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 […]

A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping psi : B_1