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 [50]; 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 [50]; summary and our conclusions are given in the final section, Section 5.

It can be defined implicitly by considering a graded

commutative algebra with

(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 [4] 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.

[3] 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 [21]; or equivalently, [L.sub.k](K) + 1 is the regularity of the face ideal of K [20].

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.