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This book treats the interaction between discrete convex geometry, commutative ring theory, algebraic K-theory, and algebraic geometry. The basic mathematical objects are lattice polytopes, rational cones, affine monoids, the algebras derived from them, and toric varieties. The book discusses several properties and invariants of these objects, such as efficient generation, unimodular triangulations and covers, basic theory of monoid rings, isomorphism problems and automorphism groups, homological properties and enumerative combinatorics. The last part is an extensive treatment of the K-theory of monoid rings, with extensions to toric varieties and their intersection theory. This monograph has been written with a view towards graduate students and researchers who want to study the cross-connections of algebra and discrete convex geometry. While the text has been written from an algebraist's view point, also specialists in lattice polytopes and related objects will find an up-to-date discussion of affine monoids and their combinatorial structure. Though the authors do not explicitly formulate algorithms, the book takes a constructive approach wherever possible. Winfried Bruns is Professor of Mathematics at Universität Osnabrück. Joseph Gubeladze is Professor of Mathematics at San Francisco State University.

K-theory. --- Polytopes. --- Rings (Algebra). --- Polytopes --- K-theory --- Rings (Algebra) --- Geometry --- Mathematics --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Mathematics. --- Algebra. --- Commutative algebra. --- Commutative rings. --- Convex geometry. --- Discrete geometry. --- Commutative Rings and Algebras. --- K-Theory. --- Convex and Discrete Geometry. --- Algebraic topology --- Homology theory --- Combinatorial geometry --- Algebra --- Mathematical analysis --- Math --- Science --- Algebraic fields --- Hyperspace --- Topology --- Discrete groups. --- Groups, Discrete --- Infinite groups --- Discrete mathematics --- Convex geometry .

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Polycyclic groups are built from cyclic groups in a specific way. They arise in many contexts within group theory itself but also more generally in algebra, for example in the theory of Noetherian rings. They also touch on some aspects of topology, geometry and number theory. The first half of this book develops the standard group theoretic techniques for studying polycyclic groups and the basic properties of these groups. The second half then focuses specifically on the ring theoretic properties of polycyclic groups and their applications, often to purely group theoretic situations. The book is not intended to be encyclopedic. Instead, it is a study manual for graduate students and researchers coming into contact with polycyclic groups, where the main lines of the subject can be learned from scratch by any reader who has been exposed to some undergraduate algebra, especially groups, rings and vector spaces. Thus the book has been kept short and readable with a view that it can be read and worked through from cover to cover. At the end of each topic covered there is a description without proofs, but with full references, of further developments in the area. The book then concludes with an extensive bibliography of items relating to polycyclic groups.

Graph theory. --- Polycyclic groups. --- Rings (Algebra). --- Solvable groups. --- Polycyclic groups --- Solvable groups --- Graph theory --- Rings (Algebra) --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Graphs, Theory of --- Theory of graphs --- Extremal problems --- Mathematics. --- Associative rings. --- Commutative algebra. --- Commutative rings. --- Group theory. --- Group Theory and Generalizations. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Algebraic fields --- Combinatorial analysis --- Topology --- Infinite groups --- Nilpotent groups --- Algebra. --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics)

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Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

Singularities (Mathematics) --- 512.761 --- Geometry, Algebraic --- Singularities. Singular points of algebraic varieties --- 512.761 Singularities. Singular points of algebraic varieties --- Adjunction formula. --- Algebraic closure. --- Algebraic geometry. --- Algebraic space. --- Algebraic surface. --- Algebraic variety. --- Approximation. --- Asymptotic analysis. --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Birational geometry. --- C0. --- Canonical singularity. --- Codimension. --- Cohomology. --- Commutative algebra. --- Complex analysis. --- Complex manifold. --- Computability. --- Continuous function. --- Coordinate system. --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension. --- Divisor. --- Du Val singularity. --- Dual graph. --- Embedding. --- Equation. --- Equivalence relation. --- Euclidean algorithm. --- Factorization. --- Functor. --- General position. --- Generic point. --- Geometric genus. --- Geometry. --- Hyperplane. --- Hypersurface. --- Integral domain. --- Intersection (set theory). --- Intersection number (graph theory). --- Intersection theory. --- Irreducible component. --- Isolated singularity. --- Laurent series. --- Line bundle. --- Linear space (geometry). --- Linear subspace. --- Mathematical induction. --- Mathematics. --- Maximal ideal. --- Morphism. --- Newton polygon. --- Noetherian ring. --- Noetherian. --- Open problem. --- Open set. --- P-adic number. --- Pairwise. --- Parametric equation. --- Partial derivative. --- Plane curve. --- Polynomial. --- Power series. --- Principal ideal. --- Principalization (algebra). --- Projective space. --- Projective variety. --- Proper morphism. --- Puiseux series. --- Quasi-projective variety. --- Rational function. --- Regular local ring. --- Resolution of singularities. --- Riemann surface. --- Ring theory. --- Ruler. --- Scientific notation. --- Sheaf (mathematics). --- Singularity theory. --- Smooth morphism. --- Smoothness. --- Special case. --- Subring. --- Summation. --- Surjective function. --- Tangent cone. --- Tangent space. --- Tangent. --- Taylor series. --- Theorem. --- Topology. --- Toric variety. --- Transversal (geometry). --- Variable (mathematics). --- Weierstrass preparation theorem. --- Weierstrass theorem. --- Zero set. --- Differential geometry. Global analysis