The investigator is engaged in research in various areas of set theory. One of his main ongoing research interests concerns the theory of the model L(R) with the axiom of determinacy. The sets of reals in L(R) form a far-reaching extension of the projective sets, and the goal of obtaining a more or less complete theory of this model with the axiom of determinacy (which is a natural strong set theoretic axiom for this model) is one of his central interests. Secondly, he is working on problems connected with the theory of equivalence relations in ZFC. Recently, R. Dougherty, A.S. Kechris (who introduced the term "descriptive dynamics" for this area) and the investigator obtained some results which suggest the beginnings of a possible classification for the Borel equivalence relations with countable classes. There are many open problems here, such as identifying a canonical cofinal hierarchy of equivalence relations. Thirdly, some problems of descriptive set theory of a more classical nature have arisen recently. D. Mauldin and the investigator have recently obtained a non-uniformization type result for co-analytic sets. There are open problems here, such as how far these results can be extended in ZF, as well as other problems in descriptive set theory which he will work on. Finally, he would like to devote some time to problems which have arisen from the inner model theory for large cardinals, such as questions about iteration trees. There is nothing more basic in mathematics than set theory. It is arguable that the natural numbers (0,1,2,3,...) are no more fundamental. For this reason it is perhaps surprising to find questions about these fundamental objects under active investigation and to see that even the choice of axioms for a theory of sets is controversial.
