  We announce some results which might bring a new insight into the
classification of algebraic solutions to the sixth Painleve equation. The main
results consist of the rationality of parameters, trigonometric Diophantine
conditions, and what the author calls the Tetrahedral Theorem regarding the
absence of algebraic solutions in certain situations. The method is based on
fruitful interactions between the moduli theoretical formulation of Painleve VI
and dynamics on character varieties via the Riemann-Hilbert correspondence.
