Solomyak will investigate several topics in operator theory related to ergodic theory. The first has connections with K- theory of operator algebras, and centers on the so-called "adic transformations" of Vershik acting on paths of a Bratelli diagram. These transformations define measure-preserving systems whose ergodic and spectral properties will be studied. The proposed approach involves dimension groups, harmonic analysis of measures, and special digit expansions of real numbers. These expansions will be considered in the second part of the project. The third topic concerns the problem of determining the invariant subspace structure for Volterra convolution operators. 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.
