This talk briefly explains why finding the vacuum solutions of gauged maximal supergravity models is a mathematically challenging problem, and which techniques exist that are powerful enough to actually solve it. As these techniques require joint effort between physicists and computer scientists, the problem is explained in a way that should be accessible to both groups. In models of extended supergravity with more than one gravitino, the symmetry group transforming the gravitini into one another can be promoted to a local symmetry. In order to maintain supersymmetry, such a deformation mandates the introduction of a potential for the scalar fields whose stationary points correspond to vacua with spontaneously broken symmetry. The most famous such model is the SO(8)-gauged N=8 supergravity in four dimensions by de Wit and Nicolai, which also can be obtained by dimensional reduction of M-theory on the 7-sphere. The mathematical problem in the determination of the vacua lies in the complexity of the potential, which is a function on a high-dimensional Riemannian symmetric space based on an E-series exceptional Lie group. While previous group theoretic arguments allowed a partial investigation of the problem only, there are powerful semi-numeric computational techniques that indeed seem to be strong enough to solve the problem in full. These methods are explained using as an example the most challenging supergravity potential, that of N=16 in D=3 with SO(8)xSO(8) gauge group, and some first data on new vacua of N=8 supergravity in D=4 are presented.