Explore the fascinating intersection of Diophantine approximation, fractals, and homogeneous flows in this 48-minute lecture by Osama Khalil. Delve into the theory of Diophantine approximation, underpinned by Dirichlet's fundamental theorem, and examine the prevalence of points with exceptional behavior. Investigate badly approximable, very well approximable, and Dirichlet-improvable points as key exceptional sets. Discover the connections between these concepts and the recurrence behavior of certain flows on homogeneous spaces, as established by Dani and Kleinbock-Margulis. Learn about new results providing a sharp upper bound on the Hausdorff dimension of divergent orbits of specific diagonal flows emanating from fractals on the space of lattices. Gain insights into the theory of projections of self-similar measures and their relevance to the topic. This lecture, part of the Hausdorff Trimester Program "Dynamics: Topology and Numbers" conference, offers a deep dive into advanced mathematical concepts at the intersection of number theory, dynamical systems, and fractal geometry. Read more