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The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra-namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches, one by Ahmed Abbes and Michel Gros, the other by Takeshi Tsuji. The authors mainly focus on generalized representations of the fundamental group that are p-adically close to the trivial representation.The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The authors show the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the volume contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored. The authors present a new approach based on a generalization of P. Deligne's covanishing topos.
Group theory. --- p-adic groups. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Groups, p-adic --- Group theory --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Dolbeault generalized representation. --- Dolbeault module. --- Dolbeault representation. --- Faltings cohomology. --- Faltings extension. --- Faltings ringed topos. --- Faltings site. --- Faltings topos. --- Galois cohomology. --- Gerd Faltings. --- Higgs bundle. --- Higgs bundles. --- Higgs crystals. --- Higgs envelopes. --- Higgs isocrystal. --- HiggsДate algebra. --- HodgeДate representation. --- HodgeДate structure. --- HodgeДate theory. --- Hyodo's theory. --- Koszul complex. --- additive categories. --- adic module. --- almost faithfully flat descent. --- almost faithfully flat module. --- almost flat module. --- almost isomorphism. --- almost tale covering. --- almost tale extension. --- cohomology. --- covanishing topos. --- crystalline-type topos. --- deformation. --- discrete AЇ-module. --- finite tale site. --- fundamental group. --- generalized covanishing topos. --- generalized representation. --- inverse limit. --- linear algebra. --- locally irreducible scheme. --- morphism. --- overconvergence. --- p-adic Hodge theory. --- p-adic Simpson correspondence. --- p-adic field. --- period ring. --- ringed covanishing topos. --- ringed total topos. --- small generalized representation. --- small representation. --- solvable Higgs module. --- tale cohomology. --- tale fundamental group. --- torsor.
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Recent advancements in computer technology have allowed for designers to have direct control over the production process through the help of computer-based tools, creating the possibility of a completely integrated design and manufacturing process. Over the last few decades, "artificial intelligence" (AI) techniques, such as machine learing and deep learning, have been topics of interest in computer-based design and manufacturing research fields. However, efforts to develop computer-based AI to handle big data in design and manufacturing have not yet been successful. This Special Issue aims to collect novel articles covering artificial intelligence-based design, manufacturing, and data-driven design. It will comprise academics, researchers, mechanical, manufacturing, production and industrial engineers and professionals related to engineering design and manufacturing.
History of engineering & technology --- product service system (PSS) --- availability --- field repair kit --- gradient-based algorithm --- robust genetic algorithm --- warpage --- design of experiments --- fringe pattern --- birefringence --- automatic design --- intelligent optimization method --- CFD --- fluid machinery --- pumps --- multi-function console --- data-driven design --- mismatch equation --- anthropometric measures --- algorithmic approach --- optimal design --- stretchable antenna-based strain sensor --- structural optimization --- structural health monitoring --- dimension reduction --- entropy-based correlation coefficient --- multidisciplinary design and analysis --- uncertainty-integrated and machine learning-based surrogate modeling --- additive manufacturing --- complexity --- modular design --- part consolidation --- product recovery --- product image design --- Kansei Engineering --- integrated decision system --- qualitative decision model --- quantitative decision model --- train seats --- measurement-assisted assembly --- coordination space --- assemblability --- small displacement torsor --- Kriging --- lower confidence bounding --- entropy theory --- product design --- simulation-based design optimization --- convolutional neural network --- object detection --- piping and instrument diagram --- unsupervised learning --- n/a
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Recent advancements in computer technology have allowed for designers to have direct control over the production process through the help of computer-based tools, creating the possibility of a completely integrated design and manufacturing process. Over the last few decades, "artificial intelligence" (AI) techniques, such as machine learing and deep learning, have been topics of interest in computer-based design and manufacturing research fields. However, efforts to develop computer-based AI to handle big data in design and manufacturing have not yet been successful. This Special Issue aims to collect novel articles covering artificial intelligence-based design, manufacturing, and data-driven design. It will comprise academics, researchers, mechanical, manufacturing, production and industrial engineers and professionals related to engineering design and manufacturing.
