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The issue "Mathematical Logic and Its Applications 2020" contains articles related to the following three directions: Descriptive Set Theory (3 articles). Solutions for long-standing problems, including those of A. Tarski and H. Friedman, are presented. Exact combinatorial optimization algorithms, in which the complexity relative to the source data is characterized by a low, or even first degree, polynomial (1 article). III. Applications of mathematical logic and the theory of algorithms (2 articles). The first article deals with the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability; for these purposes, non-standard analysis is used. The second article provides a quantitative description of the balance and adaptive resource of a human. Submissions are invited for the next issue "Mathematical Logic and Its Applications 2021"

definability --- nonconstructible reals --- projective hierarchy --- generic models --- almost disjoint forcing --- Harvey Friedman’s problem --- almost-disjoint forcing --- universal algebraic geometry --- affine algebraic geometry --- elementary equivalence --- isotypic algebras --- first order rigidity --- Ind-group --- affine spaces --- automorphisms --- free associative algebras --- Weyl algebra automorphisms --- polynomial symplectomorphisms --- deformation quantization --- infinite prime number --- semi-inner automorphism --- embeddability of varieties --- undecidability --- noncommutative Gröbner-Shirshov basis --- finitely presented algebraic systems --- algorithmic unsolvability --- turing machine --- discrete optimization --- exact algorithm --- additively exact algorithm --- graph transformation --- graph of degree 2 --- chain-cycle graph --- operation cost --- minimization of total cost --- definability of definable --- tarski problem --- type theoretic hierarchy --- adaptive resource --- resilience --- resistance --- tipping point --- theory of catastrophes

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This book presents a classification of all (complex)irreducible representations of a reductive group withconnected centre, over a finite field. To achieve this,the author uses etale intersection cohomology, anddetailed information on representations of Weylgroups.

512 --- Characters of groups --- Finite fields (Algebra) --- Finite groups --- Groups, Finite --- Group theory --- Modules (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Characters, Group --- Group characters --- Groups, Characters of --- Representations of groups --- Rings (Algebra) --- Algebra --- 512 Algebra --- Finite groups. --- Characters of groups. --- Addition. --- Algebra representation. --- Algebraic closure. --- Algebraic group. --- Algebraic variety. --- Algebraically closed field. --- Bijection. --- Borel subgroup. --- Cartan subalgebra. --- Character table. --- Character theory. --- Characteristic function (probability theory). --- Characteristic polynomial. --- Class function (algebra). --- Classical group. --- Coefficient. --- Cohomology with compact support. --- Cohomology. --- Combination. --- Complex number. --- Computation. --- Conjugacy class. --- Connected component (graph theory). --- Coxeter group. --- Cyclic group. --- Cyclotomic polynomial. --- David Kazhdan. --- Dense set. --- Derived category. --- Diagram (category theory). --- Dimension. --- Direct sum. --- Disjoint sets. --- Disjoint union. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equivalence class. --- Equivalence relation. --- Existential quantification. --- Explicit formula. --- Explicit formulae (L-function). --- Fiber bundle. --- Finite field. --- Finite group. --- Fourier transform. --- Green's function. --- Group (mathematics). --- Group action. --- Group representation. --- Harish-Chandra. --- Hecke algebra. --- Identity element. --- Integer. --- Irreducible representation. --- Isomorphism class. --- Jordan decomposition. --- Line bundle. --- Linear combination. --- Local system. --- Mathematical induction. --- Maximal torus. --- Module (mathematics). --- Monodromy. --- Morphism. --- Orthonormal basis. --- P-adic number. --- Parametrization. --- Parity (mathematics). --- Partially ordered set. --- Perverse sheaf. --- Pointwise. --- Polynomial. --- Quantity. --- Rational point. --- Reductive group. --- Ree group. --- Schubert variety. --- Scientific notation. --- Semisimple Lie algebra. --- Sheaf (mathematics). --- Simple group. --- Simple module. --- Special case. --- Standard basis. --- Subset. --- Subtraction. --- Summation. --- Surjective function. --- Symmetric group. --- Tensor product. --- Theorem. --- Two-dimensional space. --- Unipotent representation. --- Vector bundle. --- Vector space. --- Verma module. --- Weil conjecture. --- Weyl group. --- Zariski topology.

