Compressive sensing is a novel field in signal processing at the interface of applied mathematics, electrical engineering and computer science, which caught significant interest over the past five years. It provides a fundamentally new approach to signal acquisition and processing that has large potential for many applications. Compressive sensing (sparse recovery) predicts the surprising phenomenon that many sparse signals (i.e. many real-world signals) can be recovered from what was previously believed to be highly incomplete measurements (information) using computationally efficient algorithms. In the past year exciting new developments emerged on the heels of compressive sensing: low rank matrix recovery (matrix completion); as well as a novel approach for the recovery of high-dimensional functions.We plan to pursue the following research directions:- Compressive Sensing (sparse recovery): We aim at a rigorous analysis of certain measurement matrices.- Low rank matrix recovery: First results predict that low rank matrices can be recovered from incomplete linear information using convex optimization.- Low rank tensor recovery: We plan to extend methods and mathematical results from low rank matrix recovery to tensors. This field is presently completely open.- Recovery of high-dimensional functions: In order to reduce the huge computational burden usually observed in the computational treatment of high-dimensional functions, a recent novel approach assumes that the function of interest actually depends only on a small number of variables. Preliminary results suggest that compressive sensingand low rank matrix recovery tools can be applied to the efficient recovery of such functions.We plan to develop computational methods for all these topics and to derive rigorous mathematical results on their performance. With the experience I gained over the pastyears, I strongly believe that I have the necessary competence to pursue this project.