I will discuss some recent progress on the freeness problem for groups of 2x2 rational matrices generated by two parabolic matrices. In particular, I will discuss recent progress on determining the structural properties of such groups (beyond freeness) and when they have finite index in the finitely presented group SL(2,Z[1/m]), for appropriately chosen m.

Using the invariant splitting principle, we construct an infinite family of exotic pairs of contractible 4-manifolds which survive one stabilization. We argue that some of them are potential candidates for surviving two stabilizations.

We introduce bordered Floer theory and its involutive version, as well as their applications to knot complements. We will sketch the proof that invariant splittings of CFK and those of CFD correspond to each other under the Lipshitz-Ozsvath-Thurston correspondence, via invariant splitting principle, which is an ongoing work with Gary Guth.

The Julia set of a (hyperbolic) rational map naturally comes embedded in the Riemann sphere, and thus has a Hausdorff dimension. But the Hausdorff dimension varies if we tweak the parameters slightly. Is there a "best" representative or more invariant dimension? One answer comes from looking at quasi-symmetries; the \emph{conformal dimension} of the Julia set is the minimum Hausdorff dimension of any metri quasi-symmetric to the original. We characterize the Ahlfors-regular conformal dimension of Julia sets of rational maps using graphical energies arising from a natural combinatorial description. (Ahlfors-regular is a dynamically natural extra condition on the metric.) This is joint work with Kevin Pilgrim.