Final Activity Report Summary - IDEA (Integrable Difference Equations and their Applications) Since antiquity people asked the question whether the substance of the universe was discrete or continuous. It is a paradox that corpuscle theory in the Newtonian approach led to the development of differential equations by demanding the notion of continuity in their definition. In contrast, the theory of difference, i.e. discrete, equations remains a less developed subject, even in this day and age, in spite of the recent explosion of interest in this theory which was caused by its significance for computer science and information technology. The subject of the project lied on the interface between the theory of discrete equations on the one hand and the theory of nonlinear integrable equations on the other. In the domain of differential equations such systems were often referred to as soliton equations, which once again highlighted an interesting connection with corpuscle theory, since solitons are particle-like solutions of nonlinear partial differential equations which remarkably also exhibit particle-like collision properties. In the discrete domain, analogues of systems exhibiting such solutions were developed in the past two decades and these discoveries very much brought back the issue of difference equations into the limelight. The main objective of the project was to develop further the theory of nonlinear integrable difference equations and push them in the direction of equations relevant to the theory of gravity, i.e. Einsteins theory of general relativity, where under special circumstances the governing equations were known to be integrable, such as the celebrated Ernst equation which described gravitational waves. One of the main guiding principles of relativity is the freedom of changing the variables entering the equations, and this principle has been a major deficiency of the discrete theory. The most important achievement of the project was related to this issue. We showed how to compensate the apparent lack of possibility to changing the independent variables in the case of difference integrable equations. This would eventually make possible the raise of the discrete theory, by means of integrable examples, to the same level as the continuous theory, in the ultimate endeavour to build a fundamental theory describing the physics of discrete space-time.