Derived localisation in algebra and homotopy theory

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We begin to get acquainted with localisation (albeit not under this name) while in high school. Consider the set N of natural numbers: 0, 1,2,3,... Two natural numbers could be added, but they cannot always be subtracted: 1-2 is not a natural number. We say that N forms a monoid; it is commutative because a sum of two natural numbers does not depend on the order in which they are added. It is useful to have an algebraic structure that accommodates subtraction as well as addition; this is how the set of integers Z is constructed out of natural numbers; essentially, negatives are simply added to the existing elements. We say that Z is a (still commutative) group and it is a group completion of the monoid N.There are other similar examples: consider Z with the operation of multiplication (rather than addition as above). Again, it is a commutative monoid and its completion is the set Q of rational numbers.Note that in the last example Z supports two structures: addition and multiplication, suitably compatible. A formalization of this is called a (commutative) ring. On the other hand, Q is more than a ring: it is a field, which means that all non-zero element of it are invertible. The field Q is called the field of fractions of Z, because its elements could indeed be viewed as fraction with integer numerator and denominator.Group completions of monoids and fields of fractions of rings are examples of localization. In the present project we concentrate on localization of rings, or or more general structures called differential graded rings. Given a commutative ring A and a collection S of elements in A one can try to formally invert it, or localise at S. For example, the localization of Z at the set of all non-zero integers produces Q. This is one of the most fundamental and simple procedures in commutative algebra; it serves as an underpinning of advanced fields of pure mathematics, such as algebraic geometry, and is indispensable as a tool for proving theorems. Commutative localisation has been well understood for a long time.In contrast, our understanding of localization of noncommutative rings (such as a ring of square matrces) is more patchy; various useful constructions, such as forming a field of fractions, are either impossible or only hold under severe constraints. On the other hand, noncommutative localization is very important, for example it could be said that the whole subject of homotopy theory (the study of those properties of spaces which do not change under continuous deformation) revolves around localization of a certain category (which is a generalization of a noncommutative algebra).The main insight of the present project, which builds on a recent work by the proposers, is that once the category of noncommutative rings is suitably extended, the formal properties of commutative localization are almost completely restored.The goal of the present project is to exploit the consequences of this idea, extend it suitably and derive consequences in algebra, topology and category theory.