Lecture notes on Geometric Crystals and

These lecture notes present a comprehensive overview of geometric crystals and their combinatorial analogues, a topic that has gained traction in the field of mathematics over the last two decades. The primary objective is to delve into the foundations of geometric and unipotent crystals, with a focus on their interrelations and applications. The methodology includes both theoretical explorations of birational Weyl group actions on affine algebraic varieties as well as practical examples to illustrate these concepts. The authors discuss constructs and properties of W-equivariant algebro-geometric distributions and connect them to conjectures arising from local Langlands conjectures. Results demonstrate that solving these problems not only advances the understanding of crystal structures but also provides insights into deeper algebraic frameworks. The notes are supported by rigorous references from previous works in the field, showcasing a lineage of research that underpins the presented findings. This work serves as a valuable resource for researchers and graduate students aiming to navigate the complexities of geometric crystal theory and its combinatorial counterparts.