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Algebraic geometry --- Arithmetical algebraic geometry. --- Riemann-Roch theorems. --- Géométrie algébrique arithmétique. --- Arithmetical algebraic geometry --- Riemann-Roch theorems --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory
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B.L. van der Waerden: Démonstration algébrique du théorème de Riemann-Roch.- F. Severi: Del teorema di Riemann-Roch per curve, superficie e varietà. Le origini storiche e lo stato attuale.- F. Hirzebruch: Arithmetic genera and the theorem of Riemann-Roch.
Algebra. --- Algebraic varieties. --- Geometry. --- Geometry, Algebraic. --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Riemann-Roch theorems --- Geometry, Algebraic --- Theorems, Riemann-Roch --- Mathematics. --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic functions --- Geometry, algebraic. --- Algebraic geometry
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The book focuses on the educational perspective of Riemann-Roch spaces and the computation of algebraic structures connected to the Riemann-Roch theorem, which is a useful tool in fields of complex analysis and algebraic geometry. On one hand, the theorem connects the Riemann surface with its topological genus, and on the other it allows us to compute the algebraic function field spaces. In the first part of this book, algebraic structures and some of their properties are presented. The second part shows efficient algorithms and examples connected to Riemann-Roch spaces. What is important, a variety of examples with codes of algorithms are given in the book, covering the majority of the cases.
Mathematics --- Physical Sciences & Mathematics --- Calculus --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Riemann-Roch spaces, Diophantine equations, integral domains.
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Algebraic geometry --- Geometry, Algebraic --- Riemann-Roch theorems --- 512.73 --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Geometry, Algebraic. --- Riemann-Roch theorems. --- 512.73 Cohomology theory of algebraic varieties and schemes --- Rieman-Roch (Theorema's van). --- Géométrie algébrique --- Rieman-Roch (Théorèmes de). --- Meetkunde (Algebraïsche).
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The arithmetic Riemann-Roch Theorem has been shown recently by Bismut-Gillet-Soul. The proof mixes algebra, arithmetic, and analysis. The purpose of this book is to give a concise introduction to the necessary techniques, and to present a simplified and extended version of the proof. It should enable mathematicians with a background in arithmetic algebraic geometry to understand some basic techniques in the rapidly evolving field of Arakelov-theory.
Algebraic geometry --- Algebraïsche meetkunde --- Geometry [Algebraic ] --- Géométrie algébrique --- Meetkunde [Algebraïsche ] --- Riemann-Roch theorema's --- Riemann-Roch thoerems --- Theoremes de Riemann-Roch --- Geometry, Algebraic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Geometry, Algebraic --- Geometry --- Addition. --- Adjoint. --- Alexander Grothendieck. --- Algebraic geometry. --- Analytic torsion. --- Arakelov theory. --- Asymptote. --- Asymptotic expansion. --- Asymptotic formula. --- Big O notation. --- Cartesian coordinate system. --- Characteristic class. --- Chern class. --- Chow group. --- Closed immersion. --- Codimension. --- Coherent sheaf. --- Cohomology. --- Combination. --- Commutator. --- Computation. --- Covariant derivative. --- Curvature. --- Derivative. --- Determinant. --- Diagonal. --- Differentiable manifold. --- Differential form. --- Dimension (vector space). --- Divisor. --- Domain of a function. --- Dual basis. --- E6 (mathematics). --- Eigenvalues and eigenvectors. --- Embedding. --- Endomorphism. --- Exact sequence. --- Exponential function. --- Generic point. --- Heat kernel. --- Injective function. --- Intersection theory. --- K-group. --- Levi-Civita connection. --- Line bundle. --- Linear algebra. --- Local coordinates. --- Mathematical induction. --- Morphism. --- Natural number. --- Neighbourhood (mathematics). --- Parameter. --- Projective space. --- Pullback (category theory). --- Pullback (differential geometry). --- Pullback. --- Riemannian manifold. --- Riemann–Roch theorem. --- Self-adjoint operator. --- Smoothness. --- Sobolev space. --- Stochastic calculus. --- Summation. --- Supertrace. --- Theorem. --- Transition function. --- Upper half-plane. --- Vector bundle. --- Volume form.
