Explore the intricacies of random matrix models in this high-energy theory seminar presented by Henry Lin from Princeton University. Delve into the fundamentals of matrix models, their importance, and the concept of matrix integrals. Examine the complexities of two-matrix models and loop equations. Investigate the search space and positivity constraints, including the Hamburg moment problem and single finite constraints. Analyze multicut solutions and the relaxed positivity constraint, with a focus on single matrix explanation. Discover potential future directions in the field, including double cut and Isin model applications. Gain valuable insights into solving random matrix models using positivity techniques in this comprehensive lecture from the Institute for Advanced Study.