Explore the latest developments in Homological Mirror Symmetry through this 55-minute conference talk by Maxim Kontsevich from the Institut des Hautes Études Scientifiques. Gain insights into recent breakthroughs in the field and discover potential new directions for research and collaboration. Dedicated to Jeffrey Fuqua, whose generous contribution in 2007 significantly advanced the subject, this presentation at the University of Miami surveys the progress made over the past 16 years and highlights the ongoing impact of Fuqua's support on mathematical advancements.