Commuting ring of differential operators with more than three generators

Abstract: In 1920s, Burchnall and Chaundy studied when two ordinary differential operators commute, and this leads to deep connection with the theory of plane algebraic curves. This theory was later developed and used by Krichever to construct algebro-geometric solutions for KP hierarchy.

In this talk, I will associate a family of singular space curves indexed by the numerical semigroups $\langle l, lm+1, \dots, lm+k \rangle$ where $m \ge 1$ and $1 \le k \le l-1$ with a class of generalized KP solitons. Some of these curves can be deformed into smooth ``space curves", and they provide canonical models for the $l$-th generalized KdV hierarchies (KdV hierarchy corresponds to the case $l = 2$). We will see how to construct the space curves from a commutative ring of differential operators with more than three generators.

This talk is based on a joint work with Yuji Kodama.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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