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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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Beginning with a general discussion of bordism, Professors Madsen and Milgram present the homotopy theory of the surgery classifying spaces and the classifying spaces for the various required bundle theories. The next part covers more recent work on the maps between these spaces and the properties of the PL and Top characteristic classes, and includes integrality theorems for topological and PL manifolds. Later chapters treat the integral cohomology of BPL and Btop. The authors conclude with a discussion of the PL and topological cobordism rings and a construction of the torsion-free generators.

Algebraic topology --- 515.16 --- Classifying spaces --- Cobordism theory --- Manifolds (Mathematics) --- Surgery (Topology) --- Differential topology --- Homotopy equivalences --- Topology --- Geometry, Differential --- Spaces, Classifying --- Fiber bundles (Mathematics) --- Fiber spaces (Mathematics) --- Topology of manifolds --- Classifying spaces. --- Cobordism theory. --- Manifolds (Mathematics). --- Surgery (Topology). --- 515.16 Topology of manifolds --- Bijection. --- Calculation. --- Characteristic class. --- Classification theorem. --- Classifying space. --- Closed manifold. --- Cobordism. --- Coefficient. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex projective space. --- Connected sum. --- Corollary. --- Cup product. --- Diagram (category theory). --- Differentiable manifold. --- Disjoint union. --- Disk (mathematics). --- Effective method. --- Eilenberg–Moore spectral sequence. --- Elaboration. --- Equivalence class. --- Exact sequence. --- Exterior algebra. --- Fiber bundle. --- Fibration. --- Function composition. --- H-space. --- Homeomorphism. --- Homomorphism. --- Homotopy fiber. --- Homotopy group. --- Homotopy. --- Hopf algebra. --- Iterative method. --- Loop space. --- Manifold. --- Massey product. --- N-sphere. --- Normal bundle. --- Obstruction theory. --- Pairing. --- Permutation. --- Piecewise linear manifold. --- Piecewise linear. --- Polynomial. --- Prime number. --- Projective space. --- Sequence. --- Simply connected space. --- Special case. --- Spin structure. --- Steenrod algebra. --- Subset. --- Summation. --- Tensor product. --- Theorem. --- Topological group. --- Topological manifold. --- Topology. --- Total order. --- Variétés topologiques --- Topologie differentielle

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A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

Algebraic geometry --- Analytic spaces --- SINGULARITIES (Mathematics) --- 512.76 --- Singularities (Mathematics) --- Geometry, Algebraic --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Birational geometry. Mappings etc. --- Analytic spaces. --- Singularities (Mathematics). --- 512.76 Birational geometry. Mappings etc. --- Birational geometry. Mappings etc --- Analytic function. --- Analytic set. --- Analytic space. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Calculation. --- Chern class. --- Codimension. --- Coefficient. --- Cohomology. --- Compact Riemann surface. --- Complex manifold. --- Computation. --- Connected component (graph theory). --- Continuous function. --- Contradiction. --- Coordinate system. --- Corollary. --- Covering space. --- Dimension. --- Disjoint union. --- Divisor. --- Dual graph. --- Elliptic curve. --- Elliptic function. --- Embedding. --- Existential quantification. --- Factorization. --- Fiber bundle. --- Finite set. --- Formal power series. --- Hausdorff space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Intersection (set theory). --- Intersection number (graph theory). --- Inverse limit. --- Irreducible component. --- Isolated singularity. --- Iteration. --- Lattice (group). --- Line bundle. --- Linear combination. --- Line–line intersection. --- Local coordinates. --- Local ring. --- Mathematical induction. --- Maximal ideal. --- Meromorphic function. --- Monic polynomial. --- Nilpotent. --- Normal bundle. --- Open set. --- Parameter. --- Plane curve. --- Pole (complex analysis). --- Power series. --- Presheaf (category theory). --- Projective line. --- Quadratic transformation. --- Quantity. --- Riemann surface. --- Riemann–Roch theorem. --- Several complex variables. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Transition function. --- Two-dimensional space. --- Variable (mathematics). --- Zero divisor. --- Zero of a function. --- Zero set. --- Variétés complexes --- Espaces analytiques