The dynamical Mordell-Lang conjecture in characteristic zero predicts that if f : X --> X is a map of algebraic varieties over a field K of characteristic zero, Y subset X is a closed subvariety and a in X(K) is a K-rational point on X, then the return set { n in N : f^n(a) in Y(K) } is a finite union of points and arithmetic progressions. For K a field of characteristic p > 0, it is necessary to allow for finite unions with sets of the form { […]

(Joint work with A. Chernikov)Let V subseteq C^3 be a complex variety of dimension 2.The Elekes-Szabo Theorem says that if V contains `too many' points on n x n x n Cartesian products then V has a special form: either V contains a cylinder over a curve or V is related to the graph of the multiplication of an algebraic group.In this talk we generalize the Elekes-Szabo Theorem to relations on strongly minimal sets interpretable in distal structures.

To every group G is associated an associative algebra, namely its group ring kG. Which (geometric) properties are reflected in (algebraic) properties of kG? I will survey some results and conjectures in this area, concentrating on specific examples: growth, amenability, torsion, and filtrations.

About fifteen years ago, Thomas Scanlon and I gave a description of sets that arise as the intersection of a subvariety with a finitely generated subgroup inside a semiabelian variety over a finite field. Inspired by later work of Derksen on the positive characteristic Skolem-Mahler-Lech theorem, which turns out to be a special case, Jason Bell and I have recently recast those results in terms of finite automata. I will report on this work, as well as on the work-in-progress it has engendered on an effective version of the isotrivial […]