Activities This Week

The Zilber-Pink conjecture is a far reaching and widely open conjecture in the field of unlikely intersections generalizing many previous results in the area such as the Andre-Oort conjecture. We discuss this conjecture and how some cases of it can be established for curves in $\mathcal{A}_g$, the moduli space of principally polarized g-dimensional abelian varieties, following the Pila-Zannier strategy and bounds for the values of the Weil height at certain exceptional points of the curve.

We consider the family of periodic measures for the full diagonal action on the space of unimodular lattices. This family is important and natural due to its tight relation to class groups in number fields. We show that many natural families of measures on the space of lattices can be approximated using this family (in the weak sense). E.g., we show that for any 0<c\leq 1, the measure cm_{X_n} can be approximated this way, where m_{X_n} denotes the Haar probability measure on X_n. Moreover, we show that non ergodic measures can be approximated. Our proof is based on the equidistribution of Hecke neighbors and on constructions of special number fields. We will discuss the results, alternative ways to attack the problem, and our method of proof. This talk is based on a joint work with Omri Solan.