Foreman's proposal involves applying tools from mathematical logic to questions in dynamical systems. Many dynamical systems, while completely determinate, appear to have elements of random behavior. This phenomenon can be described explicitly when there is a time-invariant probability measure on the system. Such a description could state that the system is measure theoretically isomorphic to a particular known process, such as a Bernoulli process. From this point of view it is natural to try to attempt to classify dynamical systems measure-theoretically. The hope would be to have a "library" of possible measure preserving systems and be able to describe an arbitrary system measure theoretically as one in the library. This project, while very successful in its early stages, runs into insuperable obstacles for deep logical reasons. Foreman's previous work with his co-authors showed that the isomorphism problem for ergodic measure preserving systems is inherently too complex from a logical point of view to admit a classification. Foreman's proposed work involves extending these anti-classification results to differentiable systems on compact manifolds and to identify those systems for which the isomorphism problem is tractable.
Many natural systems evolve over time according to definite rules. Much of science involves discovering these rules and describing them, perhaps by a system of equations. These rules discuss how individual points in a system behave, and are often completely deterministic. However, what can be actually observed (for example due to round-off error) in these systems are sets of points. At this level the qualitative behavior of a dynamical system can be apparently random. This led to the project of classifying the possible behavior statistically so that the qualitative behavior of natural systems could be catalogued. There were many successes in the program in its early stages. Recently however, it turns out that there are reasons related to mathematical logic that the program cannot, in principle, work. Foreman's proposed research explores extending the impossibility results to concrete settings and finding large collections of systems for which there are good classifications of their statistical behavior.
NSF award 07010310 was successful on many fronts. Intellectual merit: Progress was made in two general directions. The first solved a classical problem in Ergodic Theory dating to the 1930’s when von Neumann proposed classifying the ergodic measure preserving transformations up to conjugacy. The second direction was in combinatorial set theory, where there were advances in both singular cardinal combinatorics and in understanding the relationships between generic and conventional large cardinals. In 1932, von Neumann formulated the problem of classifying the ergodic measure preserving transformations of standard measure spaces up to measure theoretic isomorphism (conjugacy). Much work was done on this problem, including the development of spectral invariants (by Koopman, Von Neumann, Halmos and others) and the use of entropy to classify measure preserving transformations (including the work of Ornstein showing the entropy is a complete invariant for Bernoulli shifts). As this grant started, the Foreman, together with Dan Rudolph and Benjamin Weiss showed that the isomorphism relation is a complete analytic equivalence relation. In particular it is not Borel. (This paper was published in the Annals of Mathematics.) This left open several directions of research. One is to locate the conjugacy relation in the quasi-ordering of analytic equivalence relations under the Borel reducibility. Progress on this had already been made with a theorem of Foreman and Weiss that showed that the relation is turbulent. In particular, they showed it is not reducible to any orbit equivalence relation of a countable group. Foreman extended this further by showing that—in fact—the orbit equivalence relation of any countable group is Borel reducible to the conjugacy relation. Another extension adapts the abstract anti-classification result to concrete situations. Foreman and Weiss showed that the results that hold for general ergodic measure preserving transformations can be specialized to hold measure preserving diffeomorphisms (C-infinity) of the disk or the torus. This result can be split as follows: the first part is a canonical symbolic representation of all (untwisted) Anosov-Katok diffeomorphisms. The resulting class of symbolic measure preserving systems is dubbed "circular systems." The second part shows that there is a functor that takes odometer-based systems to circular systems and ergodic joinings to ergodic joinings. The third part adapts the Foreman-Rudolph-Weiss construction to the case of circular systems using the aforementioned functor. A final collection of results in this direction have to do with the invariance of genericity of classes of measure preserving systems. Foreman and Weiss isolated the notion of a model for the measure preserving systems and showed that all models have the same generic classes. This generalizes results of Rudoph, Glasner and Aaronson. The second direction of investigation was in Set Theory. Here Foreman continued his work on Singular Cardinal Combinatoris with a paper on Diagonal Prikry Sequences (with Cummings), and a paper that distinguishes between various notions of mutual stationarity (with Cummings and Schimmerling). The other work Foreman did was relating generic large cardinals with conventional large cardinals. He showed that there were "Chang Conjecture" properties of the first few uncountable cardinals that are equiconsistent with huge cardinals. This used another advance: a general technique for calculating the quotient algebra of a generic elementary embedding in a forcing extension. This technique is called The Duality Theorem. Foreman also showed that a classical theorem of Kunen forbidding saturated ideals in certain intervals was sharp. Broader Impact: Foreman gave several invited lectures at meetings. Most prominent was the Distinguished Lecture Series that Foreman gave at the Fields Institue in November 2012. In addition Foreman was co-editor of the 2400 page, 3-volume Handbook of Set Theory that appeared in 2010. One of the articles (264 pages) was written by Foreman and independently refereed. The grant supported 4 graduate students who got their Ph.D.’s. It also gave partial support for travel, meetings and speakers for 2 postdocs. This includes 2 women. One graduate student was involved in math education outreach activities.
