Explore Sarason's Ha-plitz product problem in this 53-minute lecture by Kehe Zhu from the State University of New York at Albany. Delivered as part of the Focus Program on Analytic Function Spaces and their Applications at the Fields Institute, delve into various aspects of this mathematical challenge. Begin with an introduction to the relevant spaces and the problem itself, then progress through topics such as bounded operators, harmonic analysis, and partial results. Examine the Fox base and wire operators, and investigate companion problems and non-trivial issues. Discover surprising elements of the problem before concluding with a comprehensive summary of the findings and implications in this advanced mathematical exploration.