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Let $\lambda$ be a limit of Woodin cardinals. It was shown by the second author that the pointclass of ${<\lambda}$-homogeneously Suslin sets has the scale property. We give a new proof of this fact, which avoids the use of stationary tower forcing., Comment: 11 pages

Assume ZF + AD + V=L(R). Let $[\alpha,\beta]$ be a $\Sigma_1$ gap with $J_\alpha(R)$ admissible. We analyze $J_\beta(R)$ as a natural form of ``derived model'' of a premouse $P$, where $P$ is found in a generic extension of $V$. In particular, we will have $\mathcal{P}(R)\cap J_\beta(R)=\mathcal{P}(R)\cap D$, and if $J_\beta(R)\models$``$\Theta$ exists'', then $J_\beta(R)$ and $D$ in fact have the same universe. This analysis will be employed in further work, yet to appear, toward a resolution of a conjecture of Rudominer and Steel on the nature of $(L(R))^M$, for $\omega$-small mice $M$. We also establish some preliminary work toward this conjecture in the present paper., Comment: 128 pages

We develop the theory of meta-iteration trees, that is, iteration trees whose base "model" is itself an ordinary iteration tree. We prove a comparison theorem for meta-iteration strategies parallel to the one for ordinary iteration strategies, and use it to show that the iteration strategy component of a mouse pair condenses to itself under weak tree embeddings. These constitute a class of embeddings between iteration trees that is significantly larger than the class of embeddings mentioned in the definition of mouse pair. We then use this very strong hull condensation property of mouse pairs to show that every iterate of a mouse pair is an iterate via a single $\lambda$-tight, normal iteration tree, and that the associated tail strategies are independent of how the iterate was reached., Comment: 105 pages

We obtain a partial result on the following conjecture. Conjecture. Let (P, {\Sigma}) be a projectum stable mouse pair, and let \kappa be a cardinal of V such that \kappa < o(M_\infty(P, {\Sigma})); then the following are equivalent: (1) \kappa is a Suslin cardinal, (2) \kappa is a cutpoint of M_\infty(P, {\Sigma}).

In this paper, we prove a fine condensation theorem. This is quite similar to condensation theorems for pure extender mice in the literature, except that condensation for iteration strategies has been added to the mix.

Recall that the Mouse Set Conjecture says that under AD++V=L(P(R)), a real is ordinal definable if and only if it belongs to an iterable mouse. The Mouse Set Conjecture for sets of reals says that under the same theory, a set of reals is ordinal definable from a real if and only if it belongs to a mouse over the reals. We prove that the Mouse Set Conjecture implies the Mouse Set Conjecture for sets of reals.

Let M be a fine structural mouse. Let D be a fully backgrounded L[E]-construction computed inside an iterable coarse premouse S. We describe a process comparing M with D, through forming iteration trees on M and on S. We then prove that this process succeeds.

By forcing with $\mathbb{P}_{\rm max}$ over strong models of determinacy, we obtain models where different square principles at $\omega_2$ and $\omega_3$ fail. In particular, we obtain a model of $2^{\aleph_0}=2^{\aleph_1}=\aleph_2 + \lnot\square(\omega_2) + \lnot\square(\omega_3)$., Comment: Revised version, incorporating the referee's suggestions

There is a fascinating interplay and overlap between recursion theory and descriptive set theory. A particularly beautiful source of such interaction has been Martin's conjecture on Turing invariant functions. This longstanding open problem in recursion theory has connected to many problems in descriptive set theory, particularly in the theory of countable Borel equivalence relations. In this paper, we shall give an overview of some work that has been done on Martin's conjecture, and applications that it has had in descriptive set theory. We will present a long unpublished result of Slaman and Steel that arithmetic equivalence is a universal countable Borel equivalence relation. This theorem has interesting corollaries for the theory of universal countable Borel equivalence relations in general. We end with some open problems, and directions for future research., Comment: Corrected typos

If T is an iteration tree on K and F is a countably certified extender that coheres with the final model of T, then F is on the extender sequence of the final model of T. Several applications of maximality are proved, including: o K computes successors of weakly compact cardinals correctly. o K^c is an iterate of K. o (with Mitchell) If alpha is a cardinal > aleph_1, then K-restriction-alpha is universal for mice of height alpha. Other results in this paper, when combined with work of Woodin, imply: o If square-kappa-finite fails and kappa is a singular, strong limit cardinal, then Inductive Determinacy holds. o If square-kappa-finite fails and kappa is a weakly compact cardinal, then L(R)-determinacy holds.

A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model. Assume that kappa is a countably closed cardinal and that alpha is a successor cardinal of K with kappa < alpha < kappa^+. Then cf( alpha ) = kappa. In particular, K computes successors of countably closed singular cardinals correctly. (The hypothesis of countable closure is not required; see "Weak covering without countable closure", W. J. Mitchell and E. Schimmerling, Math. Res. Lett., Vol. 2, No. 5, Sept. 1995.)

We introduce two new iteration games: the game G, which is a strengthening of the weak iteration game, and the game G+, which is somewhat stronger than G but weaker than the full iteration game of length omega_1. For a countable M elementarily embeddable in some V_{eta}, we can show that II wins G(M,omega_1) and that I does not win the G+(M).