Mathematical descriptions of dynamical systems
Set Theory is fundamental to the study of mathematics and its application to the behaviour of systems. It is based on the concept of membership, that one set of objects (in mathematics referring to numbers, points, functions, etc.) is a member of another set much as the first grade of elementary school is a ‘member’ of the elementary school set. A subset of Set Theory, Descriptive Set Theory is an area of mathematics concerned with the study of the structure of definable sets of real numbers (as opposed to integers). Manipulation of these sets, essentially combining them in different ways, is the essence of operator algebras. The term refers to algebraic structures in which a linear operator combines any two vectors to form a third vector (the simple analogy is the set of integers in which the multiplication ‘operator’ acting on any two integers yields a third integer). Recently, great progress has been made at the interface of operator algebras, Descriptive Set Theory and Ergodic Theory. Ergodic theory is concerned with the behaviour of dynamical systems over very long time intervals and has its origins in theorems of von Neumann, among others. Generally speaking, an ergodic system ‘forgets’ its initial state, always exhibiting the same time average behaviour (statistically and qualitatively speaking) when allowed to run for long periods of time regardless of initial conditions. European researchers supported by funding of the ‘Descriptive set theory and operator algebras’ (DSTOA) project set out to address mathematical problems that combine Descriptive Set Theory and operator algebra with consideration of their application and relevance to ergodic systems. Numerous results have been achieved so far, in particular regarding rigidity results for von Neumann algebras and so-called von Neumann equivalence arising from so-called measure-preserving ergodic a.e. free group actions. Continued research regarding group actions and rigidity phenomena and their associated von Neumann algebras should enhance mathematical descriptions of ergodicity with eventual future applications to industrially relevant processes.
