# Complexity of random functions of many variables: from geometry to statistical physics and deep learning algorithms.

Functions of many variables may be very complex. Think for instance of the following simple question: How hard is it to find the minimum of a cubic polynomial of many variables? If you chose the cubic polynomial randomly, it is very hard.

I will survey recent work describing this phenomenon and its consequences, first from a geometric point of view. I will illustrate this phenomenon first in the case of random functions on the high-dimensional sphere. These random functions happen to be the energy landscapes of important models of statistical physics of disordered media, i.e spherical spin glasses. I will then show how this could be extended to the random landscapes of statistics of large data sets, and for instance of deep learning algorithms, which are at the heart of many of the recent progress in Data Science. The common mathematical field underlying these different questions is given by Random Matrix Theory, through the classical tool of random geometry, i.e. the Kac-Rice formulae.

The relevant work is joint with mathematical colleagues (Auffinger, Cerny, Jagannath, Subag, Zeitouni) or physicists (Biroli, Cammarota) and Computer Scientists (Yann Le Cun and his team at Facebook).