Workshop on Lie Theory and Integrable Systems
October 26, 2023

We discuss recent construction of Heun operators as bilinear combinations of two generators of the Askey-Wilson algebra (as well as of its degenerate cases). This construction is related to an important "band and time limiting" problem in Fourier analysis. Classical mechanical analogs of the Heun operators give rise to several families of dynamical systems having explicit solutions in terms of elliptic functions.

It is well known that for a meromorphic linear system with only regular singularities, any formal solution is necessarily convergent. It is less well known that for meromorphic linear systems with irregular singularities, a prescribed asymptotics at an irregular singular point determine different fundamental solutions in different sectorial regions surrounding the singular point. The transition matrices between the preferred solutions in the different sectoral regions are known as the Stokes matrices. This talk shows a relation between Stokes matrices and various structures appearing in integrability. It then explains that how the theory of quantum groups, Yangians, crystal basis and so on can be used to study the Stokes phenomenon.

I will speak about a local analog of the Deligne-Riemann-Roch theorem for line bundles on a family of smooth projective curves. First, I recall the Deligne-Riemann-Roch theorem. Then I will speak about its local analog. The two parts for this local analog of the Deligne-Riemann-Roch theorem consist of the central extensions of the group that is the semidirect product of the group of invertible functions on the formal punctured disc and the group of automorphisms on this disc. These central extensions are by the multiplicative group. The theorem is that these central extensions are equivalent over the ground field of rational numbers. The talk is based on my reсent preprint arXiv:2308.0649.

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.