Mathematical Sciences: Mathematical Physics of Crystals, andTiling

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9531584 Radin This project has to do with classifying the tilings of Euclidean space produceable by copies of a finite set of distinct tiles. The investigator plans to apply his recent results on rotation symmetric tilings of the plane and space. The methods to be used are a combination of traditional ergodic theory and new techniques from aperiodic tilings. There are fundamental mathematical questions in common between the two areas of investigation: determining the causes of crystalline symmetry in low temperature matter, and classifying the tilings of space by copies of finitely many tiles. These are both optimization problems; the crystal problem is concerned with determining the basic mechanisms which force low temperature and high pressure matter to have the high degree of crystalline order and symmetry which is found experimentally.