product service system (PSS) --- availability --- field repair kit --- gradient-based algorithm --- robust genetic algorithm --- warpage --- design of experiments --- fringe pattern --- birefringence --- automatic design --- intelligent optimization method --- CFD --- fluid machinery --- pumps --- multi-function console --- data-driven design --- mismatch equation --- anthropometric measures --- algorithmic approach --- optimal design --- stretchable antenna-based strain sensor --- structural optimization --- structural health monitoring --- dimension reduction --- entropy-based correlation coefficient --- multidisciplinary design and analysis --- uncertainty-integrated and machine learning-based surrogate modeling --- additive manufacturing --- complexity --- modular design --- part consolidation --- product recovery --- product image design --- Kansei Engineering --- integrated decision system --- qualitative decision model --- quantitative decision model --- train seats --- measurement-assisted assembly --- coordination space --- assemblability --- small displacement torsor --- Kriging --- lower confidence bounding --- entropy theory --- product design --- simulation-based design optimization --- convolutional neural network --- object detection --- piping and instrument diagram --- unsupervised learning --- n/a
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Recent advancements in computer technology have allowed for designers to have direct control over the production process through the help of computer-based tools, creating the possibility of a completely integrated design and manufacturing process. Over the last few decades, "artificial intelligence" (AI) techniques, such as machine learing and deep learning, have been topics of interest in computer-based design and manufacturing research fields. However, efforts to develop computer-based AI to handle big data in design and manufacturing have not yet been successful. This Special Issue aims to collect novel articles covering artificial intelligence-based design, manufacturing, and data-driven design. It will comprise academics, researchers, mechanical, manufacturing, production and industrial engineers and professionals related to engineering design and manufacturing.
History of engineering & technology --- product service system (PSS) --- availability --- field repair kit --- gradient-based algorithm --- robust genetic algorithm --- warpage --- design of experiments --- fringe pattern --- birefringence --- automatic design --- intelligent optimization method --- CFD --- fluid machinery --- pumps --- multi-function console --- data-driven design --- mismatch equation --- anthropometric measures --- algorithmic approach --- optimal design --- stretchable antenna-based strain sensor --- structural optimization --- structural health monitoring --- dimension reduction --- entropy-based correlation coefficient --- multidisciplinary design and analysis --- uncertainty-integrated and machine learning-based surrogate modeling --- additive manufacturing --- complexity --- modular design --- part consolidation --- product recovery --- product image design --- Kansei Engineering --- integrated decision system --- qualitative decision model --- quantitative decision model --- train seats --- measurement-assisted assembly --- coordination space --- assemblability --- small displacement torsor --- Kriging --- lower confidence bounding --- entropy theory --- product design --- simulation-based design optimization --- convolutional neural network --- object detection --- piping and instrument diagram --- unsupervised learning --- product service system (PSS) --- availability --- field repair kit --- gradient-based algorithm --- robust genetic algorithm --- warpage --- design of experiments --- fringe pattern --- birefringence --- automatic design --- intelligent optimization method --- CFD --- fluid machinery --- pumps --- multi-function console --- data-driven design --- mismatch equation --- anthropometric measures --- algorithmic approach --- optimal design --- stretchable antenna-based strain sensor --- structural optimization --- structural health monitoring --- dimension reduction --- entropy-based correlation coefficient --- multidisciplinary design and analysis --- uncertainty-integrated and machine learning-based surrogate modeling --- additive manufacturing --- complexity --- modular design --- part consolidation --- product recovery --- product image design --- Kansei Engineering --- integrated decision system --- qualitative decision model --- quantitative decision model --- train seats --- measurement-assisted assembly --- coordination space --- assemblability --- small displacement torsor --- Kriging --- lower confidence bounding --- entropy theory --- product design --- simulation-based design optimization --- convolutional neural network --- object detection --- piping and instrument diagram --- unsupervised learning
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In this monograph p-adic period domains are associated to arbitrary reductive groups. Using the concept of rigid-analytic period maps the relation of p-adic period domains to moduli space of p-divisible groups is investigated. In addition, non-archimedean uniformization theorems for general Shimura varieties are established. The exposition includes background material on Grothendieck's "mysterious functor" (Fontaine theory), on moduli problems of p-divisible groups, on rigid analytic spaces, and on the theory of Shimura varieties, as well as an exposition of some aspects of Drinfelds' original construction. In addition, the material is illustrated throughout the book with numerous examples.
