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Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, May 21-25, 1973. The papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmüller spaces, Jacobian varieties, and quasiconformal mappings. These topics are intertwined, representing a common meeting of algebra, geometry, and analysis.

Group theory --- Complex analysis --- Number theory --- RIEMANN SURFACES --- Discontinuous groups --- congresses --- Congresses --- Riemann surfaces --- Congresses. --- Groupes discontinus --- Combinatorial topology --- Functions of complex variables --- Surfaces, Riemann --- Functions --- Abelian variety. --- Adjunction (field theory). --- Affine space. --- Algebraic curve. --- Algebraic structure. --- Analytic function. --- Arithmetic genus. --- Automorphism. --- Bernhard Riemann. --- Boundary (topology). --- Cauchy sequence. --- Cauchy–Schwarz inequality. --- Cayley–Hamilton theorem. --- Closed geodesic. --- Combination. --- Commutative diagram. --- Commutator subgroup. --- Compact Riemann surface. --- Complex dimension. --- Complex manifold. --- Complex multiplication. --- Complex space. --- Complex torus. --- Congruence subgroup. --- Conjugacy class. --- Convex set. --- Cyclic group. --- Degeneracy (mathematics). --- Diagram (category theory). --- Diffeomorphism. --- Differential form. --- Dimension (vector space). --- Disjoint sets. --- E7 (mathematics). --- Endomorphism. --- Equation. --- Equivalence class. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Finite group. --- Finitely generated group. --- Fuchsian group. --- Fundamental domain. --- Fundamental lemma (Langlands program). --- Fundamental polygon. --- Galois extension. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homomorphism. --- Hurwitz's theorem (number theory). --- Inclusion map. --- Inequality (mathematics). --- Inner automorphism. --- Intersection (set theory). --- Irreducibility (mathematics). --- Isomorphism class. --- Isomorphism theorem. --- Jacobian variety. --- Jordan curve theorem. --- Kleinian group. --- Limit point. --- Mapping class group. --- Metric space. --- Monodromy. --- Monomorphism. --- Möbius transformation. --- Non-Euclidean geometry. --- Orthogonal trajectory. --- Permutation. --- Polynomial. --- Power series. --- Projective variety. --- Quadratic differential. --- Quadric. --- Quasi-projective variety. --- Quasiconformal mapping. --- Quotient space (topology). --- Rectangle. --- Riemann mapping theorem. --- Riemann surface. --- Schwarzian derivative. --- Simply connected space. --- Simultaneous equations. --- Special case. --- Subgroup. --- Subsequence. --- Surjective function. --- Symmetric space. --- Tangent space. --- Teichmüller space. --- Theorem. --- Topological space. --- Topology. --- Uniqueness theorem. --- Unit disk. --- Variable (mathematics). --- Winding number. --- Word problem (mathematics). --- RIEMANN SURFACES - congresses --- Discontinuous groups - Congresses --- Geometrie algebrique --- Fonctions d'une variable complexe --- Surfaces de riemann

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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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Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.

Algebraic geometry --- Ordered algebraic structures --- Associative rings --- Abelian groups --- Functor theory --- Anneaux associatifs --- Groupes abéliens --- Foncteurs, Théorie des --- 512.73 --- 515.14 --- Functorial representation --- Algebra, Homological --- Categories (Mathematics) --- Functional analysis --- Transformations (Mathematics) --- Commutative groups --- Group theory --- Rings (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Algebraic topology --- Abelian groups. --- Associative rings. --- Functor theory. --- 515.14 Algebraic topology --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes abéliens --- Foncteurs, Théorie des --- Abelian group. --- Absolute value. --- Addition. --- Algebraic K-theory. --- Algebraic equation. --- Algebraic integer. --- Banach algebra. --- Basis (linear algebra). --- Big O notation. --- Circle group. --- Coefficient. --- Commutative property. --- Commutative ring. --- Commutator. --- Complex number. --- Computation. --- Congruence subgroup. --- Coprime integers. --- Cyclic group. --- Dedekind domain. --- Direct limit. --- Direct proof. --- Direct sum. --- Discrete valuation. --- Division algebra. --- Division ring. --- Elementary matrix. --- Elliptic function. --- Exact sequence. --- Existential quantification. --- Exterior algebra. --- Factorization. --- Finite group. --- Free abelian group. --- Function (mathematics). --- Fundamental group. --- Galois extension. --- Galois group. --- General linear group. --- Group extension. --- Hausdorff space. --- Homological algebra. --- Homomorphism. --- Homotopy. --- Ideal (ring theory). --- Ideal class group. --- Identity element. --- Identity matrix. --- Integral domain. --- Invertible matrix. --- Isomorphism class. --- K-theory. --- Kummer theory. --- Lattice (group). --- Left inverse. --- Local field. --- Local ring. --- Mathematics. --- Matsumoto's theorem. --- Maximal ideal. --- Meromorphic function. --- Monomial. --- Natural number. --- Noetherian. --- Normal subgroup. --- Number theory. --- Open set. --- Picard group. --- Polynomial. --- Prime element. --- Prime ideal. --- Projective module. --- Quadratic form. --- Quaternion. --- Quotient ring. --- Rational number. --- Real number. --- Right inverse. --- Ring of integers. --- Root of unity. --- Schur multiplier. --- Scientific notation. --- Simple algebra. --- Special case. --- Special linear group. --- Subgroup. --- Summation. --- Surjective function. --- Tensor product. --- Theorem. --- Topological K-theory. --- Topological group. --- Topological space. --- Topology. --- Torsion group. --- Variable (mathematics). --- Vector space. --- Wedderburn's theorem. --- Weierstrass function. --- Whitehead torsion. --- K-théorie

