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This volume collects contributions by leading experts in the area
of commutative algebra related to the INdAM meeting "Homological
and Computational Methods in Commutative Algebra" held in Cortona
(Italy) from May 30 to June 3, 2016 . The conference and this
volume are dedicated to Winfried Bruns on the occasion of his 70th
birthday. In particular, the topics of this book strongly reflect
the variety of Winfried Bruns' research interests and his great
impact on commutative algebra as well as its applications to
related fields. The authors discuss recent and relevant
developments in algebraic geometry, commutative algebra,
computational algebra, discrete geometry and homological algebra.
The book offers a unique resource, both for young and more
experienced researchers seeking comprehensive overviews and
extensive bibliographic references.
This volume collects contributions by leading experts in the area
of commutative algebra related to the INdAM meeting "Homological
and Computational Methods in Commutative Algebra" held in Cortona
(Italy) from May 30 to June 3, 2016 . The conference and this
volume are dedicated to Winfried Bruns on the occasion of his 70th
birthday. In particular, the topics of this book strongly reflect
the variety of Winfried Bruns' research interests and his great
impact on commutative algebra as well as its applications to
related fields. The authors discuss recent and relevant
developments in algebraic geometry, commutative algebra,
computational algebra, discrete geometry and homological algebra.
The book offers a unique resource, both for young and more
experienced researchers seeking comprehensive overviews and
extensive bibliographic references.
For every mathematician, ring theory and K-theory are intimately
connected: al- braic K-theory is largely the K-theory of rings. At
?rst sight, polytopes, by their very nature, must appear alien to
surveyors of this heartland of algebra. But in the presence of a
discrete structure, polytopes de?ne a?ne monoids, and, in their
turn, a?ne monoids give rise to monoid algebras. Teir spectra are
the building blocks of toric varieties, an area that has developed
rapidly in the last four decades. From a purely systematic
viewpoint, "monoids" should therefore replace "po- topes" in the
title of the book. However, such a change would conceal the
geometric ?avor that we have tried to preserve through all
chapters. Before delving into a description of the contents we
would like to mention three general features of the book: (?) the
exhibiting of interactions of convex geometry, ring theory, and
K-theory is not the only goal; we present some of the central
results in each of these ?elds; (?) the exposition is of
constructive (i. e., algorithmic) nature at many places throughout
the text-there is no doubt that one of the driving forces behind
the current popularity of combinatorial geometry is the quest for
visualization and computation; (? ) despite the large amount of
information from various ?elds, we have strived to keep the
polytopal perspective as the major organizational principle.
For every mathematician, ring theory and K-theory are intimately
connected: al- braic K-theory is largely the K-theory of rings. At
?rst sight, polytopes, by their very nature, must appear alien to
surveyors of this heartland of algebra. But in the presence of a
discrete structure, polytopes de?ne a?ne monoids, and, in their
turn, a?ne monoids give rise to monoid algebras. Teir spectra are
the building blocks of toric varieties, an area that has developed
rapidly in the last four decades. From a purely systematic
viewpoint, "monoids" should therefore replace "po- topes" in the
title of the book. However, such a change would conceal the
geometric ?avor that we have tried to preserve through all
chapters. Before delving into a description of the contents we
would like to mention three general features of the book: (?) the
exhibiting of interactions of convex geometry, ring theory, and
K-theory is not the only goal; we present some of the central
results in each of these ?elds; (?) the exposition is of
constructive (i. e., algorithmic) nature at many places throughout
the text-there is no doubt that one of the driving forces behind
the current popularity of combinatorial geometry is the quest for
visualization and computation; (? ) despite the large amount of
information from various ?elds, we have strived to keep the
polytopal perspective as the major organizational principle.
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