p-adic groups --- p-divisible groups --- Moduli theory --- 512.7 --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Groups, p-divisible --- Group schemes (Mathematics) --- Groups, p-adic --- Group theory --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- p-divisible groups. --- Moduli theory. --- p-adic groups. --- Abelian variety. --- Addition. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic number field. --- Algebraic space. --- Algebraically closed field. --- Artinian ring. --- Automorphism. --- Base change. --- Basis (linear algebra). --- Big O notation. --- Bilinear form. --- Canonical map. --- Cohomology. --- Cokernel. --- Commutative algebra. --- Commutative ring. --- Complex multiplication. --- Conjecture. --- Covering space. --- Degenerate bilinear form. --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Duality (mathematics). --- Elementary function. --- Epimorphism. --- Equation. --- Existential quantification. --- Fiber bundle. --- Field of fractions. --- Finite field. --- Formal scheme. --- Functor. --- Galois group. --- General linear group. --- Geometric invariant theory. --- Hensel's lemma. --- Homomorphism. --- Initial and terminal objects. --- Inner automorphism. --- Integral domain. --- Irreducible component. --- Isogeny. --- Isomorphism class. --- Linear algebra. --- Linear algebraic group. --- Local ring. --- Local system. --- Mathematical induction. --- Maximal ideal. --- Maximal torus. --- Module (mathematics). --- Moduli space. --- Monomorphism. --- Morita equivalence. --- Morphism. --- Multiplicative group. --- Noetherian ring. --- Open set. --- Orthogonal basis. --- Orthogonal complement. --- Orthonormal basis. --- P-adic number. --- Parity (mathematics). --- Period mapping. --- Prime element. --- Prime number. --- Projective line. --- Projective space. --- Quaternion algebra. --- Reductive group. --- Residue field. --- Rigid analytic space. --- Semisimple algebra. --- Sheaf (mathematics). --- Shimura variety. --- Special case. --- Subalgebra. --- Subgroup. --- Subset. --- Summation. --- Supersingular elliptic curve. --- Support (mathematics). --- Surjective function. --- Symmetric bilinear form. --- Symmetric space. --- Tate module. --- Tensor algebra. --- Tensor product. --- Theorem. --- Topological ring. --- Topology. --- Torsor (algebraic geometry). --- Uniformization theorem. --- Uniformization. --- Unitary group. --- Weil group. --- Zariski topology.
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Arithmetic and Geometry presents highlights of recent work in arithmetic algebraic geometry by some of the world's leading mathematicians. Together, these 2016 lectures-which were delivered in celebration of the tenth anniversary of the annual summer workshops in Alpbach, Austria-provide an introduction to high-level research on three topics: Shimura varieties, hyperelliptic continued fractions and generalized Jacobians, and Faltings height and L-functions. The book consists of notes, written by young researchers, on three sets of lectures or minicourses given at Alpbach.The first course, taught by Peter Scholze, contains his recent results dealing with the local Langlands conjecture. The fundamental question is whether for a given datum there exists a so-called local Shimura variety. In some cases, they exist in the category of rigid analytic spaces; in others, one has to use Scholze's perfectoid spaces.The second course, taught by Umberto Zannier, addresses the famous Pell equation-not in the classical setting but rather with the so-called polynomial Pell equation, where the integers are replaced by polynomials in one variable with complex coefficients, which leads to the study of hyperelliptic continued fractions and generalized Jacobians.The third course, taught by Shou-Wu Zhang, originates in the Chowla-Selberg formula, which was taken up by Gross and Zagier to relate values of the L-function for elliptic curves with the height of Heegner points on the curves. Zhang, X. Yuan, and Wei Zhang prove the Gross-Zagier formula on Shimura curves and verify the Colmez conjecture on average.
Arithmetical algebraic geometry. --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Abelian variety. --- Algebraic geometry. --- Algebraic independence. --- Algebraic space. --- Analytic number theory. --- Arbitrarily large. --- Automorphic form. --- Automorphism. --- Base change. --- Big O notation. --- Class number formula. --- Cohomology. --- Complex multiplication. --- Computation. --- Conjecture. --- Conjugacy class. --- Continued fraction. --- Cusp form. --- Diagram (category theory). --- Dimension. --- Diophantine equation. --- Diophantine geometry. --- Discriminant. --- Divisible group. --- Double coset. --- Eisenstein series. --- Endomorphism. --- Equation. --- Existential quantification. --- Exponential map (Riemannian geometry). --- Fiber bundle. --- Floor and ceiling functions. --- Formal group. --- Formal power series. --- Formal scheme. --- Fundamental group. --- Geometric Langlands correspondence. --- Geometry. --- Heegner point. --- Hodge structure. --- Hodge theory. --- Homomorphism. --- I0. --- Integer. --- Intersection number. --- Irreducible component. --- Isogeny. --- Isomorphism class. --- Jacobian variety. --- L-function. --- Langlands dual group. --- Laurent series. --- Linear combination. --- Local system. --- Logarithmic derivative. --- Logarithmic form. --- Mathematics. --- Modular form. --- Moduli space. --- Monotonic function. --- Natural topology. --- P-adic analysis. --- P-adic number. --- Pell's equation. --- Perverse sheaf. --- Polylogarithm. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Prime number. --- Projective space. --- Quaternion algebra. --- Rational point. --- Real number. --- Reductive group. --- Rigid analytic space. --- Roth's theorem. --- Series expansion. --- Shafarevich conjecture. --- Sheaf (mathematics). --- Shimura variety. --- Siegel zero. --- Special case. --- Stack (mathematics). --- Subset. --- Summation. --- Szpiro's conjecture. --- Tate conjecture. --- Tate module. --- Taylor series. --- Theorem. --- Theta function. --- Topological ring. --- Topology. --- Torsor (algebraic geometry). --- Upper and lower bounds. --- Vector bundle. --- Weil group. --- Witt vector. --- Zariski topology.