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The results established in this book constitute a new departure in ergodic theory and a significant expansion of its scope. Traditional ergodic theorems focused on amenable groups, and relied on the existence of an asymptotically invariant sequence in the group, the resulting maximal inequalities based on covering arguments, and the transference principle. Here, Alexander Gorodnik and Amos Nevo develop a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. Simple general conditions on the spectral theory of the group and the regularity of the averaging sets are formulated, which suffice to guarantee convergence to the ergodic mean. In particular, this approach gives a complete solution to the problem of establishing mean and pointwise ergodic theorems for the natural averages on semisimple algebraic groups and on their discrete lattice subgroups. Furthermore, an explicit quantitative rate of convergence to the ergodic mean is established in many cases. The topic of this volume lies at the intersection of several mathematical fields of fundamental importance. These include ergodic theory and dynamics of non-amenable groups, harmonic analysis on semisimple algebraic groups and their homogeneous spaces, quantitative non-Euclidean lattice point counting problems and their application to number theory, as well as equidistribution and non-commutative Diophantine approximation. Many examples and applications are provided in the text, demonstrating the usefulness of the results established.

Dynamics. --- Ergodic theory. --- Harmonic analysis. --- Lattice theory. --- Lie groups. --- Ergodic theory --- Lie groups --- Lattice theory --- Harmonic analysis --- Dynamics --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamical systems --- Kinetics --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Groups, Lie --- Ergodic transformations --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical --- Lie algebras --- Symmetric spaces --- Topological groups --- Continuous groups --- Mathematical physics --- Measure theory --- Absolute continuity. --- Algebraic group. --- Amenable group. --- Asymptote. --- Asymptotic analysis. --- Asymptotic expansion. --- Automorphism. --- Borel set. --- Bounded function. --- Bounded operator. --- Bounded set (topological vector space). --- Congruence subgroup. --- Continuous function. --- Convergence of random variables. --- Convolution. --- Coset. --- Counting problem (complexity). --- Counting. --- Differentiable function. --- Dimension (vector space). --- Diophantine approximation. --- Direct integral. --- Direct product. --- Discrete group. --- Embedding. --- Equidistribution theorem. --- Ergodicity. --- Estimation. --- Explicit formulae (L-function). --- Family of sets. --- Haar measure. --- Hilbert space. --- Hyperbolic space. --- Induced representation. --- Infimum and supremum. --- Initial condition. --- Interpolation theorem. --- Invariance principle (linguistics). --- Invariant measure. --- Irreducible representation. --- Isometry group. --- Iwasawa group. --- Lattice (group). --- Lie algebra. --- Linear algebraic group. --- Linear space (geometry). --- Lipschitz continuity. --- Mass distribution. --- Mathematical induction. --- Maximal compact subgroup. --- Maximal ergodic theorem. --- Measure (mathematics). --- Mellin transform. --- Metric space. --- Monotonic function. --- Neighbourhood (mathematics). --- Normal subgroup. --- Number theory. --- One-parameter group. --- Operator norm. --- Orthogonal complement. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Pointwise convergence. --- Pointwise. --- Principal homogeneous space. --- Principal series representation. --- Probability measure. --- Probability space. --- Probability. --- Rate of convergence. --- Regular representation. --- Representation theory. --- Resolution of singularities. --- Sobolev space. --- Special case. --- Spectral gap. --- Spectral method. --- Spectral theory. --- Square (algebra). --- Subgroup. --- Subsequence. --- Subset. --- Symmetric space. --- Tensor algebra. --- Tensor product. --- Theorem. --- Transfer principle. --- Unit sphere. --- Unit vector. --- Unitary group. --- Unitary representation. --- Upper and lower bounds. --- Variable (mathematics). --- Vector group. --- Vector space. --- Volume form. --- Word metric.