The Cluster of Excellence is probing and thoroughly studying this entire dynamic range through ten Research Areas. Each Research Area is driven by recent developments and a broad vision. It starts from a strong local expertise and experienced teams, and comes with specific and ambitious goals for the years ahead. The theme of the Hausdorff Center is mathematics, of which we share a broad and dynamic vision. At its center stands the classical core: the mathematics of famous and fruitful conjectures, the mathematics that is in symbiosis with theoretical physics, and which continually reveals to us deep and unexpected connections. Beyond this core, mathematics drives, and is driven by, the successful trend towards quantitative modeling in the natural and social sciences. The steadily improving performance of computers opens new perspectives for model simulations and the direct transfer of mathematics into technological applications.