The goal of the talk is to give an overview of some results about algebraic and geometric structures in orbits of polynomial dynamical systems.
More precisely, given a polynomial $f$ over a field $K$ and a structural set $S \subset K$ defined in terms unrelated to $f$, it is natural to expect that the orbits of $f$ have a finite intersection with $S$. For example, when $K$ is a number field, we will consider $S$ to be the set of integral points, the set of units, or a finitely generated group. If $S$ is an orbit of another polynomial this is known as a problem about orbit intersections, which has recently been studied by Ghioca, Tucker and Zieve. One can also consider the multivariate generalisation of this question. For example, if the set $S$ is an algebraic variety, this falls within the so-called dynamical Mordell-Lang conjecture.
We are interested in finiteness results or, failing this, in bounding the frequency of such intersections, both in the zero and positive characteristics, as well as for both univariate and multivariate cases. We shall also discuss several open questions in this direction.