Mathematical Conversations

In high dimensions, what does it look like when we take the intersection of a set of random half-spaces with either the sphere or the Hamming cube? This is one phrasing of the so-called perceptron problem, whose study originated with a toy model of a very simple neural network. In this talk I will survey the (non-rigorous) physics solution from the 70s and 80s, and some more recent mathematical progress.