Commuting scalar partial differential (and not only) operators and moduli spaces of torsion-free sheaves.
Alexander Zheglov (Moscow State University, now a visitor of SRMC in PKU)
Abstract: In my talk I’ll give an overview of the results obtained by me, as well as jointly with co-authors, related to the problem of classifying commuting (scalar) differential, or more generally, differential-difference or integral-differential operators in several variables. The problem, under some reasonable restrictions, essentially reduces to the description of projective algebraic varieties that have a non-empty moduli space of torsion-free sheaves with a fixed Hilbert polynomial.
More precisely, it turns out to be possible to classify the so-called quasi-elliptic rings, which describe a wide class of operator rings appeared in the theory of (quantum) integrable systems. They are contained in a certain non-commutative “universal” ring - a purely algebraic analogue of the ring of pseudodifferential operators on a manifold and admit (under some weak restrictions) a convenient algebraic-geometric description. This description is a natural generalization of the classification of rings of commuting ordinary differential or difference operators, described in the works of Krichever, Novikov, Drinfeld, Mumford, Mulase. Moreover, already in the case of dimension two there are significant restrictions on the geometry of spectral manifolds.
mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry
Audience: general audience
Comments: Workshop on Lie theory and integrable systems at BIMSA
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.
| Organizers: | Niсolai Reshetikhin, Andrii Liashyk, Ivan Sechin, Andrey Tsiganov* |
| *contact for this listing |