubini polynomials --- p-adic integral on ?p --- generating functions --- q-Euler number --- acoustic wave equation --- congruence --- trivariate Fibonacci polynomials --- stochastic thermodynamics --- fermionic p-adic integrals --- Laguerre polynomials --- fluctuation theorem --- Bernoulli numbers and polynomials --- w-torsion Fubini polynomials --- non-equilibrium free energy --- hypergeometric functions 1F1 and 2F1 --- recursive formula --- Chebyshev polynomials of the first --- second --- central complete Bell polynomials --- Apostol-type Frobenius–Euler polynomials --- sums of finite products --- q-Euler polynomial --- symmetric identities --- stability --- fermionic p-adic q-integral on ?p --- Gegenbauer polynomials --- continued fraction --- thermodynamics of information --- well-posedness --- fermionic p-adic integral on ?p --- catalan numbers --- classical Gauss sums --- three-variable Hermite polynomials --- q-Changhee polynomials --- Catalan numbers --- two variable q-Bernstein polynomials --- q-Euler polynomials --- analytic method --- representation --- mutual information --- Fibonacci --- Legendre polynomials --- Gegenbauer --- generalized Bernoulli polynomials and numbers of arbitrary complex order --- Lucas --- elementary method --- new sequence --- third --- the degenerate Laplace transform --- computational formula --- operational connection --- sums of finite products of Chebyshev polynomials of the third and fourth kinds --- Changhee polynomials --- linear form in logarithms