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In this thesis we present the results of four years of research on some aspects of p-adic geometry. A first result is on definable Lipschitz extensions of p-adic functions, a second result is on differentiation in P-minimal structures. Both results will appear in the form of an article in a peer-reviewed mathematical journal.Firstly, we prove a definable version of Kirszbrauns theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, let K be a finite field extension of the field of p-adic numbers, then we prove that every definable family of λ-Lipschitz functions on a subset of K extends to a definable family of λ-Lipschitz functions on K.Secondly, we prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.The basic facts of p-adic fields are reviewed in the first chapter of this thesis. Model theory and its applications are reviewed in the second chapter. The third chapter contains the new results on definable Lipschitz extensions of p-adic functions, and the forth chapter those on differentiation in P-minimal structures. Finally, the fifth chapter contains a discussion of the main results in this thesis, together with a look at future research.
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There is a well-known correspondence between the objects of algebra and geometry: a space gives rise to a function algebra; a vector bundle over the space corresponds to a projective module over this algebra; cohomology can be read off the de Rham complex; and so on. In this book Yuri Manin addresses a variety of instances in which the application of commutative algebra cannot be used to describe geometric objects, emphasizing the recent upsurge of activity in studying noncommutative rings as if they were function rings on "noncommutative spaces." Manin begins by summarizing and giving examples of some of the ideas that led to the new concepts of noncommutative geometry, such as Connes' noncommutative de Rham complex, supergeometry, and quantum groups. He then discusses supersymmetric algebraic curves that arose in connection with superstring theory; examines superhomogeneous spaces, their Schubert cells, and superanalogues of Weyl groups; and provides an introduction to quantum groups. This book is intended for mathematicians and physicists with some background in Lie groups and complex geometry.Originally published in 1991.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.
Geometry, Algebraic --- Noncommutative rings --- Geometry, Algebraic. --- Noncommutative rings. --- Non-commutative rings --- Associative rings --- Algebraic geometry --- Geometry
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Commutative algebra, combinatorics, and algebraic geometry are thriving areas of mathematical research with a rich history of interaction. Connections Between Algebra, Combinatorics, and Geometry contains lecture notes, along with exercises and solutions, from the Workshop on Connections Between Algebra and Geometry held at the University of Regina from May 29-June 1, 2012. It also contains research and survey papers from academics invited to participate in the companion Special Session on Interactions Between Algebraic Geometry and Commutative Algebra, which was part of the CMS Summer Meeting at the University of Regina held June 2–3, 2012, and the meeting Further Connections Between Algebra and Geometry, which was held at the North Dakota State University, February 23, 2013. This volume highlights three mini-courses in the areas of commutative algebra and algebraic geometry: differential graded commutative algebra, secant varieties, and fat points and symbolic powers. It will serve as a useful resource for graduate students and researchers who wish to expand their knowledge of commutative algebra, algebraic geometry, combinatorics, and the intricacies of their intersection. .
Commutative algebra. --- Combinatorial analysis. --- Geometry. --- Algebra --- Mathematics --- Euclid's Elements --- Combinatorics --- Mathematical analysis --- Algebra. --- Geometry, algebraic. --- Commutative Rings and Algebras. --- Algebraic Geometry. --- Algebraic geometry --- Geometry --- Commutative rings. --- Algebraic geometry. --- Rings (Algebra)
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Combinatorics and Algebraic Geometry have enjoyed a fruitful interplay since the nineteenth century. Classical interactions include invariant theory, theta functions, and enumerative geometry. The aim of this volume is to introduce recent developments in combinatorial algebraic geometry and to approach algebraic geometry with a view towards applications, such as tensor calculus and algebraic statistics. A common theme is the study of algebraic varieties endowed with a rich combinatorial structure. Relevant techniques include polyhedral geometry, free resolutions, multilinear algebra, projective duality and compactifications.
Geometry, Algebraic --- Combinatorial geometry --- Combinatorial analysis --- Geometric combinatorics --- Geometrical combinatorics --- Discrete geometry --- Geometry, algebraic. --- Combinatorics. --- Algebra. --- Algebraic Geometry. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Combinatorics --- Algebra --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Rings (Algebra)
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The content in Chapter 1-3 is a fairly standard one-semester course on local rings with the goal to reach the fact that a regular local ring is a unique factorization domain. The homological machinery is also supported by Cohen-Macaulay rings and depth. In Chapters 4-6 the methods of injective modules, Matlis duality and local cohomology are discussed. Chapters 7-9 are not so standard and introduce the reader to the generalizations of modules to complexes of modules. Some of Professor Iversen's results are given in Chapter 9. Chapter 10 is about Serre's intersection conjecture. The graded case
Local rings. --- Injective modules (Algebra) --- Intersection homology theory. --- Homology theory --- Modules (Algebra) --- Rings, Local --- Commutative rings --- Injective modules (Algebra).
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This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions. Specific topics include: · Homological dimensions of Prüfer-like rings · Quasi complete rings · Total graphs of rings · Properties of prime ideals over various rings · Bases for integer-valued polynomials · Boolean subrings · The portable property of domains · Probabilistic topics in Intn(D) · Closure operations in Zariski-Riemann spaces of valuation domains · Stability of domains · Non-Noetherian grade · Homotopy in integer-valued polynomials · Localizations of global properties of rings · Topics in integral closure · Monoids and submonoids of domains The book includes twenty articles written by many of the most prominent researchers in the field. Most contributions are authored by attendees of the conference in commutative algebra held at the Graz University of Technology in December 2012. There is also a small collection of invited articles authored by those who did not attend the conference. Following the model of the Graz conference, the volume contains a number of comprehensive survey articles along with related research articles featuring recent results that have not yet been published elsewhere.
