Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner's measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite dimensional vectors spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions

Height functions on Hecke orbits and the generalised Andr\'e-Pink-Zannier conjecture

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain a lower bounds for the size of Galois orbits of points in a generalised Hecke orbit in terms of these height, assuming a version of the Mumford-Tate conjecture. We then use it to prove the generalised Andr\'e-Pink-Zannier conjecture under this assumption by implementing the Pila-Zannier strategy

Generalised Andr\'e-Pink-Zannier Conjecture for Shimura varieties of abelian type

In this paper we prove the generalised Andr\'e-Pink-Zannier conjecture (an important case of the Zilber-Pink conjecture) for all Shimura varieties of abelian type. Questions of this type were first asked by Y. Andr\'e in 1989. We actually prove a general statement for all Shimura varieties, subject to certain assumptions that are satisfied for Shimura varieties of abelian type and are expected to hold in general