This  concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem)  with the fundamental group of the variety.
 In particular, this leads to new examples (of surfaces) of  failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties).

Dynatomic curves parametrize n-periodic orbits of a one-parameter family of polynomial dynamical systems. These curves lack the structure of their arithmetic-geometric analogues (modular curves of level n) but can be studied dynamically.  Morton and Silverman conjectured a dynamical analogue of the uniform boundedness conjecture (theorems of Mazur, Merel), asserting uniform bounds for the number of rational periodic points for such a family.  I will discuss recent work towards the function field version of their conjecture, including results on the reduction mod p of dynatomic curves for the quadratic polynomial family z^2+c.

Building on advancements in computer vision we now have an array of visual tracking methods that allow the reliable estimation of cellular motion in high-throughput settings as well as more complex biological specimens. In many cases the underlying assumptions of these methods are still not well defined and result in failures when analysing large scale experiments.

Using organotypic co-culture systems we can now mimic more physiologically relevant microenvironments in vitro.  The robust analysis of cellular dynamics in such complex biological systems remains an open challenge. I will attempt to outline some of these challenges and provide some very preliminary results on analysing more complex cellular behaviours.