Abstract: It is a basic open question in geometric measure theory to understand regularity of a stationary integral varifold. Even a.e. regularity remains an open question. The central issue is analyzing the varifold near a point Z with a tangent plane of multiplicity q > 1. The talk will focus on the analytic machinery developed recently to address this question when q=2 for two cases: (i) the varifold corresponds to a Lipschitz two-valued graph of arbitrary codimension (joint work with S. Becker-Kahn) (ii) the varifold has codimension 1, stable regular part and no density 3/2 points near Z (this last condition is met if e.g. the varifold corresponds to a rectifiable current).