Emergence of Collective Behavior from the Quantum Many-Body Problem

The many-body Schrödinger equation may be considered a triumph of reductionism: a vast variety of phenomena has been reduced to a single linear partial differential equation. Unfortunately the Schrödinger equation can in general not be solved efficiently, since entanglement suffers from the curse of exponentially growing dimensionality. So this triumph has created a new challenge: revealing the macroscopic behavior from the microscopic equation. I will present recent advances on the emergence of nonlinear equations through the analysis of scaling limits, and then discuss current challenges in the description of interacting fermionic quantum systems. In particular I will discuss the role of non-trivial correlations in the ground state.