Integral Hasse principle and strong approximation for Markoff surfaces

In 2017 Ghosh and Sarnak conjectured how often the integral Hasse
principle should fail in a family of Markoff surfaces with evidence based
on numerical experiments. Using reduction theory they were able to obtain a
lower bound which is of magnitude still far away from the expected size. In
this talk we shall explain what can be achieved with techniques from
Arithmetic geometry. In particular, we get sharp bounds for the number of
failures of the integral Hasse principle explained by the Brauer-Manin
obstruction and a lower bound for the number of failures which are not
explained by the Brauer-Manin obstruction. This talk is based on a join
work with Dan Loughran.