9501176 Cherlin The project concerns the interaction of group theory (finite and algebraic) with model theory and set theory, as well as with some combinatorial issues that involve unusual combinatorial geometries. Cherlin will investigate a class of groups arising in model theory, whose simple sections are conjectured to be algebraic groups. Within this class, the class of "tame omega-stable groups" provides a test case in which it is conjectured -- the Borovik philosophy -- that a modification of the strategy used in the classification of the finite simple groups will suffice to prove the conjecture. This amounts in practice to doing somewhat more than the classification of simple algebraic groups over algebraically closed fields, using methods of finite group theory, and very little algebraic geometry beyond rudimentary properties of dimension. The Borovik philosophy has been fleshed out fairly clearly in odd characteristic, and requires similar scrutiny in characteristic two. (The definitions of even and odd characteristic are group theoretic but quite different from the ones used in finite group theory.) Thomas will continue the study of the cofinalities of permutation groups. This investigation goes back to a question of Serre, the precise answer of which is heavily dependent on set theoretic hypotheses. Results in this area tend to rely heavily on delicate set theoretic forcing constructions and also involve character theoretic computations relating to rapid generation by conjugacy classes. There are also relationships, largely obscure, between these group theoretic cofinalities and more standard set theoretic cardinal invariants attached to the continuum. Most recently it has been seen that there are also solid connections to pcf theory, an area of set theory developed by Saharon Shelah. This theory is seen as a new approach to cardinal arithmetic that is less affected by independence phenomena than the classical app roach. Model theory attempts to classify mathematical theories. The "best" theories are those whose models (realizations) can be given structural classifications. Twenty years ago Zilber showed that all such theories can be built from familiar algebraic structures: groups. Cherlin and Zilber then conjectured a classification scheme for the groups that arise in this way. One promising route is to mine the literature on finite simple groups. As a result of the combined labors of about one hundred mathematicians in journal articles filling about 10,000 highly condensed journal pages, the finite simple groups were classified. A parallel strategy is developed for a large class of the infinite groups arising in model theory. This should require considerably less than 100 mathematicians or 10,000 journal pages, for two reasons: (1) much of the work in the finite case relates to the properties of 26 bizarre finite groups which are not involved to any significant extent in the infinite case; (2) the somewhat haphazard evolution of the original proof in the finite case, over more than two decades, can be avoided on the basis of lessons learned from that proof. It is nonetheless a large-scale project, and large-scale planning will be critical to its success. At the present time the plan, though still incomplete, is already being implemented. Cherlin will address the urgent need for a "top-level" analysis to serve as a guide for the mathematicians working on this strategy in parts of the U.S. and also in South America and Europe. Set theory has developed rapidly and steadily since Paul Cohen settled some of the oldest questions in the subject in his 1963 breakthrough. Many areas of mathematics involve surprisingly deep questions in set theory. Recently this has been shown by Thomas to be the case in the study of the structure of those infinite groups which are best approximated by finite groups. Work in this area has two quite different aspects which mesh toge ther well. This subject serves as a site in which to develop new set theoretic methods, providing a new link between computational methods in algebra and set theory. ***
