Almost-Prime Times in Horospherical Flows

There is a rich connection between homogeneous dynamics and number theory. Often in such applications it is desirable for dynamical results to be effective (i.e. the rate of convergence for dynamical phenomena are known). In the first part of this talk, I will provide the necessary background and relevant history to state an effective equidistribution result for horospherical flows on the space of unimodular lattices in $\mathbb{R}^n$. I will then describe an application to studying the distribution of almost-prime times (integer times having fewer than a fixed number of prime factors) in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture. In the second part of the talk I will describe some of the ingredients and key steps that go into proving these results.