Smooth random functions of very many variables can be topologically very complex, and thus it can be terribly hard to find their minimum.One does not need to look very far for such an example: pick at random a homogeneous polynomial of degree p (with p larger than 3) of a large number of variables and restrict it to the (high-dimensional) unit sphere. Important examples of such functions include many Hamiltonians of statistical mechanics in disordered media (as Spin Glasses or Random Interfaces in high disorder). They can also include the loss functions of high dimensional inference problems, and naturally the landscapes defined by Machine Learning.

We will cover some of the recent progress in our understanding of both questions: the statics or geometric question about the topological complexity and the transition to simple landscapes (the so-called topological trivialization), as well as the dynamics and optimization questions.