Explore a detailed mathematics seminar where Professor Pramod Achar delves into the fundamental relationships between four derived categories as established in the influential ABG paper. Learn about the intricate connections between quantum group representations at unity roots, quantum Borel representations with small quantum Borel triviality, coherent sheaves on Springer resolution, and perverse sheaves on affine Grassmannian. Discover the foundational aspects of these categories through Ext-calculations, understand the construction of functors between them, and follow the comprehensive proof outline demonstrating their equivalence.