Non-Relativistic Intersecting Branes, Newton-Cartan Geometry and AdS/CFT

We discuss non-relativistic variants of four-dimensional ${\cal N}$=4 super-Yang-Mills theory obtained from generalised Newton-Cartan geometric limits of D3-branes in ten-dimensional spacetime. We argue that the natural interpretation of these limits is that they correspond to non-relativistic D1-branes or D3-branes intersecting the original D3-branes. The resulting gauge theories have dynamics that reduce to quantum mechanics on monopole moduli space or two-dimensional sigma-models on Hitchin moduli space respectively. We show that these theories possess interesting infinite-dimensional symmetries and we discuss the dual $AdS$ geometries.