Kashiwara demystifies crystal bases and categorifications

August 8, 2018, 5:11 pm

Kashiwara demystifies crystal bases and categorifications

Considered the founding father of algebraic analysis, Japanese mathematician Masaki Kashiwara led a packed plenary audience through his theory of “Crystal Bases & Categorifications” this afternoon, having paid homage to his supervisor Mikio Sato.

Awarded the coveted Chern Medal last Tuesday at ICM 2018 for his contributions to mathematics in a career spanning nearly half a century, Kashiwara guided the assembled international mathematicians through a map of evolution through quantum group theory, crystal bases (a concept pioneered by him in 1990, that allows questions in representation theory to be answered in terms of ‘combinatorics’), and global bases, through the dichotomy of Quiver Hecke algebras and cluster algebras, and leading to his monoidal categorifications of cluster algebras.

Read more:

The Japanese researcher explained how combinatorial objects can be lifted to objects with more structure, involving categories, so that combinatorial problems are shown as different structures in the higher categorical context.

Despite his expertise on representation theory, Geordie Henderson from the University of Sydney, who delivered a plenary at ICM earlier today, said he was surprised. “I’m very familiar with the work and it’s extraordinary because somehow the origin of cluster algebras was exactly this question. Cluster algebras are applied to many different things, but now it has actually come back to the origins of clusters algebras, which is very lovely to see.” Henderson described the R matrix technique outlined during the plenary as “a very beautiful idea.”

Oxford University’s Kevin McGerty said Kashiwara’s work could be understood as “physics giving insight to mathematics.” Fascinated with today’s plenary, McGerty said the Chern medalist “built a bridge between two quite different parts of mathematics – representation theory and combinatorial.” He said most would never imagine this existing, and that most “wouldn’t have dreamed of it being as powerful as it has been and being able to capture as much of representation theory as it actually does.”

Listing the many awards held by colleague and compatriot, including the Iyanaga (1981), Asahi (1988), Japanese Academy (1988), and Kyoto Maths (2018) awards, Tohoku University’s Motoko Kotani said: “I’m very lucky that I was born Japanese, because it has allowed me to read his articles in Japanese.”