Explore the intricacies of Heegaard Floer homology and its applications in topology across dimensions 3.5 to 2 in this comprehensive lecture. Delivered by Jen HOM from Georgia Tech, USA, as part of the Frontiers in Geometry and Topology Summer School, delve into advanced mathematical concepts and their implications in the field. Gain insights into cutting-edge research and theoretical frameworks that bridge the gap between different dimensional topological spaces.