The Mathematical Challenge of Perceptrons and Related Mean-Field Spin Glasses

A seemingly simple variation of the standard Sherrington-Kirkpatrick model is the bipartite SK-model. There, the set of spin variables is split into two groups, and random pair interactions are only present between variables in different groups. Somewhat surprisingly, a full mathematical understanding of this model is presently lacking, and there are contradicting conjectures about the form of a variational formula for the free energy, one should expect. The model is a very special case of a big class of models, named perceptrons which are of considerable interest for neural networks, machine learning, and statistics. The talk will attempt to give an overview of what is mathematically known about this class of models, and explain the source of their mathematical difficulties.