### Compatibility and Crystal Microstructure

John Ball

Mathematical Institute, University of Oxford

The lecture will describe various ideas related to compatibility and incompatibility of gradients that provide information concerning the formation and morphology of martensitic microstructures in single crystals and polycrystals. From a mathematical perspective this involves consideration of the structure of gradient Young measures, the quasiconvexity condition of the multi-dimensional calculus of variations, quasiregular maps, and generalizations of the Hadamard jump condition. A typical question is whether in a given experimental situation complex microstructures such as those involving layers within layers are preferred to simpler ones.