Explore a comprehensive overview of recent developments in the invariant subspace problem through this 55-minute lecture by Bernard Chevreau from the University of Bordeaux. Delve into the historical context, proof techniques, and key concepts such as reflexivity and hypercyclic operators. Examine the Scott Brown method, classical interpolation, and universal operators while gaining insights into the dual algebra techniques and halfspace considerations. Understand the significance of this problem in the field of analytic function spaces and its applications as part of the Focus Program at the Fields Institute.