We will introduce the "lightface" projective hierarchy and examine it both from syntactical and semantical aspect. "Lightface" \Sigma^0_1" sets are effective versions of open sets. We also prove that lightface \Sigma^0_1 sets of reals can be represented as sets of branches of recursive trees, and lithtface \Sigma^1_1 sets can be represented as projections of recursive trees.

I will discuss the Diagonal Reflection Principle (DRP), which is a highly simultaneous form of stationary set reflection that follows from strong forcing axioms like $PFA^{+\omega_1}$. DRP can be viewed as a weaker version of the statement "there is a normal ideal with completeness $\omega_2$ whose associated poset is proper (i.e. preserves stationary subsets of $[X]^\omega$ for all $X$)". In the presence of sufficiently absolute partitions of $cof(\omega)$, such ideals yield generic embeddings $j: V \to_G ult(V,G)$ with critical point $\omega_2$ such that large portions of $j$ are visible to $V$.

We use the extent to which the combinatorial principle "weak
square" holds to quantify the influence of Martin's Maximum
on the combinatorics of singular cardinals.
(joint work with Menachem Magidor)

Which consistent statements can be forced to be true?
It is shown that "resectionable" \Sigma_1 statements about parameters
in H_{\omega_2} which are "honestly consistent" can be forced
to be true in a stationary set preserving extension, and we also show
that a strong form of BMM, according to which all "honestly consistent"
\Sigma_1 statements about parameters in H_{\omega_2} are true,
is consistent. We also give some applications.