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This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential equations in one complex variable with polynomial coefficients, and the theory of one parameter families of exponential sums over finite fields. After reviewing some results from representation theory, the book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G). The final part of the book is devoted to comparison theorems relating G and G of suitably "corresponding" situations, which provide a systematic explanation of the remarkable "coincidences" found "by hand" in the hypergeometric case.
Exponential sums. --- Differential equations. --- Adjoint representation. --- Algebraic geometry. --- Algebraic integer. --- Algebraically closed field. --- Automorphism. --- Base change. --- Bernard Dwork. --- Big O notation. --- Bijection. --- Calculation. --- Characteristic polynomial. --- Codimension. --- Coefficient. --- Cohomology. --- Comparison theorem. --- Complex manifold. --- Conjugacy class. --- Connected component (graph theory). --- Convolution. --- Determinant. --- Diagram (category theory). --- Differential Galois theory. --- Differential equation. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Divisor. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Equation. --- Euler characteristic. --- Existential quantification. --- Exponential sum. --- Fiber bundle. --- Field of fractions. --- Finite field. --- Formal power series. --- Fourier transform. --- Fundamental group. --- Fundamental representation. --- Galois extension. --- Galois group. --- Gauss sum. --- Generic point. --- Group theory. --- Homomorphism. --- Hypergeometric function. --- Identity component. --- Identity element. --- Integer. --- Irreducibility (mathematics). --- Irreducible representation. --- Isogeny. --- Isomorphism class. --- L-function. --- Laurent polynomial. --- Lie algebra. --- Logarithm. --- Mathematical induction. --- Matrix coefficient. --- Maximal compact subgroup. --- Maximal torus. --- Mellin transform. --- Monic polynomial. --- Monodromy theorem. --- Monodromy. --- Monomial. --- Natural number. --- Normal subgroup. --- P-adic number. --- Permutation. --- Polynomial. --- Prime number. --- Pullback. --- Quotient group. --- Reductive group. --- Regular singular point. --- Representation theory. --- Ring homomorphism. --- Root of unity. --- Scientific notation. --- Set (mathematics). --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Summation. --- Surjective function. --- Symmetric group. --- Tensor product. --- Theorem. --- Theory. --- Three-dimensional space (mathematics). --- Torsor (algebraic geometry). --- Trichotomy (mathematics). --- Unitarian trick. --- Unitary group. --- Variable (mathematics).
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One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series.Originally published in 1980.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Ordered algebraic structures --- 512.73 --- 512.66 --- Geometry, Algebraic --- Homology theory --- Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebraic geometry --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- Geometry, Algebraic. --- Homology theory. --- Sheaf theory. --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Abelian category. --- Abelian group. --- Adjoint functors. --- Affine variety. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cycle. --- Algebraic equation. --- Algebraic space. --- Algebraically closed field. --- Artinian. --- Automorphism. --- Base change. --- Brauer group. --- CW complex. --- Cardinal number. --- Category of sets. --- Central simple algebra. --- Chow's lemma. --- Closed immersion. --- Codimension. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Complex number. --- Dedekind domain. --- Derived category. --- Diagram (category theory). --- Direct limit. --- Discrete valuation ring. --- Divisor. --- Epimorphism. --- Equivalence class. --- Existential quantification. --- Fibration. --- Field of fractions. --- Fine topology (potential theory). --- Finite field. --- Finite morphism. --- Flat morphism. --- Functor. --- Fundamental class. --- Fundamental group. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Generic point. --- Group scheme. --- Gysin sequence. --- Henselian ring. --- Identity element. --- Inclusion map. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Invertible sheaf. --- Isomorphism class. --- Lefschetz pencil. --- Local ring. --- Maximal ideal. --- Module (mathematics). --- Morphism of schemes. --- Morphism. --- Noetherian. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime ideal. --- Prime number. --- Principal homogeneous space. --- Profinite group. --- Projection (mathematics). --- Projective variety. --- Quasi-compact morphism. --- Residue field. --- Riemann surface. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Stein factorization. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Surjective function. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Torsion sheaf. --- Torsor (algebraic geometry). --- Vector bundle. --- Weil conjecture. --- Yoneda lemma. --- Zariski topology. --- Zariski's main theorem. --- Geometrie algebrique --- Cohomologie
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