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Lie groups --- Symmetric spaces --- Algebraic varieties --- Embeddings (Mathematics) --- 512.81 --- Imbeddings (Mathematics) --- Geometry, Algebraic --- Immersions (Mathematics) --- Varieties, Algebraic --- Linear algebraic groups --- Spaces, Symmetric --- Geometry, Differential --- Groups, Lie --- Lie algebras --- Topological groups --- Algebraic varieties. --- Lie groups. --- Symmetric spaces. --- Embeddings (Mathematics). --- 512.81 Lie groups
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An overview of developments in the past 15 years of adjunction theory, the study of the interplay between the intrinsic geometry of a projective variety and the geometry connected with some embedding of the variety into a projective space. Topics include consequences of positivity, the Hilbert schem
Adjunction theory. --- Algebraic varieties. --- Embeddings (Mathematics) --- Projective spaces. --- Embeddings (Mathematics). --- Adjunction theory --- Algebraic varieties --- Projective spaces --- 512.7 --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Spaces, Projective --- Geometry, Projective --- Imbeddings (Mathematics) --- Geometry, Algebraic --- Immersions (Mathematics) --- Varieties, Algebraic --- Linear algebraic groups
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Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, distance geometry, combinatorial matrix theory, statistical physics, VLSI design etc. A main feature of this book is its interdisciplinarity. The book contains a wealth of results, from different mathematical disciplines, which are presented here in a unified and comprehensive manner. Geometric representations and methods turn out to be the linking theme. This book will provide a unique and invaluable source for researchers and graduate students. From the Reviews: "This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the interdisciplinarity of these fields as it gathers methods and results from polytope theory, geometry of numbers, probability theory, design and graph theory around two objects, cuts and metrics. [… ] The book is very nicely written [… ] The book is also very well structured. With knowledge about the relevant terms, one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. [… ] In my opinion, the book is a beautiful piece of work. The longer one works with it, the more beautiful it becomes." Robert Weismantel, Optima 56 (1997) "… In short, this is a very interesting book which is nice to have." Alexander I. Barvinok, MR 1460488 (98g:52001) "… This is a large and fascinating book. As befits a book which contains material relevant to so many areas of mathematics (and related disciplines such as statistics, physics, computing science, and economics), it is self-contained and written in a readable style. Moreover, the index, bibliography, and table of contents are all that they should be in such a work; it is easy to find as much or as little introductory material as needed." R.Dawson, Zentralblatt MATH Database 0885.52001.
Discrete mathematics --- Graph theory --- Metric spaces --- Embeddings (Mathematics) --- Théorie des graphes --- Espaces métriques --- Plongements (Mathématiques) --- Graph theory. --- Metric spaces. --- Embeddings (Mathematics). --- Théorie des graphes --- Espaces métriques --- Plongements (Mathématiques) --- Discrete mathematics. --- Geometry. --- Combinatorics. --- Convex geometry . --- Discrete geometry. --- Number theory. --- Discrete Mathematics. --- Graph Theory. --- Convex and Discrete Geometry. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Geometry --- Combinatorial geometry --- Combinatorics --- Mathematical analysis --- Graphs, Theory of --- Theory of graphs --- Combinatorial analysis --- Topology --- Mathematics --- Euclid's Elements --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Extremal problems
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The Bidual of C(X) I
Analytical spaces --- Banach spaces --- Duality theory (Mathematics) --- Embeddings (Mathematics) --- 515.12 --- 517.986 --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Topological algebras. Theory of infinite-dimensional representations --- 515.12 General topology --- General topology --- Imbeddings (Mathematics) --- Geometry, Algebraic --- Immersions (Mathematics) --- Algebra --- Mathematical analysis --- Topology --- Functions of complex variables --- Generalized spaces --- Banach spaces.
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517.982 --- 517.982 Linear spaces with topology and order or other structures --- Linear spaces with topology and order or other structures --- Integral representations --- Representations, Integral --- Imbedding theorems --- Theorems, Embedding --- Theorems, Imbedding --- Embeddings (Mathematics) --- Functions of several real variables. --- Invariant embedding. --- Functions of several real variables --- Lebesgue integration --- Invariant imbedding --- Invariant imbedding. --- Embedding theorems --- Functions of several complex variables --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Complex variables --- Several complex variables, Functions of --- Functions of complex variables --- Functional analysis --- Invariant embedding
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Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and Riemann-Roch translate into simple facts about polytopes, toric varieties provide a marvelous source of examples in algebraic geometry. In the other direction, general facts from algebraic geometry have implications for such polytopes, such as to the problem of the number of lattice points they contain. In spite of the fact that toric varieties are very special in the spectrum of all algebraic varieties, they provide a remarkably useful testing ground for general theories. The aim of this mini-course is to develop the foundations of the study of toric varieties, with examples, and describe some of these relations and applications. The text concludes with Stanley's theorem characterizing the numbers of simplicies in each dimension in a convex simplicial polytope. Although some general theorems are "ed without proof, the concrete interpretations via simplicial geometry should make the text accessible to beginners in algebraic geometry.
Algebraic geometry --- Differential geometry. Global analysis --- 512.7 --- Algebraic geometry. Commutative rings and algebras --- Toric varieties. --- 512.7 Algebraic geometry. Commutative rings and algebras --- Toric varieties --- Embeddings, Torus --- Torus embeddings --- Varieties, Toric --- Algebraic varieties --- Addition. --- Affine plane. --- Affine space. --- Affine variety. --- Alexander Grothendieck. --- Alexander duality. --- Algebraic curve. --- Algebraic group. --- Atiyah–Singer index theorem. --- Automorphism. --- Betti number. --- Big O notation. --- Characteristic class. --- Chern class. --- Chow group. --- Codimension. --- Cohomology. --- Combinatorics. --- Commutative property. --- Complete intersection. --- Convex polytope. --- Convex set. --- Coprime integers. --- Cotangent space. --- Dedekind sum. --- Dimension (vector space). --- Dimension. --- Direct proof. --- Discrete valuation ring. --- Discrete valuation. --- Disjoint union. --- Divisor (algebraic geometry). --- Divisor. --- Dual basis. --- Dual space. --- Equation. --- Equivalence class. --- Equivariant K-theory. --- Euler characteristic. --- Exact sequence. --- Explicit formula. --- Facet (geometry). --- Fundamental group. --- Graded ring. --- Grassmannian. --- H-vector. --- Hirzebruch surface. --- Hodge theory. --- Homogeneous coordinates. --- Homomorphism. --- Hypersurface. --- Intersection theory. --- Invertible matrix. --- Invertible sheaf. --- Isoperimetric inequality. --- Lattice (group). --- Leray spectral sequence. --- Limit point. --- Line bundle. --- Line segment. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mixed volume. --- Moduli space. --- Moment map. --- Monotonic function. --- Natural number. --- Newton polygon. --- Open set. --- Picard group. --- Pick's theorem. --- Polytope. --- Projective space. --- Quadric. --- Quotient space (topology). --- Regular sequence. --- Relative interior. --- Resolution of singularities. --- Restriction (mathematics). --- Resultant. --- Riemann–Roch theorem. --- Serre duality. --- Sign (mathematics). --- Simplex. --- Simplicial complex. --- Simultaneous equations. --- Spectral sequence. --- Subgroup. --- Subset. --- Summation. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Toric variety. --- Unit disk. --- Vector space. --- Weil conjecture. --- Zariski topology.
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