Mathematical Sciences: The Mathematical Physics of Crystals

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Radin will study mathematical problems relating the causes of crystalline symmetry in low temperature matter and classification of the dynamical systems associated with uniquely ergodic tilings of Euclidean space. The methods to be used are a combination of traditional ergodic theory and new techniques from aperiodic tilings. It is expected that the recent results on rotation symmetric tilings will open significant new research in the study of material structure and in the classification of dynamical systems. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.