Explore the dynamics of Lorenz maps in this 54-minute lecture from the Simons Semester on Dynamics. Delve into the world of piecewise monotone interval maps with a single discontinuity, examining their connections to the Lorenz attractor, number theory, and fractal geometry. Focus on renormalizations of expanding Lorenz maps, investigating return maps to smaller intervals around the discontinuity. Learn about Y. Ding's 2011 paper on the correspondence between renormalizations and proper completely invariant closed sets, and discover why this result is not universally applicable. Examine counterexamples and efforts to correct Ding's statement, based on joint work with Piotr Oprocha. Gain insights into the complex behavior of these mathematical objects and their significance in dynamical systems theory.