Given a lightface $\Sigma^1_2$ set of reals A we present the construction of a tree on $\omega\times\omega_1$ such that A is the projection of T. Moreover, the tree T is an element of any transitive model of ZF-PowerSetAxiom that has $\omega_1$ as element.

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.

There are several examples in the literature where compactness properties of a cardinal $\kappa$ imply "bad" behavior of certain generic ultrapowers with critical point $\kappa$; particularly generic ultrapowers associated with tower forcings (Woodin's stationary tower forcing is an example of a tower forcing). I will discuss instances of this phenomenon due to Burke, Foreman-Magidor, and Cox-Viale.

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 $V$.

In recent years there have been striking applications of infinitary
methods in finite combinatorics. I will survey some of these methods,
notably the theory of "flag algebras" due to Razborov.