Geometry, topology and emergent simplicity

Complex systems involve many varied working parts and display structured, yet complicated phenomena that are difficult to understand. Can we make sense of complexity without knowing the detailed inner workings of the system? What robust predictions can we make and are there simple emergent laws to describe such phenomena? One powerful and old approach is to use tools from geometry and topology to address this question. Using both historical and contemporary examples I will dicsuss how these tools afford simple lessons in complex systems and highlight some future exciting frontiers, particularly in relation to biology.