Mathematical Sciences: Problems in Descriptive Set Theory, Ergodic Theory and Set Theory

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9500494 Foreman Foreman will work on two types of problems in this research project. The first kind lies within ergodic theory and has to do with conjugacy of invertible measure-preserving transformations. In particular, Foreman will use the tools of descriptive set theory to attempt to determine whether every invertible ergodic transformation with finite entropy is conjugate to a smooth transformation on a compact smooth manifold. The second kind of problem concerns the strength and consequences of very large ideals on cardinals such as the second uncountable cardinal. Foreman intends to use these ideals to investigate partition relations. In the early twentieth century, ergodic theory arose as a technique for studying complicated models of physical phenomenon by looking at the statistical behavior of the associated flows. Dynamical systems arising in physics with otherwise intractible behavior were amenable to this kind of analysis. A prominent problem in ergodic theory deals with the extent to which the general statistical behavior of measure preserving systems models the behavior of physically realistic situations. Foreman will continue to investigate these problems using the tools of descriptive set theory, an unconventional approach which brings logic to bear on the problem. ***