Joel Kamnitzer- Perfect Bases in Representation Theory- Three Mountains and Their Springs

Explore the intricacies of perfect bases in representation theory through this 41-minute lecture by Joel Kamnitzer, presented at the International Mathematical Union. Delve into the combinatorial descriptions of tensor product multiplicities for semisimple groups, focusing on three known examples of bases compatible with Chevalley generator actions. Discover how these bases originate from geometric or algebraic "mountains" and converge on the same combinatorial shadow: the crystal B(∞) and Mirković–Vilonen polytopes. Learn about the introduction of measures supported on these polytopes to differentiate between the three bases. Examine the interaction between these bases and the cluster structure on the coordinate ring of the maximal unipotent subgroup. Access accompanying slides for visual support of the concepts discussed in this advanced mathematical exploration.