In this report, I will introduce the large data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension d>=2. This is achieved by introducing several new ideas: (i) a time discretization method to establish the existence of smooth solutions, (ii) constructing the orthonormal frame by a parallel transport method and a lifting criterion, (iii) introducing intrinsic fractional function spaces, (iv) deriving a difference equation to prove the uniqueness result. This is based on joint work with Daniel Tataru.