Commutative algebra. --- Mathematics. --- Commutative rings. --- Algebraic topology. --- Algebra --- Topology --- Rings (Algebra) --- Math --- Science --- Algebra. --- Commutative Rings and Algebras. --- Category Theory, Homological Algebra. --- Algebraic Topology. --- Mathematics --- Mathematical analysis --- Category theory (Mathematics). --- Homological algebra. --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory
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This volume is a sequel to “Manis Valuation and Prüfer Extensions I,” LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prüfer-Manis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter’s work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called “Kronecker extensions,” where PM valuations are put to use in arbitrary commutative ring extensions in a way that ultimately goes back to the work of Leopold Kronecker.
Commutative algebra. --- Commutative rings. --- Prüfer rings. --- Commutative semihereditary domains --- Commutative semihereditary entire rings --- Domains, Commutative semihereditary --- Domains, Prüfer --- Entire rings, Commutative semihereditary --- Prüfer domains --- Prüfer's domains --- Prüfer's rings --- Semihereditary domains, Commutative --- Semihereditary entire rings, Commutative --- Rings (Algebra) --- Algebra --- Algebra. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis
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Investigating the correspondence between systems of partial differential equations and their analytic solutions using a formal approach, this monograph presents algorithms to determine the set of analytic solutions of such a system and conversely to find differential equations whose set of solutions coincides with a given parametrized set of analytic functions. After giving a detailed introduction to Janet bases and Thomas decomposition, the problem of finding an implicit description of certain sets of analytic functions in terms of differential equations is addressed. Effective methods of varying generality are developed to solve the differential elimination problems that arise in this context. In particular, it is demonstrated how the symbolic solution of partial differential equations profits from the study of the implicitization problem. For instance, certain families of exact solutions of the Navier-Stokes equations can be computed.
Mathematics. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Algebra. --- Field theory (Physics). --- Partial differential equations. --- Field Theory and Polynomials. --- Commutative Rings and Algebras. --- Associative Rings and Algebras. --- Partial Differential Equations. --- Differential equations, partial. --- Partial differential equations --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Differential equations, Partial. --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Algebra
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This book is an introduction to singularities for graduate students and researchers. It is said that algebraic geometry originated in the seventeenth century with the famous work Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences by Descartes. In that book he introduced coordinates to the study of geometry. After its publication, research on algebraic varieties developed steadily. Many beautiful results emerged in mathematicians’ works. Most of them were about non-singular varieties. Singularities were considered “bad” objects that interfered with knowledge of the structure of an algebraic variety. In the past three decades, however, it has become clear that singularities are necessary for us to have a good description of the framework of varieties. For example, it is impossible to formulate minimal model theory for higher-dimensional cases without singularities. Another example is that the moduli spaces of varieties have natural compactification, the boundaries of which correspond to singular varieties. A remarkable fact is that the study of singularities is developing and people are beginning to see that singularities are interesting and can be handled by human beings. This book is a handy introduction to singularities for anyone interested in singularities. The focus is on an isolated singularity in an algebraic variety. After preparation of varieties, sheaves, and homological algebra, some known results about 2-dim ensional isolated singularities are introduced. Then a classification of higher-dimensional isolated singularities is shown according to plurigenera and the behavior of singularities under a deformation is studied.
Mathematics. --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Singularities (Mathematics) --- Geometry, Algebraic. --- Algebra. --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Math --- Science --- Geometry, Algebraic --- Geometry, algebraic. --- Algebra --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Algebraic fields
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The book is primarily intended as a textbook on modern algebra for undergraduate mathematics students. It is also useful for those who are interested in supplementary reading at a higher level. The text is designed in such a way that it encourages independent thinking and motivates students towards further study. The book covers all major topics in group, ring, vector space and module theory that are usually contained in a standard modern algebra text. In addition, it studies semigroup, group action, Hopf's group, topological groups and Lie groups with their actions, applications of ring theory to algebraic geometry, and defines Zariski topology, as well as applications of module theory to structure theory of rings and homological algebra. Algebraic aspects of classical number theory and algebraic number theory are also discussed with an eye to developing modern cryptography. Topics on applications to algebraic topology, category theory, algebraic geometry, algebraic number theory, cryptography and theoretical computer science interlink the subject with different areas. Each chapter discusses individual topics, starting from the basics, with the help of illustrative examples. This comprehensive text with a broad variety of concepts, applications, examples, exercises and historical notes represents a valuable and unique resource. .
Number theory. --- Algebra. --- Mathematics. --- Group theory. --- Commutative Rings and Algebras. --- Group Theory and Generalizations. --- Category Theory, Homological Algebra. --- Applications of Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Number study --- Numbers, Theory of --- Groups, Theory of --- Substitutions (Mathematics) --- Math --- Category theory (Mathematics). --- Homological algebra. --- Commutative algebra. --- Commutative rings. --- Applied mathematics. --- Engineering mathematics. --- Number Theory. --- Mathematical analysis --- Science --- Engineering --- Engineering analysis --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Rings (Algebra)
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