The Perceptron capacity model is a mean field model to study spin glass, and is fundamental in the theory of neural networks. In this talk, we will discuss its underlying mathematical problem, which concerns measuring typically how much of the discrete cube {-1, 1}^N (or the unit sphere) is left when one intersects the set with many random half-spaces. We first introduce notations, phrase the question in the language of statistical mechanics, then describe results in the special case where both the number of random half-spaces and the dimension N are large. Speaker(s): Han Le (University of Michigan)