سیامک یاسمی

How important is the theory of categories? From the early days of its creation until today, this question has occupied the minds of mathematicians and different answers have been provided: For some categories are merely a tool. For others, they are a fundamental part of today’s mathematics. Generally, questions like these do not have definite and absolute answers. At least for this particular question, the mathematics community has not reached a common answer, and probably will never reach a resolution. Our goal in writing this article is to discuss the origins and developments of the theory of categories and also discuss its role in mathematics.

The idea to create an academy of scientists from the developing world first discussed among a small group of internationally renowned researchers in October 1981 during a meeting in Rome, Italy. Under the leadership of Abdus Salam, the Pakistani physicist and Nobel laureate, the world academy of sciences for the developing country (TWAS) was founded in Trieste, Italy, in 1983. Along with some useful information about TWAS, the question of “why we need such a non-governmental organization” is discussed.

There has been many researches in the field of combinational commutative algebra; however, what makes this research distinct from the previous ones is its focus on monomial ideals and their connection to graphs. The present study aims at various goals one of which is analyzing the existence of linear resolution of such ideals. Concerning combinatorial commutative algebra there are numerous methods for creating connection between combinational objects and algebric objects‎. ‎This article is to study such objects through corresponding monomial ideals with graphs and simplicial complexes‎. ‎The most significant treatable ideals of simplicial complex include edge ideal and Stanly-Reisner ideal‎. Let r be a positive integer. A monomial ideal I is said to be linear in the first r steps, if for some integer d , β_(i.i+j) (I)=0 for all 0≤i