Combinatorial Aspects of Commutative Algebra and Algebraic Geometry

Combinatorial Aspects of Commutative Algebra and Algebraic Geometry

Author: Gunnar Fløystad

Publisher: Springer Science & Business Media

Published: 2011-05-16

Total Pages: 186

ISBN-13: 3642194923

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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.


Geometric And Combinatorial Aspects Of Commutative Algebra

Geometric And Combinatorial Aspects Of Commutative Algebra

Author: Jurgen Herzog

Publisher: CRC Press

Published: 2001-03-06

Total Pages: 424

ISBN-13: 9780203908013

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This work is based on the lectures presented at the International Conference of Commutative Algebra and Algebraic Geometry held in Messina, Italy. It discusses developments and advances in commutative algebra, algebraic geometry, and combinatorics - highlighting the theory of projective schemes, the geometry of curves, determinantal and stable idea


Combinatorial Structures in Algebra and Geometry

Combinatorial Structures in Algebra and Geometry

Author: Dumitru I. Stamate

Publisher: Springer Nature

Published: 2020-09-01

Total Pages: 182

ISBN-13: 3030521117

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This proceedings volume presents selected, peer-reviewed contributions from the 26th National School on Algebra, which was held in Constanța, Romania, on August 26-September 1, 2018. The works cover three fields of mathematics: algebra, geometry and discrete mathematics, discussing the latest developments in the theory of monomial ideals, algebras of graphs and local positivity of line bundles. Whereas interactions between algebra and geometry go back at least to Hilbert, the ties to combinatorics are much more recent and are subject of immense interest at the forefront of contemporary mathematics research. Transplanting methods between different branches of mathematics has proved very fruitful in the past – for example, the application of fixed point theorems in topology to solving nonlinear differential equations in analysis. Similarly, combinatorial structures, e.g., Newton-Okounkov bodies, have led to significant advances in our understanding of the asymptotic properties of line bundles in geometry and multiplier ideals in algebra. This book is intended for advanced graduate students, young scientists and established researchers with an interest in the overlaps between different fields of mathematics. A volume for the 24th edition of this conference was previously published with Springer under the title "Multigraded Algebra and Applications" (ISBN 978-3-319-90493-1).


Combinatorial Commutative Algebra

Combinatorial Commutative Algebra

Author: Ezra Miller

Publisher: Springer Science & Business Media

Published: 2005-06-21

Total Pages: 442

ISBN-13: 9780387237077

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Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs


Algebraic Combinatorics and Coinvariant Spaces

Algebraic Combinatorics and Coinvariant Spaces

Author: Francois Bergeron

Publisher: CRC Press

Published: 2009-07-06

Total Pages: 227

ISBN-13: 1439865078

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Written for graduate students in mathematics or non-specialist mathematicians who wish to learn the basics about some of the most important current research in the field, this book provides an intensive, yet accessible, introduction to the subject of algebraic combinatorics. After recalling basic notions of combinatorics, representation theory, and


Combinatorial Aspects of Commutative Algebra

Combinatorial Aspects of Commutative Algebra

Author: Viviana Ene

Publisher: American Mathematical Soc.

Published: 2009-11-25

Total Pages: 194

ISBN-13: 0821847589

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This volume contains the proceedings of the Exploratory Workshop on Combinatorial Commutative Algebra and Computer Algebra, which took place in Mangalia, Romania on May 29-31, 2008. It includes research papers and surveys reflecting some of the current trends in the development of combinatorial commutative algebra and related fields. This volume focuses on the presentation of the newest research results in minimal resolutions of polynomial ideals (combinatorial techniques and applications), Stanley-Reisner theory and Alexander duality, and applications of commutative algebra and of combinatorial and computational techniques in algebraic geometry and topology. Both the algebraic and combinatorial perspectives are well represented and some open problems in the above directions have been included.


Lectures in Geometric Combinatorics

Lectures in Geometric Combinatorics

Author: Rekha R. Thomas

Publisher: American Mathematical Soc.

Published: 2006

Total Pages: 156

ISBN-13: 9780821841402

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This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the statepolytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Grobner bases of toric ideals and other methods fromcommutative algebra. The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational aspects of the theory of polytopes.


Binomial Ideals

Binomial Ideals

Author: Jürgen Herzog

Publisher: Springer

Published: 2018-09-28

Total Pages: 332

ISBN-13: 3319953494

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This textbook provides an introduction to the combinatorial and statistical aspects of commutative algebra with an emphasis on binomial ideals. In addition to thorough coverage of the basic concepts and theory, it explores current trends, results, and applications of binomial ideals to other areas of mathematics. The book begins with a brief, self-contained overview of the modern theory of Gröbner bases and the necessary algebraic and homological concepts from commutative algebra. Binomials and binomial ideals are then considered in detail, along with a short introduction to convex polytopes. Chapters in the remainder of the text can be read independently and explore specific aspects of the theory of binomial ideals, including edge rings and edge polytopes, join-meet ideals of finite lattices, binomial edge ideals, ideals generated by 2-minors, and binomial ideals arising from statistics. Each chapter concludes with a set of exercises and a list of related topics and results that will complement and offer a better understanding of the material presented. Binomial Ideals is suitable for graduate students in courses on commutative algebra, algebraic combinatorics, and statistics. Additionally, researchers interested in any of these areas but familiar with only the basic facts of commutative algebra will find it to be a valuable resource.


Algebraic and Combinatorial Aspects of Tropical Geometry

Algebraic and Combinatorial Aspects of Tropical Geometry

Author: Erwan Brugalle

Publisher: American Mathematical Soc.

Published: 2013-05-23

Total Pages: 363

ISBN-13: 0821891464

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This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. Tropical geometry is a new and rapidly developing field of mat


Commutative Algebra

Commutative Algebra

Author: Alberto Corso

Publisher: CRC Press

Published: 2005-08-15

Total Pages: 289

ISBN-13: 1420028324

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Packed with contributions from international experts, Commutative Algebra: Geometric, Homological, Combinatorial, and Computational Aspects features new research results that borrow methods from neighboring fields such as combinatorics, homological algebra, polyhedral geometry, symbolic computation, and topology. This book consists of articles pres