We show that given $\omega$ many supercompact cardinals, there is a
generic extension in which there are no Aronszajn trees at
$\aleph_{\omega+1}$. This is an improvement of the large cardinal
assumptions. The previous hypothesis was a huge cardinal and $\omega$ many
supercompact cardinals above it, in Magidor-Shelah.

I will present the proofs of some recent results of Viale
and Weiss. Weiss introduced the notion of a slender function in his
dissertation: roughly, a function $M \mapsto F(M) \subset M$ (where
$M$ models a fragment of set theory) is slender iff for every
countable $Z \in M$, $Z \cap F(M) \in M$; i.e. $M$ can see countable
fragments of $F(M)$. Viale and Weiss proved that under the Proper
Forcing Axiom, for every regular $\theta \ge \omega_2$, there are
stationarily many $M \in P_{\omega_2}(H_{(2^\theta)^+})$ which
``catch'' $F(M \cap H_\theta)$ whenever $F$ is slender (i.e. whenever
$F$ is slender then there is some $X_F \in M$ such that $F(M \cap
H_\theta) = M \cap X_F$). The stationarity of this collection implies
many of the known consequences of PFA; e.g. failure of weak square at
every regular $\theta \ge \omega_2$; and separating internally
approachable sets from sets of uniform uncountable cofinality.

The graph G_0 was introduced by Kechris-Solecki-Todorcevic in the late 90s,
and has since turned into an essential object in descriptive set theory. In
joint work with Richard Ketchersid, we prove a version of the G_0-dichotomy
in models of AD^+. This is then used to establish that the quotient by the
equivalence relation E_0 is a successor of R, a result previously known
under AD_R, but (perhaps surprisingly) not in L(R).