Discriminants Resultants and Multidimensional Determinants

Author: Israel M. Gelfand
Publisher: Springer Science & Business Media
ISBN: 0817647716
Format: PDF, Docs
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"This book revives and vastly expands the classical theory of resultants and discriminants. Most of the main new results of the book have been published earlier in more than a dozen joint papers of the authors. The book nicely complements these original papers with many examples illustrating both old and new results of the theory."—Mathematical Reviews

Introduction to Toric Varieties AM 131

Author: William Fulton
Publisher: Princeton University Press
ISBN: 1400882524
Format: PDF, Docs
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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.

Algorithmic Arithmetic Geometry and Coding Theory

Author: Stéphane Ballet
Publisher: American Mathematical Soc.
ISBN: 1470414619
Format: PDF, Kindle
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This volume contains the proceedings of the 14th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held June 3-7, 2013, at CIRM, Marseille, France. These international conferences, held every two years, have been a major event in the area of algorithmic and applied arithmetic geometry for more than 20 years. This volume contains 13 original research articles covering geometric error correcting codes, and algorithmic and explicit arithmetic geometry of curves and higher dimensional varieties. Tools used in these articles include classical algebraic geometry of curves, varieties and Jacobians, Suslin homology, Monsky-Washnitzer cohomology, and -functions of modular forms.

Topics on Real and Complex Singularities

Author: Satoshi Koike
Publisher: World Scientific
ISBN: 9814596051
Format: PDF, ePub, Mobi
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A phenomenon which appears in nature, or human behavior, can sometimes be explained by saying that a certain potential function is maximized, or minimized. For example, the Hamiltonian mechanics, soapy films, size of an atom, business management, etc. In mathematics, a point where a given function attains an extreme value is called a critical point, or a singular point. The purpose of singularity theory is to explore the properties of singular points of functions and mappings. This is a volume on the proceedings of the fourth Japanese–Australian Workshop on Real and Complex Singularities held in Kobe, Japan. It consists of 11 original articles on singularities. Readers will be introduced to some important new notions for characterizations of singularities and several interesting results are delivered. In addition, current approaches to classical topics and state-of-the-art effective computational methods of invariants of singularities are also presented. This volume will be useful not only to the singularity theory specialists but also to general mathematicians. Contents:On the CR Hamiltonian Flows and CR Yamabe Problem (T Akahori)An Example of the Reduction of a Single Ordinary Differential Equation to a System, and the Restricted Fuchsian Relation (K Ando)Fronts of Weighted Cones (T Fukui and M Hasegawa)Involutive Deformations of the Regular Part of a Normal Surface (A Harris and K Miyajima)Connected Components of Regular Fibers of Differentiable Maps (J T Hiratuka and O Saeki)The Reconstruction and Recognition Problems for Homogeneous Hypersurface Singularities (A V Isaev)Openings of Differentiable Map-Germs and Unfoldings (G Ishikawa)Non Concentration of Curvature near Singular Points of Two Variable Analytic Functions (S Koike, T-C Kuo and L Paunescu)Saito Free Divisors in Four Dimensional Affine Space and Reflection Groups of Rank Four (J Sekiguchi)Holonomic Systems of Differential Equations of Rank Two with Singularities along Saito Free Divisors of Simple Type (J Sekiguchi)Parametric Local Cohomology Classes and Tjurina Stratifications for μ-Constant Deformations of Quasi-Homogeneous Singularities (S Tajima) Readership: Mathematicians in singularity theory or in adjacent areas; advanced undergraduates and graduate students in mathematics; non-experts interested in singularity theory and its applications. Key Features:Contains applications of the singularity theory to other mathematical fieldsNew topics in singularity theory, e.g. the relationship between free divisors and holonomic systems, openings of differentiable map-germs, non-concentration of curvatureIncludes articles by prize-winning researchers like Kimio Miyajima and Osamu SaekiKeywords:Singularities;CR Structure;Deformation Theory;Free Divisor;Concentration of Curvature;Holonomic System;Front;Opening

Developments and Retrospectives in Lie Theory

Author: Geoffrey Mason
Publisher: Springer
ISBN: 3319098047
Format: PDF, Mobi
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The Lie Theory Workshop, founded by Joe Wolf (UC, Berkeley), has been running for over two decades. These workshops have been sponsored by the NSF, noting the talks have been seminal in describing new perspectives in the field covering broad areas of current research. At the beginning, the top universities in California and Utah hosted the meetings which continue to run on a quarterly basis. Experts in representation theory/Lie theory from various parts of the US, Europe, Asia (China, Japan, Singapore, Russia), Canada, and South and Central America were routinely invited to give talks at these meetings. Nowadays, the workshops are also hosted at universities in Louisiana, Virginia, and Oklahoma. The contributors to this volume have all participated in these Lie theory workshops and include in this volume expository articles which cover representation theory from the algebraic, geometric, analytic, and topological perspectives with also important connections to math physics. These survey articles, review and update the prominent seminal series of workshops in representation/Lie theory mentioned-above, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, number theory, and mathematical physics. Many of the contributors have had prominent roles in both the classical and modern developments of Lie theory and its applications.

An Introduction to Manifolds

Author: Loring W. Tu
Publisher: Springer Science & Business Media
ISBN: 1441974008
Format: PDF
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Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.

Algebra

Author: I.M. Gelfand
Publisher: Springer Science & Business Media
ISBN: 9780817636777
Format: PDF
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This book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation.

Introduction to Tropical Geometry

Author: Diane Maclagan
Publisher: American Mathematical Soc.
ISBN: 0821851985
Format: PDF, Mobi
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Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.

Algorithmic Number Theory

Author: Claus Fieker
Publisher: Springer Science & Business Media
ISBN: 3540438637
Format: PDF, Kindle
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Self-organized criticality (SOC) has become a magic word in various scientific disciplines; it provides a framework for understanding complexity and scale invariance in systems showing irregular fluctuations. In the first 10 years after Per Bak and his co-workers presented their seminal idea, more than 2000 papers on this topic appeared. Seismology has been a field in earth sciences where the SOC concept has already deepened the understanding, but there seem to be much more examples in earth sciences where applying the SOC concept may be fruitful. After introducing the reader into the basics of fractals, chaos and SOC, the book presents established and new applications of SOC in earth sciences, namely earthquakes, forest fires, landslides and drainage networks.

Regular Sequences and Resultants

Author: Gunter Scheja
Publisher: CRC Press
ISBN: 1439863652
Format: PDF, Docs
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This carefully prepared manuscript presents elimination theory in weighted projective spaces over arbitrary noetherian commutative base rings. Elimination theory is a classical topic in commutative algebra and algebraic geometry, and it has become of renewed importance recently in the context of applied and computational algebra. This monograph provides a valuable complement to sparse elimination theory in that it presents in careful detail the algebraic difficulties from working over general base rings. This is essential for applications in arithmetic geometry and many other places. Necessary tools concerning monoids of weights, generic polynomials and regular sequences are treated independently in the first part of the book. Many supplements added to each chapter provide extra details and insightful examples. Necessary tools concerning monoids of weights, generic polynomials and regular sequences are treated independently in the first part of the book. Many supplements added to each chapter provide extra details and insightful examples.