# Mathematics Colloquia and Seminars

I discuss recent work with Moritz Reintjes in which we derive the {\it Regularity Transformation equations}, a system of nonlinear {\it elliptic} equations with matrix valued differential forms as unknowns. We prove that existence of solutions to the RT-equations is equivalent to the existence of a coordinate transformation sufficient to smooth a crinkled map of spacetime to optimal metric regularity in General Relativity. We then give an existence theorem for the RT-equations based on elliptic regularity in $L^p$ spaces, and as a consequence show that if a connection and its curvature tensor are both in $W^{m,p}$, $m\geq1$, $p>n,$ then there always exists a coordinate transformation with Jacobian in $W^{m+1,p}$, such that in the new coordinate system, the connection is in $W^{m+1,p}$. This tells us that we can solve the Einstein equations in coordinate systems in which the equations are simpler and solution metrics are one order below optimal, and still be guaranteed the existence of other coordinate systems in which the metric exhibits optimal regularity, i.e., two derivatives above its curvature tensor. When connection and curvature are in $L^{\infty}$, the RT-equations reduce the open problem of {\it regularity singularities} at GR shock waves to an approachable problem in elliptic regularity theory, a topic of authors' current research. The starting point for the derivation of the RT-equations is the {\it Riemann-flat condition}, a geometric condition for metric smoothing introduced previously by the authors.