Explore the third lecture in a six-part series focusing on the mixing properties of horocycle flows and their time-changes, with particular emphasis on the "mixing via shearing" method and equidistribution of transverse curves. Delve into advanced mathematical concepts including homogeneous unipotent flows and uniformly parabolic dynamics, examining how smooth time-changes affect these systems. Learn about Ratner's result on mixing rates while understanding the fundamental principles of geometric observation, bounded integrals, and equidistribution of curves. Master the mathematical framework surrounding smooth Archomogeneous flows and sigma change variables through detailed proofs and practical applications. Part of the Simons Semester on Dynamics, this advanced mathematics lecture builds upon previous discussions of homogeneous flows, Haar measure, and PSL2(R) action on the hyperbolic plane.