Local structure theorems for algebraic foliations

It is gradually being realised that the correct definition of foliation in algebraic geometry is through formal stacks. After giving some motivation lying behind this change in perspective, we will describe a correspondence between classical foliations and certain stacks. We will then see how Frobenius theorem, a local structure theorem for smooth foliations, can be seen as the existence of a minimal presentation of a stack. Finally, we will discuss how the local structure theorem of Alper, Hall and Rydh for algebraic stacks may yield a local structure theorem for certain singular foliations.