Kashiwara's theory of crystal bases has been fruitful to many different areas of mathematics. A classical application has been to representation theory (where it originates), which leads to another combinatorial proof of the Cauchy identity, where the famous Robinson-Schensted-Knuth (RSK) bijection becomes an isomorphism of crystals. However, its discrete (and combinatorial) nature makes it hard to work with sometimes, so we transform its description into piecewise-linear functions. This description allows us to construct a rational lifting, enabling a more continuous theory of crystals that can lead to further connections. This was formalized in the work of Berestein and Kazhdan with their theory of geometric and unipotent crystals. The RSK bijection can also be formulated into piecewise-linear functions, which has a rational lifting to an isomorphism of algebraic varieties called geometric RSK (gRSK) with nice properties. In this talk, we discuss the background on geometric crystals and gRSK, including the two different descriptions due to Noumi-Yamada and O'Connell-Seppäläinen-Zygouras.

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