Poonen's Closed Point Sieve has proven to be a powerful technique for producing structural and combinatorial results for varieties over finite fields. In this talk we discuss three results which come, in part, as a consequence of this technique. First we will discuss semiample Bertini Theorems over finite fields and examine the probability with which a semiample complete intersection is smooth. Next we apply the Closed Point Sieve to compute the probability with which a high degree projective hypersurface over $\mathbb{F}_{2^k}$ is locally Frobenius split (a characteristic $p$ analog of log canonical singularities). In doing so we show that most such hypersurfaces are only mildly singular. The final part, which is based on joint work with Kiran Kedlaya and James Upton, discusses $p$-adic coefficient objects in rigid cohomology. Namely, we show (under a geometric tameness hypothesis) that the overconvergence of a Frobenius isocrystal can be detected by the restriction of that isocrystal to the collection of smooth curves on a variety.

We present recent results about finite jet determination of CR maps of positive codimension from real-analytic CR manifolds into real-algebraic subsets in complex space, or more generally Nash subsets. One instance of such results is the unique jet determination of germs of CR maps from minimal real-analytic CR submanifolds in $\C^N$ into Nash subsets in $\C^{N'}$ of D'Angelo finite type, for arbitrary $N,N'\geq 2$. This is joint work with B. Lamel.