Non-accumulation of periodic torus orbits

The `linearization’ technique is a powerful method in homogeneous dynamics to control the time a unipotent orbit spends in the vicinity of a closed homogeneous subset. This method relies on the polynomial nature of a unipotent flow and does not extend to diagonalizable actions.

I will describe a new arithmetic approach to bound the accumulation of periodic orbits of higher-rank diagonalizable groups. This method plays a major role in the recent progress on the Michel-Venkatesh mixing conjecture.