Explore the mathematical foundations of self-affine systems in this 45-minute lecture from the Simons Semester on Dynamics series. Delve into the dimension theory of iterated function systems (IFS), moving beyond well-understood conformal transformations to examine the more complex realm of non-conformal maps, particularly affine transformations. Learn how to calculate the Hausdorff dimension of planar self-affine sets under specific conditions of strong separation of cylinders and strong irreducibility of linear parts. Study both foundational work by Falconer and recent mathematical developments through collaborative research with Antti Kaenmaki, Mike Hochman, and Ariel Rapaport.