  We describe three algorithms to determine the stable, semistable, and
torus-polystable loci of the GIT quotient of a projective variety by a
reductive group. The algorithms are efficient when the group is semisimple. By
using an implementation of our algorithms for simple groups, we provide several
applications to the moduli theory of algebraic varieties, including the
K-moduli of algebraic varieties, the moduli of algebraic curves and the Mukai
models of the moduli space of curves for low genus. We also discuss a number of
potential improvements and some natural open problems arising from this work.
