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This book is an introduction to singularities for graduate students and researchers. Algebraic geometry is said to have originated in the seventeenth century with the famous workDiscours 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. First, mostly non-singular varieties were studied. 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. 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-dimensional 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. In the second edition, brief descriptions about recent remarkable developments of the researches are added as the last chapter.

Geometry, algebraic. --- Algebra. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Algebraic geometry --- Geometry --- Singularities (Mathematics) --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Rings (Algebra) --- Algebra --- Algebraic rings --- Ring theory --- Algebraic fields

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This book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.

Algebra. --- Geometry, algebraic. --- Differential equations, partial. --- Commutative Rings and Algebras. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Partial Differential Equations. --- Algebraic geometry --- Geometry --- Mathematics --- Mathematical analysis --- Partial differential equations --- Commutative algebra. --- Algebra --- Commutative rings. --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Partial differential equations. --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra)

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This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises). In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading. A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.

Mathematics. --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Algebra. --- Field theory (Physics). --- Group theory. --- Number theory. --- Field Theory and Polynomials. --- Number Theory. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Group Theory and Generalizations. --- Geometry, algebraic. --- Number study --- Numbers, Theory of --- Algebra --- Groups, Theory of --- Substitutions (Mathematics) --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Algebraic geometry --- Geometry --- Galois theory --- Rings (Algebra) --- Algebraic rings --- Ring theory --- Algebraic fields

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This carefully written textbook provides an accessible introduction to the representation theory of algebras, including representations of quivers. The book starts with basic topics on algebras and modules, covering fundamental results such as the Jordan-Hölder theorem on composition series, the Artin-Wedderburn theorem on the structure of semisimple algebras and the Krull-Schmidt theorem on indecomposable modules. The authors then go on to study representations of quivers in detail, leading to a complete proof of Gabriel's celebrated theorem characterizing the representation type of quivers in terms of Dynkin diagrams. Requiring only introductory courses on linear algebra and groups, rings and fields, this textbook is aimed at undergraduate students. With numerous examples illustrating abstract concepts, and including more than 200 exercises (with solutions to about a third of them), the book provides an example-driven introduction suitable for self-study and use alongside lecture courses.

Algebra. --- Group theory. --- Associative Rings and Algebras. --- Commutative Rings and Algebras. --- Group Theory and Generalizations. --- Category Theory, Homological Algebra. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Mathematics --- Mathematical analysis --- Representations of algebras. --- Associative rings. --- Rings (Algebra). --- Commutative algebra. --- Commutative rings. --- Category theory (Mathematics). --- Homological algebra. --- Rings (Algebra) --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic rings --- Ring theory --- Algebraic fields

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This volume resulted from presentations given at the international “Brainstorming Workshop on New Developments in Discrete Mechanics, Geometric Integration and Lie–Butcher Series”, that took place at the Instituto de Ciencias Matemáticas (ICMAT) in Madrid, Spain. It combines overview and research articles on recent and ongoing developments, as well as new research directions. Why geometric numerical integration? In their article of the same title Arieh Iserles and Reinout Quispel, two renowned experts in numerical analysis of differential equations, provide a compelling answer to this question. After this introductory chapter a collection of high-quality research articles aim at exploring recent and ongoing developments, as well as new research directions in the areas of geometric integration methods for differential equations, nonlinear systems interconnections, and discrete mechanics. One of the highlights is the unfolding of modern algebraic and combinatorial structures common to those topics, which give rise to fruitful interactions between theoretical as well as applied and computational perspectives. The volume is aimed at researchers and graduate students interested in theoretical and computational problems in geometric integration theory, nonlinear control theory, and discrete mechanics. .

Lie algebras --- Lie groups --- Numerical analysis. --- Global differential geometry. --- Systems theory. --- Topological Groups. --- Algebra. --- Numerical Analysis. --- Differential Geometry. --- Systems Theory, Control. --- Topological Groups, Lie Groups. --- Non-associative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Groups, Topological --- Continuous groups --- Geometry, Differential --- Differential geometry. --- System theory. --- Topological groups. --- Lie groups. --- Nonassociative rings. --- Rings (Algebra). --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Groups, Lie --- Symmetric spaces --- Topological groups --- Systems, Theory of --- Systems science --- Science --- Differential geometry --- Philosophy