In the past few years, conjectures have been made and partial results achieved on unlikely intersections in complex dynamics, following the program initiated by Baker-DeMarco. I will explain their dynamical generalization of the famous André-Oort conjecture on CM points in moduli spaces of abelian varieties. Ghioca, Nguyen, Ye, and myself recently proved the first complete case of this conjecture, for pairs of unicritical polynomials, and I will discuss our result and the connection to invariant subvarieties of $P1 \times P1$, and the structure of the Mandelbrot set.[-]

Fatou noticed in 1926 that certain transcendental entire functions have Julia sets in which there are curves of points that escape to infinity under iteration and he wondered whether this might hold for a more general class of functions. In 1989, Eremenko carried out an investigation of the escaping set of a transcendental entire function f, $I(f)=\left \{ z\in\mathbb{C}:\left | f^{n}\left ( z \right ) \right | \rightarrow \infty \right \}$ and produced a conjecture with a weak and a strong form. The strong form asks if every point in the escaping set of an arbitrary transcendental entire function can be joined to infinity by a curve in the escaping set.
This was answered in the negative by the 2011 paper of Rottenfusser, Rückert, Rempe, and Schleicher (RRRS) by constructing a tract that produces a function that cannot contain such a curve. In the same paper, it was also shown that if the function was of finite order, that is, log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )$ as $\left | z \right |\rightarrow \infty$, then every point in the escaping set can indeed be connected to infinity by a curve in the escaping set.
The counterexample $f$ used in the RRRS paper has growth such that log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )^{k}$ where $K > 12$ is an arbitrary constant. The question is, can this exponent, K, be decreased and can explicit calculations and counterexamples be performed and constructed that improve on this?[-]