Problems in descriptive set theory, ergodic theory and set theory
Ohio State University Distinguished Scholar (1993). The Outstanding Undergraduate in the College of Arts and Sciences, University of Colorado (1975).
1992-present: Professor of Mathematics and Philosophy, UC Irvine.
In this project Foreman proposes to work on two main types of problems. The first type of problem deals with applications of descriptive set theory to ergodic theory. In the early twentiieth century, ergodic theory arose as a technique for studying complicated differential equations by looking at the statistical behavior of the associated flows. Dynamical systems arising in physics with otherwise intractible behavior were amendable to thsi kind of analysis. A prominent problem in ergodic theory deals with the extent that general statistical behavior of measure preserving systems models the behavior of flows on compact C<> manifolds with smooth densities. A precise statement of this problem is to ask whether evry flow of finite entropy can be realized as a smooth flowon a compact C<> manifold. In recent papers Foreman has constructed two new classes of zero entropy flows. Foreman proposes to study the problem of realizability of flows as differentiable flows using techniques developed in these papers together with techniques developed recently by Kechris et. al. for studying equivalence relations determined by Polish group actions. The second family of problems Foreman proposes to study are those related to the theory of saturated ideals and their consequences. Foreman proposes to extend his work that yielded the consistency of a countably complete, uniform, <>-dense ideal on <> to similar constructions that should yield <>-dense idealson <> for each natural number n. He hopes that he will be able to find more applications of these ideals to combinatorial problems on the <>'s such as the problem of whether it is consistent for all <>.
(with M. Magidor) "Large Cardinals and definable counterexamples to the continuum hypothesis". Journal of Pure and Applied Logic, Vol.76 (1995), pp 47-97.
"Amenable groups and invariant means". Journal of Functional Analysis, Vol.126 (1) (1994), pp 7-25.
(with F. Beleznay) "The Collection of distal flows is not Borel". American Journal of Mathematics, Vol.117(1) (1995), pp 203-239.
(with R. Dougherty) "Banach-Tarski Decompositions using sets with the property of Baire", Proceedings of the National Academy of Sciences, USA, 89 (1992) 10726-10728.
(with R. Dougherty) "Banach-Tarski Decompositions Using Sets with the Property of Baire", Journal of the American Mathematical Society, v.7, No 1 (1994) 75-124.
(with M. Magidor) "A very weak square principle". Journal of Symbolic Logic, Vol.62(1), (1997), pp 175-196.
(with J. Baumgartner and O. Spinas) "On the spectrum of T-invariants". Journal of Algebra, Vol.189 (1997), pp 406-418.
(with F. Beleznay) "The complexity of the measure distal flows". Journal of Ergodic Theory and Dynamical Systems, Vol.16 (1996), pp 929-962.
(with P. Eklof and S. Shelah) "On invariants for w1-seperable groups". Trans. Am. Math. Soc., Vol.347(11) (1995), pp 4385-4402.
Editor, Journal of Symbolic Logic.
Editor, Notre Dame Journal for Symbolic Logic.