Speaker: A. Zheglov (MSU, SRMC)
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.
Considering such rings as subrings of a certain complete non-commutative ring $\hat{D}_n^{sym}$, the normal forms of
differential operators mentioned in the title are obtained after
conjugation by some invertible operator
(«Schur operator»), calculated with the help of one of the operators in a ring.
Normal forms of {\it commuting} operators are polynomials with constant
coefficients in the differentiation, integration and shift operators,
which have a finite order in each variable, and can be effectively
calculated for any given commuting operators.
I’ll talk about some recent applications of the theory of normal forms: an effective
parametrisation of torsion free sheaves with vanishing cohomologies on a
projective curve, and the Dixmier conjecture for the first Weyl algebra.
It is expected that the study of normal forms of commuting operators can
subsequently help in the description of the moduli space of torsion free sheaves with a
fixed Hilbert polynomial on the spectral manifold of a ring of commuting
operators of arbitrary dimension, as well as in solving the problem of
finding explicit examples of difference or differential-difference commuting operators.
Time: 13:00 ~ 16:00, Wednesday, December 18, 2024
Location: Room 1610, SIMIS, Block A, No. 657 Songhu Road, Yangpu District, Shanghai
English 
简体中文