Explore a 51-minute lecture on the complexity of log-concave poset inequalities presented by Swee Hong Chan from Rutgers University at IPAM's Integrability and Algebraic Combinatorics Workshop. Delve into the concept of log-concavity in mathematics, examining its occurrence in various contexts from simple binomial coefficients to complex Stanley's inequality for linear extensions of posets. Investigate a rigorous framework that combines combinatorics, complexity theory, and geometry to assess the intrinsic difficulty of different types of log-concave inequalities. Gain insights into the joint work of Chan and Igor Pak as they explore the varying levels of complexity in proving log-concave inequalities and their implications in mathematical research.