Core model theory centers around the construction and study of canonical inner models satisfying large cardinal hypotheses. One of the most important open problems in pure set theory, the problem of constructing canonical inner models with supercompact cardinals, belongs to this realm.
In order to construct a model of the large cardinal hypothesis H, one must make an assumption of consistency strength at least that of H. It is usually simplest to assume H itself. In this case we have a model-construction which should work as far as H = ``there is a superstrong cardinal'', and can be shown to work as far as H = ``there is a Woodin limit of Woodin cardinals''. The missing ingredient is a proof of the strategic branches hypothesis, or SBH, which asserts the existence of iteration strategies for countable elementary submodels of the universe V of all sets. SBH is also crucial to one natural attempt to prove Woodin's Omega-conjecture.
In 1986, Martin and Steel isolated two basic hypotheses concerning iteration, the unique branches hypothesis UBH and the cofinal branches hypothesis CBH. They showed (roughly) that UBH implies CBH, and CBH implies SBH. They also proved a fragment of UBH sufficient to construct iteration strategies for inner models with one Woodin cardinal. This work was subsequently extended to models with Woodin limits of Woodin cardinals. On the other hand, in the late 80's Woodin refuted UBH, in work that has never been published, and is not widely known. Very recently, via a different and simpler construction, Woodin has refuted CBH. Again, this work is not written up anywhere.
Woodin's recent refutation of CBH is an important breakthrough, and one major goal of the workshop is to get a good understanding of it. Beyond that, we hope to get some idea how one might retreat from CBH, and yet still get a proof of SBH.
In many applications, it is important to construct an inner model of large cardinal hypothesis H under an assumption that is not itself a large cardinal hypothesis. This is more difficult, often involving the deep and extensive machinery associated to Jensen's Covering Theorem and its generalizations. In case H is significantly stronger than ``there is one Woodin cardinal'', the connection between large cardinals and models of the axiom of determinacy, or AD, becomes quite important. (Essentially this is because the relevant iteration strategies are coded by definable sets of reals, and constructed by induction on their Wadge complexity.) Here Woodin has made another important recent advance over the last several years, by extending Steel's analysis of HODL(R) to an analysis of HODM for arbitrary models M of AD+. It turns out that HODM is a canonical inner model with large cardinals, but not of the sort which has been previously considered. This new kind of inner model looks to be an important tool in constructing the more familiar kind, and the second major goal of the workshop is to understand these new models, and how they might be used.
Woodin has agreed to lecture on his work at the workshop. In addition, we will have one or two other lecturers (most probably including Steel) expositing parts of this work, or material upon which it rests. We shall also produce a clear written account of this work immediately after the workshop concludes. Our hope is that the wider dissemination of Woodin's work will lead to further advances on the basic problem of constructing inner models with supercompact cardinals.