Iva Halacheva - The Cactus Group, Crystals, and Perverse Equivalences

Explore the connections between the cactus group, crystals, and perverse equivalences in this advanced mathematics lecture. Delve into the construction of equivalences on derived categories using Rickard complexes, and examine how these complexes satisfy braid relations for semisimple Lie algebras. Learn about the action of the braid group and discover how the complex corresponding to the positive lift of the longest Weyl group element induces a bijection on irreducible objects and recovers the cactus group action on the corresponding crystal. Follow the progression from SL2 settings to general G examples, and understand the combinatorial aspects of the cactus group in relation to these mathematical structures.