From a July 2014 workshop in Kyoto, 15 papers explore affine and complete algebraic geometry, group actions, and commutative algebra
. The main areas they cover are algebraic varieties containing An-cylinders, algebraic varieties with fibrations, algebraic group actions and orbit stratifications on algebraic varieties, and automorphism groups and birational automorphism groups of algebraic varieties.
The paper is organized as follows: in Section 2, based on the single particle picture, we introduce our map from Lagrangian variables to Eulerian variables and briefly discuss its relation to the approach in ; in Section 3, we show how then on commutative algebra
can be taken into account via the approach given in Section 2; particularly we focus on then on commutative effects of the external potential; in Section 4, the cosmological implications of the noncommutative effects in our approach are discussed along the line in ; summary and our conclusions are given in the final section, Section 5.
It can be defined implicitly by considering a graded commutative algebra
(iii) A Hom-Jordan algebra is a Hom-algebra (A, *, [alpha]) such that (A, *) is a commutative algebra
and the Hom-Jordan identity
Moreover, this theorem can be considered as a generalization of Cohen's theorem, a famous theorem in commutative algebra
In  we listed properties of groups related to their character tables and showed how they can be generalized to Hopf algebras for which R(H) is a commutative algebra
. We summarize:
Then section 5 deals with an algebraic structure that gathers all the quadrangulations together, namely a commutative algebra
endowed with a derivation.
 Greuel, G.-M.; Pfister, G.: A Singular Introduction to Commutative Algebra
. Second edition, Springer (2007).
Readers should have a background in number theory, commutative algebra
, and the general theory of schemes.
Leray numbers also play a role in combinatorial commutative algebra
: [L.sub.k](K) turns out to be the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of K over k ; or equivalently, [L.sub.k](K) + 1 is the regularity of the face ideal of K .
Professor Jugal Kishore Verma has made substantial and fundamental contributions to theimportant area of mixed multiplicities in commutative algebra
. His research in this area has had a greatimpact in neighboring fields.