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Commutative algebra, combinatorics, and algebraic geometry are
thriving areas of mathematical research with a rich history of
interaction. Connections Between Algebra and Geometry contains
lecture notes, along with exercises and solutions, from the
Workshop on Connections Between Algebra and Geometry held at the
University of Regina from May 29-June 1, 2012. It also contains
research and survey papers from academics invited to participate in
the companion Special Session on Interactions Between Algebraic
Geometry and Commutative Algebra, which was part of the CMS Summer
Meeting at the University of Regina held June 2-3, 2012, and the
meeting Further Connections Between Algebra and Geometry, which was
held at the North Dakota State University February 23, 2013. This
volume highlights three mini-courses in the areas of commutative
algebra and algebraic geometry: differential graded commutative
algebra, secant varieties, and fat points and symbolic powers. It
will serve as a useful resource for graduate students and
researchers who wish to expand their knowledge of commutative
algebra, algebraic geometry, combinatorics, and the intricacies of
their intersection.
Commutative algebra, combinatorics, and algebraic geometry are
thriving areas of mathematical research with a rich history of
interaction. Connections Between Algebra and Geometry contains
lecture notes, along with exercises and solutions, from the
Workshop on Connections Between Algebra and Geometry held at the
University of Regina from May 29-June 1, 2012. It also contains
research and survey papers from academics invited to participate in
the companion Special Session on Interactions Between Algebraic
Geometry and Commutative Algebra, which was part of the CMS Summer
Meeting at the University of Regina held June 2-3, 2012, and the
meeting Further Connections Between Algebra and Geometry, which was
held at the North Dakota State University February 23, 2013. This
volume highlights three mini-courses in the areas of commutative
algebra and algebraic geometry: differential graded commutative
algebra, secant varieties, and fat points and symbolic powers. It
will serve as a useful resource for graduate students and
researchers who wish to expand their knowledge of commutative
algebra, algebraic geometry, combinatorics, and the intricacies of
their intersection.
This is the first of two volumes of a state-of-the-art survey
article collection which originates from three commutative algebra
sessions at the 2009 Fall Southeastern American Mathematical
Society Meeting at Florida Atlantic University. The articles reach
into diverse areas of commutative algebra and build a bridge
between Noetherian and non-Noetherian commutative algebra. These
volumes present current trends in two of the most active areas of
commutative algebra: non-noetherian rings (factorization, ideal
theory, integrality), and noetherian rings (the local theory,
graded situation, and interactions with combinatorics and
geometry). This volume contains combinatorial and homological
surveys. The combinatorial papers document some of the increasing
focus in commutative algebra recently on the interaction between
algebra and combinatorics. Specifically, one can use combinatorial
techniques to investigate resolutions and other algebraic
structures as with the papers of Floystad on Boij-Soederburg
theory, of Geramita, Harbourne and Migliore, and of Cooper on
Hilbert functions, of Clark on minimal poset resolutions and of
Mermin on simplicial resolutions. One can also utilize algebraic
invariants to understand combinatorial structures like graphs,
hypergraphs, and simplicial complexes such as in the paper of Morey
and Villarreal on edge ideals. Homological techniques have become
indispensable tools for the study of noetherian rings. These ideas
have yielded amazing levels of interaction with other fields like
algebraic topology (via differential graded techniques as well as
the foundations of homological algebra), analysis (via the study of
D-modules), and combinatorics (as described in the previous
paragraph). The homological articles the editors have included in
this volume relate mostly to how homological techniques help us
better understand rings and singularities both noetherian and
non-noetherian such as in the papers by Roberts, Yao, Hummel and
Leuschke.
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