The main problem raised in the project is to better understand finite objects arising in mathematics through their limiting behavior, and to better understand infinite objects by their finite approximations. In particular, consider limits of discrete structures, like finite graphs or processes on finite graphs and limits of continuous structures, like Riemannian manifolds or the homology and eigenfunctions on these manifolds. The importance of this project for society is that it provides a framework in which one can put an order on the universe of very big finite objects, like networks or structured large data. The overall objectives are to understand specific instances of this limiting theory, including locally symmetric spaces, processes on finite graphs, unimodular random graphs and Riemannian manifolds, invariant random subgroups of discrete and Lie groups and invariant stochastic processes on groups and graphs.