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Microstructures, phase transitions and geometry
Microstructures are abundant in nature and often crucially influence a material's or system's behaviour. The mathematical analysis of microstructures leads to simply stated but challenging mathematical problems and involves an interaction of a variety of mathematical fields including the calculus of variations, partial differential equations, real and functional analysis and differential geometry. This paper outlines some of the basic problems using the example of crystal microstructure in solid-solid phase transitions and discusses recent progress on exact solutions (obtained by Gromov's method of convex integration) and on the interaction of surface energy and fine scale geometry.