Explore a lecture on product mixing, quantum separation, and comfortable juntas in the context of the special unitary group. Delve into Gowers' problem of determining the largest subset of SU_n with specific product properties. Examine the product mixing conjecture and its implications for quantum communication protocols. Learn about a groundbreaking result that completely solves the product-mixing problem, providing sharp bounds for subsets of SU_n with Haar measure exceeding e^(-cn^(1/3)). Understand how this solution immediately yields an upper bound for Gowers' problem and its significance in the field of quantum mathematics and communication.