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Informatica --- Informatique --- Mathématiques appliquées --- Toegepaste wiskunde --- Discrete wiskunde --- Computerwiskunde

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Combinatorial commutative algebra is an active area of research with thriving connections to other fields of pure and applied mathematics. This book provides a self-contained introduction to the subject, with an emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determinantal rings. The eighteen chapters cover a broad spectrum of topics, ranging from homological invariants of monomial ideals and their polyhedral resolutions, to hands-on tools for studying algebraic varieties with group actions, such as toric varieties, flag varieties, quiver loci, and Hilbert schemes. Over 100 figures, 250 exercises, and pointers to the literature make this book appealing to both graduate students and researchers. Ezra Miller received his doctorate in 2000 from UC Berkeley. After two years at MIT in Cambridge and one year at MSRI in Berkeley, he is currently Assistant Professor at the University of Minnesota, Twin Cities. Miller was awarded an Alfred P. Sloan Dissertation Fellowship in 1999 and an NSF Postdoctoral Fellowship in 2000. Besides his mathematical interests, which include combinatorics, algebraic geometry, homological algebra, and polyhedral geometry, Miller is fond of music theory and composition, molecular biology, and ultimate frisbee. Bernd Sturmfels received doctoral degrees in 1987 from the University of Washington, Seattle and TU Darmstadt, Germany. After two postdoc years at the IMA in Minneapolis and RISC-Linz in Austria, he taught at Cornell University before joining UC Berkeley in 1995, where he is now Professor of Mathematics and Computer Science. A leading experimentalist among mathematicians, Sturmfels has authored seven books and over 130 research articles in the areas of combinatorics, algebraic geometry, symbolic computation, and their applications, and he has mentored 16 doctoral students.

Ordered algebraic structures --- Geometry --- Discrete mathematics --- algebra --- landmeetkunde --- discrete wiskunde

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The object of this monograph is to document what is most interesting about linear monoids. We show how these results ?t together into a coherent blend of semigroup theory, groups with BN-pair, representation theory, convex - ometry and algebraicgrouptheory.The intended reader is one who is familiar with some of these topics, and is willing to learn about the others. The intention of the author is to convince the reader that reductive monoids are among the darlings of algebra. We do this by systematically assembling many of the major known results with many proofs,examples and explanations. To further entice the reader, we have included many exercises. The theory of linear algebraic monoids is quite recent, originating around 1980. Both Mohan Putcha and the author began the systematic study in- pendently. But this development would not have been possible without the pioneering work of Chevalley, Borel and Tits on algebraic groups. Also, there is the related, but more general theory of spherical embeddings, developed largely by Brion, Luna and Vust. These theories were developed somewhat independently, but it is always a good idea to interpret monoid results in the combinatorial apparatus of spherical embeddings. Each chapter of this monograph is focussed on one or more of the major themes of the subject. These are: classi?cation, orbits, geometry, represen- tions, universal constructions and combinatorics. There is an inherent div- sity and richness in the subject that usually rewards a stalwart investigation.

Algebra --- Geometry --- Discrete mathematics --- algebra --- discrete wiskunde --- geometrie

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Ordered algebraic structures --- Geometry --- Discrete mathematics --- algebra --- landmeetkunde --- discrete wiskunde

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• Reference Manager
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The object of this monograph is to document what is most interesting about linear monoids. We show how these results ?t together into a coherent blend of semigroup theory, groups with BN-pair, representation theory, convex - ometry and algebraicgrouptheory.The intended reader is one who is familiar with some of these topics, and is willing to learn about the others. The intention of the author is to convince the reader that reductive monoids are among the darlings of algebra. We do this by systematically assembling many of the major known results with many proofs,examples and explanations. To further entice the reader, we have included many exercises. The theory of linear algebraic monoids is quite recent, originating around 1980. Both Mohan Putcha and the author began the systematic study in- pendently. But this development would not have been possible without the pioneering work of Chevalley, Borel and Tits on algebraic groups. Also, there is the related, but more general theory of spherical embeddings, developed largely by Brion, Luna and Vust. These theories were developed somewhat independently, but it is always a good idea to interpret monoid results in the combinatorial apparatus of spherical embeddings. Each chapter of this monograph is focussed on one or more of the major themes of the subject. These are: classi?cation, orbits, geometry, represen- tions, universal constructions and combinatorics. There is an inherent div- sity and richness in the subject that usually rewards a stalwart investigation.

Algebra --- Geometry --- Discrete mathematics --- algebra --- discrete wiskunde --- geometrie