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Kiyosi Ito, the founder of stochastic calculus, is one of the few central figures of the twentieth century mathematics who reshaped the mathematical world. Today stochastic calculus is a central research field with applications in several other mathematical disciplines, for example physics, engineering, biology, economics and finance. The Abel Symposium 2005 was organized as a tribute to the work of Kiyosi Ito on the occasion of his 90th birthday. Distinguished researchers from all over the world were invited to present the newest developments within the exciting and fast growing field of stochastic analysis. The present volume combines both papers from the invited speakers and contributions by the presenting lecturers. A special feature is the Memoirs that Kiyoshi Ito wrote for this occasion. These are valuable pages for both young and established researchers in the field.

Stochastic analysis --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Distribution (Probability theory. --- Mathematics. --- Global analysis (Mathematics). --- Mathematical statistics. --- Finance. --- Probability Theory and Stochastic Processes. --- Applications of Mathematics. --- Real Functions. --- Analysis. --- Statistical Theory and Methods. --- Quantitative Finance. --- Funding --- Funds --- Economics --- Currency question --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistical methods --- Probabilities. --- Applied mathematics. --- Engineering mathematics. --- Functions of real variables. --- Analysis (Mathematics). --- Statistics . --- Economics, Mathematical . --- Real variables --- Engineering --- Engineering analysis --- Probability --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Mathematical economics --- Econometrics --- Statistical analysis --- Statistical data --- Statistical science --- Methodology --- Itō, Kiyosi, --- Itō, K. --- Ito, Kiesi, --- Itō, Kiyoshi, --- 伊藤淸, --- 伊藤清,

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The theme of this symposium was operator algebras in a wide sense. In the last 40 years operator algebras has developed from a rather special dis- pline within functional analysis to become a central ?eld in mathematics often described as “non-commutative geometry” (see for example the book “Non-Commutative Geometry” by the Fields medalist Alain Connes). It has branched out in several sub-disciplines and made contact with other subjects like for example mathematical physics, algebraic topology, geometry, dyn- ical systems, knot theory, ergodic theory, wavelets, representations of groups and quantum groups. Norway has a relatively strong group of researchers in the subject, which contributed to the award of the ?rst symposium in the series of Abel Symposia to this group. The contributions to this volume give a state-of-the-art account of some of these sub-disciplines and the variety of topics re?ect to some extent how the subject has branched out. We are happy that some of the top researchers in the ?eld were willing to contribute. The basic ?eld of operator algebras is classi?ed within mathematics as part of functional analysis. Functional analysis treats analysis on in?nite - mensional spaces by using topological concepts. A linear map between two such spaces is called an operator. Examples are di?erential and integral - erators. An important feature is that the composition of two operators is a non-commutative operation.

C*-algebras --- Von Neumann algebras --- Algebras, Von Neumann --- Algebras, W --- Neumann algebras --- Rings of operators --- W*-algebras --- Hilbert space --- Algebras, C star --- Algebras, W star --- C star algebras --- W star algebras --- Banach algebras --- Functional analysis. --- Algebra. --- Global analysis (Mathematics). --- Operator theory. --- K-theory. --- Differentiable dynamical systems. --- Functional Analysis. --- Analysis. --- Operator Theory. --- K-Theory. --- Dynamical Systems and Ergodic Theory. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Algebraic topology --- Homology theory --- Functional analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematics --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematical analysis. --- Analysis (Mathematics). --- Dynamics. --- Ergodic theory. --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- 517.1 Mathematical analysis --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics)

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The Abel Symposium 2009 "Combinatorial aspects of Commutative Algebra and Algebraic Geometry", held at Voss, Norway, featured talks by leading researchers in the field.  This is the proceedings of the Symposium, presenting contributions on syzygies, tropical geometry, Boij-Söderberg theory, Schubert calculus, and quiver varieties. The volume also includes an introductory survey on binomial ideals with applications to hypergeometric series, combinatorial games and chemical reactions.   The contributions pose interesting problems, and offer up-to-date research on some of the most active fields of commutative algebra and algebraic geometry with a combinatorial flavour.

Combinatorial analysis -- Congresses. --- Commutative algebra -- Congresses. --- Geometry, Algebraic -- Congresses. --- Commutative algebra --- Geometry, Algebraic --- Combinatorial analysis --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Mathematics. --- Algebraic geometry. --- Commutative algebra. --- Commutative rings. --- Combinatorics. --- Commutative Rings and Algebras. --- Algebraic Geometry. --- Algebra. --- Geometry, algebraic. --- Combinatorics --- Mathematical analysis --- Algebraic geometry --- Geometry --- Rings (Algebra)

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This book features survey and research papers from The Abel Symposium 2011, held in Balestrand, Norway 2011. It examines a very active research area that has had a growing influence and profound impact in many other areas of mathematics like commutative algebra, algebraic geometry, algebraic groups and combinatorics. This volume illustrates and extends such connections with algebraic geometry, cluster algebra theory, commutative algebra, dynamical systems and triangulated categories. In addition, it includes contributions on further developments in representation theory of quivers and algebras. Algebras, Quivers and Representations is targeted at researchers and graduate students in algebra, representation theory and triangulated categories.

Mathematics --- Category theory. Homological algebra --- Ordered algebraic structures --- Algebra --- Algebraic geometry --- Differential geometry. Global analysis --- Geometry --- Ergodic theory. Information theory --- algebra --- landmeetkunde --- differentiaal geometrie --- wiskunde --- geometrie --- informatietheorie --- Representations of algebras.