Mathematical Sciences: Joinings, Combinatorics and Ergodic Theory

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Professor King will study a number of questions in ergodic theory. He will consider potential generalizations of a structure theorem based on joinings. He will investigate the "continuous selection problem", related to work of Oxtoby and Ulam. He will study two combinatorial problems: the 2-fold 3- fold mixing problem and the van den Berg-Kesten "box conjecture". 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.