Mathematics Colloquia and Seminars

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Composite materials are media that look homogeneous but in fact have complex structure (microstructure) when viewed under a microscope. These materials are finding their way into our everyday lives in objects such as skis, golf clubs, automobiles, aircraft, computers, construction components of buildings and bridges, sensors and actuators many many more. It is an important and a formidable task to predict the properties (called effective properties) of such media theoretically. The most serious obstruction in our way is the strong dependence of the effective properties of composite materials on the microstructure. So it comes as a nice surprise to come across exact formulae relating an effective tensor of a composite to the tensors of its constituents regardless of the microstructure. Such formulae have been discovered before and were rightfully regarded as rare jewels in the subject. In my talk I will describe the general theory of such formulae that we call exact relations. The new machinery allows one to harvest all exact relation in a context of virtually any coupled linear physical problem including conductivity, elasticity, piezo-electricity, and many others. The application of representation theory of rotation groups SO(2) and SO(3) makes obtaining actual exact relations feasible in rather high dimensional settings. The current work has given rise to a set of new questions in group representations, that is currently being transformed into a beautiful theory by one of my collaborators.