Numerous results in different areas of analysis describe various aspects of convex functions: properties of the determinant of the Hessian, structure of the singular set of the gradient, ``propagation of singularities'' of the gradient, and so on. The proofs of these results rely strongly on some form of convexity, or equivalently, some form of monotonicity of the gradient map. Most such results have been extended to larger classes of functions, for example, functions that can be written as differences of convex functions, but using arguments that still rely crucially on convexity.
I will describe an approach to such results that abandons all convexity assumptions and instead relies only on the Lagrangian structure of the graph of the gradient of ANY sufficiently integrable function, not necessarily convex. Here ``sufficiently integrable'' can include quite irregular functions. The key point in this approach is ultimately a rigidity property of Lagrangian integral currents. This rigidity theorem will be the main focus of the talk.