In this talk we outline the core group theoretic routine,
the "Local Certificates" algorithm, underlying the
new Graph Isomorphism test. The basic
strategy follows Luks's group-theoretic divide-and-conquer
approach (1980). We address the bottleneck of Luks's
technique via local-global interaction based on a
new group theoretic lemma.
Undergraduate-level familiarity with the basic concept of
group theory (homomorphism, kernel, quotient group,
permutation groups) will be assumed.