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Courbes algebriques. --- Fonctions zeta. --- Corps finis --- Riemann-Roch, Theoreme de --- Sommes exponentielles --- Courbes algébriques --- Fonctions zêta --- Curves, Algebraic. --- Algebraic fields. --- Functions, Zeta. --- Algebraic fields --- Curves, Algebraic --- Functions, Zeta --- Corps algébriques --- Courbes algebriques --- Fonctions zeta
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This book is devoted to computing the index of elliptic PDEs on non-compact Riemannian manifolds in the presence of local singularities and zeros, as well as polynomial growth at infinity. The classical Riemann–Roch theorem and its generalizations to elliptic equations on bounded domains and compact manifolds, due to Maz’ya, Plameneskii, Nadirashvilli, Gromov and Shubin, account for the contribution to the index due to a divisor of zeros and singularities. On the other hand, the Liouville theorems of Avellaneda, Lin, Li, Moser, Struwe, Kuchment and Pinchover provide the index of periodic elliptic equations on abelian coverings of compact manifolds with polynomial growth at infinity, i.e. in the presence of a "divisor" at infinity. A natural question is whether one can combine the Riemann–Roch and Liouville type results. This monograph shows that this can indeed be done, however the answers are more intricate than one might initially expect. Namely, the interaction between the finite divisor and the point at infinity is non-trivial. The text is targeted towards researchers in PDEs, geometric analysis, and mathematical physics.
Global analysis (Mathematics). --- Manifolds (Mathematics). --- Mathematical analysis. --- Analysis (Mathematics). --- Topology. --- Global Analysis and Analysis on Manifolds. --- Analysis. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- 517.1 Mathematical analysis --- Mathematical analysis --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential equations, Elliptic. --- Riemann-Roch theorems. --- Theorems, Riemann-Roch --- Algebraic functions --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial
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Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.
Mathematics --- Mathématiques --- Congresses --- Congrès --- 51 --- -Math --- Science --- Congresses. --- -Mathematics --- 51 Mathematics --- -51 Mathematics --- Math --- Mathématiques --- Congrès --- A priori estimate. --- Addition. --- Additive group. --- Affine space. --- Algebraic geometry. --- Algebraic group. --- Atiyah–Singer index theorem. --- Bernoulli number. --- Boundary value problem. --- Bounded operator. --- C*-algebra. --- Canonical transformation. --- Cauchy problem. --- Characteristic class. --- Clifford algebra. --- Coefficient. --- Cohomology. --- Commutative property. --- Commutative ring. --- Complex manifold. --- Complex number. --- Complex vector bundle. --- Dedekind sum. --- Degenerate bilinear form. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential operator. --- Dimension (vector space). --- Ellipse. --- Elliptic operator. --- Equation. --- Euler characteristic. --- Euler number. --- Existence theorem. --- Exotic sphere. --- Finite difference. --- Finite group. --- Fourier integral operator. --- Fourier transform. --- Fourier. --- Fredholm operator. --- Hardy space. --- Hilbert space. --- Holomorphic vector bundle. --- Homogeneous coordinates. --- Homomorphism. --- Homotopy. --- Hyperbolic partial differential equation. --- Identity component. --- Integer. --- Integral transform. --- Isomorphism class. --- John Milnor. --- K-theory. --- Lebesgue measure. --- Line bundle. --- Local ring. --- Mathematics. --- Maximal ideal. --- Modular form. --- Module (mathematics). --- Monoid. --- Normal bundle. --- Number theory. --- Open set. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Piecewise linear manifold. --- Poisson bracket. --- Polynomial ring. --- Polynomial. --- Prime number. --- Principal part. --- Projective space. --- Pseudo-differential operator. --- Quadratic form. --- Rational variety. --- Real number. --- Reciprocity law. --- Resolution of singularities. --- Riemann–Roch theorem. --- Shift operator. --- Simply connected space. --- Special case. --- Square-integrable function. --- Subalgebra. --- Submanifold. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Topology. --- Variable (mathematics). --- Vector bundle. --- Vector space. --- Winding number. --- Mathematics - Congresses
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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
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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 quoted 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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