In many models of solid mechanics fine-scaled patterns are formed. In mathematics, these are often explained as minimizers of non-convex singularly perturbed variational functionals, while in physics, they are often viewed as instabilities of homogeneous states. In this workshop we will focus on three such areas: wrinkling of thin elastic sheets, dislocations in crystals, and pattern formation in multi-well problems as they occur for example in shape-memory alloys.

Until recently mathematical results were in many cases restricted to the identification of energyscaling laws, i.e., how the energy of the ground state depends on the parameters in the system. For some problems successful efforts, often based on Fourier methods, or on advanced techniques of geometric measure theory, were made to go beyond the scaling laws, for example by identifying effective limiting functionals which characterize the microstructure. This progress is particularly interesting for the ansatz-based methods in physics, since they can help to justify the approximations and assumptions on which those methods are based.

The purpose of the workshop is twofold: on one hand the aim is to bring together mathematicians, physicists and engineers in order to make the gap between very different methods smaller. On the other hand, we will bring together different mathematics communities, which rarely meet but can greatly benefit from interchange of recent methods.