Given a family $F$ of elliptic curves defined over $Q$, we are interested in the set $H(Y)$ of curves $E$ in $F$, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on $E$ is bounded by some parameter $Y$. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set $H(Y)$. We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.

following the joint paper with L.Shaheen http://people.maths.ox.ac.uk/zilber/wLb.pdf

The representation dimension of an algebra was introduced in the early 70's by M. Auslander, with the goal of measuring how far an algebra is from having finite number of finitely generated indecomposable modules (up to isomorphism). This invariant is not well understood. For instance, it was not until 2002 that O. Iyama proved that every algebra has finite representation dimension. This was done by constructing special quasihereditary algebras. In this talk I will give an introduction to this topic and I shall briefly explain Iyama's construction.

In quite an elementary, hands-on talk, I will discuss some Ax-Lindemann type results in the setting of modular functions.  There are some very powerful results in this area due to Pila, but in nonclassical variants we have only quite weak results, for a rather silly reason to be discussed in the talk.