The Abel Prize 2025: Masaki Kashiwara

The 2025 Abel Prize has been awarded to Masaki Kashiwara

for his fundamental contributions to algebraic analysis and representation theory, in particular the development of the theory of D-modules and the discovery of crystal bases.

In particular crystal bases are fascinating tools used in the representation theory of quantum groups (and by extension, also for ordinary Lie groups). To understand what a quantum group is think about the symmetries of a square. You can rotate it by 90 degrees, reflect it, etc. These operations form a group where the order matters (rotation then reflection is different from reflection then rotation).

Now, imagine these symmetries are "fuzzy" or "quantum." The order of operations becomes crucial, and the mathematical structure describing them is a quantum group. They're not groups in the traditional sense, but they retain some key features, like having a way to combine operations and an identity element.

Quantum groups arose from studying solvable models in statistical mechanics and quantum field theory. They provide a powerful framework for understanding these systems.

We can therefore imagine the representation of a quantum group as a complicated multi-dimensional object. A crystal base is like a simplified "skeleton" or a set of key features that captures the essential structure of this object in a more manageable way.

Further readings

• Crystal bases and canonical bases for quantum Borcherds-Bozec algebras by Zhaobing Fan, Shaolong Han, Seok-Jin Kang, Young Rock Kim
• Crystal base theory is expected to further develop through links with other fields