Counting and dynamics in $\mathrm{SL}_2$

In this talk I'll discuss a lattice point count for a thin semigroup inside $\mathrm{SL}_2(\mathbb Z)$. It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes. The approach to counting is dynamical---with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play​. At finite places the necessary expansion property follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli). These are underpinned by tripling estimates in $\mathrm{SL}_2(\mathbb F_p)$ due to Helfgott. I'll try to explain in simple terms the key dynamical facts behind all of these methods. This talk is based on joint work with Hee Oh and Dale Winter.