Publications

In this article we study the Gieseker–Maruyama moduli spaces B(e, n) of stable rank 2 algebraic vector bundles with Chern classes c1 = e ∈ {−1, 0} and c2 = n ≥ 1 on the projective space P3 . We construct the two new infinite series Σ0 and Σ1 of irreducible components of the spaces B(e, n) for e = 0 and e = −1, respectively. General bundles of these components are obtained as cohomology sheaves of monads whose middle term is a rank 4 symplectic instanton bundle in case e = 0, respectively, twisted symplectic bundle in case e = −1. We show that the series Σ0 contains components for all big enough values of n (more precisely, at least for n ≥ 146). Σ0 yields the next example, after the series of instanton components, of an infinite series of components of B(0, n) satisfying this property.

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.

We present a combinatorial monomial basis (or, more precisely, a family of monomial bases) in every finite-dimensional irreducible so2n+1-module. These bases are in many ways similar to the FFLV bases for types A and C. They are also defined combinatorially via sums over Dyck paths in certain triangular grids. Our sums, however, involve weights depending on the length of the corresponding root. Accordingly, our bases also induce bases in certain degenerations of the modules but these degenerations are obtained not from the filtration by PBW degree but by a weighted version thereof.

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables.  In particular, our result proves the multivariate version of the Abel--Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive, similarly to what Sottile and White conjectured in Schubert calculus.  The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos.

Using the bilinear formalism, we consider multicomponent and matrix Kadomtsev-Petviashvili hierarchies. The main tool is the bilinear identity for the tau function realized as a vacuum expectation value of a Clifford group element composed of multicomponent fermionic operators. We also construct the Baker– Akhiezer functions and obtain auxiliary linear equations that they satisfy.

We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme M(k), k≥3 of coherent semistable rank 2 sheaves with Chern classes c_1=0, c_2=k, c_3=0 on P^3 generic points of which correspond to sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly μ-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new component of M(3).

We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl(N)-invariant R-matrix. We study two types of Bethe vectors. The first type corresponds to the original monodromy matrix. The second type is associated to a monodromy matrix closely related to the inverse of the monodromy matrix. We show that these two types of the Bethe vectors are identical up to normalization and reshuffling of the Bethe parameters. To prove this correspondence we use the current approach. This identity gives new combinatorial relations for the scalar products of the Bethe vectors.

For a point p of the complex projective plane and a triple (g,d,l) of non-negative integers  we define a Hurwitz--Severi number H(g,d,l) as the  number of generic irreducible plane curves of genus g and degree d+l having an l-fold node at p and at most ordinary nodes as  singularities at the other points, such that the projection of the curve  from p has a prescribed set of local and remote tangents and lines  passing through nodes. In the cases d+l >= g+2 and d+2l >= g+2 >  d+l we express the above Hurwitz--Severi numbers via appropriate  ordinary Hurwitz numbers. The remaining case d+2l < g+2 is still widely  open.

We consider three types of rings of supersymmetric poly-nomials: polynomial ones Λm,n, partially polynomial Λ+ym,nand Laurent supersymmetric rings Λ±m,n. For each type of rings we give their descriptions in terms of generators and relations. As a corollary we get for n ≥man isomorphism Λ+ym,n=Λ+ym,m⊗Λ+y0,n−m. It is also true for polynomial rings, but in this case the isomorphism does not preserve the grading. For each type of rings some natural basis consisting of Euler supercharacters is constructed.

We give an explicit recursive description of the Hilbert series and Gröbner bases for the family of quadratic ideals defining the jet schemes of a double point. We relate these recursions to the Rogers–Ramanujan identity and prove a conjecture of the second author, Oblomkov and Rasmussen.

In this paper we consider inside the algebra PsΔ of pseudo dierence operators, two deformations of the Lie subalgebra spanned by the positive powers of an invertible constant pseudo difference operator 0 of degree one. The rst deformation is by the group in PsΔ corresponding to the Lie subalgebra PsΔ<0 of elements of negative degree and the second by the group corresponding to the Lie subalgebra PsΔ≤0 of elements of degree.zero or lower. We require that the evolution equations of both deformations are certain compatible Lax equations that are determined by choosing a Lie subalgebra, depending of Λ_0, that complements the Lie subalgebras PsΔ<0 resp. PsΔ≤0. This yields two integrable hierarchies associated with Λ_0, where the hierarchy of the wider deformation is called the strict version of the rst because of the form of the Lax equations. For Λ_0 equal to the matrix of the shift operator the hierarchy corresponding to the simplest deformation is known as the discrete KP hierarchy. Both hierarchies are shown to possess an equivalent zero curvature form and we conclude with a discussion of the solvability of the related Cauchy problems.

We describe the correspondence of the Matsuo-Cherednik type between the quantum nn -body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N|M)GL(N|M) . The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the {\mathbb Z}_2 -grading for a fixed value of N+M , so that N+M+1 different qKZ systems of equations lead to the same n -body quantum problem. The obtained results can be viewed as a quantization of the previously described quantum-classical correspondence between the classical n -body Ruijsenaars-Schneider model and the supersymmetric GL(N|M) quantum spin chains on n sites.

We introduce the discrete time version of the spin Calogero-Moser system. The equations of motion follow from the dynamics of poles ofrational solutions to the matrix Kadomtsev-Petviashvili hierarchy with discrete time. The dynamics of poles is derived using the auxiliarylinear problem for the discrete flow

We consider a special class of quantum non-dynamical R  -matrices in the fundamental representation of GL N   with spectral parameter given by trigonometric solutions of the associative Yang-Baxter equation. In the simplest case N=2  these are the well-known 6-vertex R  -matrix and its 7-vertex deformation. The R  -matrices are used for construction of the classical relativistic integrable tops of the Euler-Arnold type. Namely, we describe the Lax pairs with spectral parameter, the inertia tensors and the Poisson structures. The latter are given by the linear Poisson-Lie brackets for the non-relativistic models, and by the classical Sklyanin type algebras in the relativistic cases. In some particular cases the tops are gauge equivalent to the Calogero-Moser-Sutherland or trigonometric Ruijsenaars-Schneider models.

A Calogero–Sutherland system with two types of interacting spin variables has been described using the Hitchin approach and quasicompact structure. Complete integrability has been established by means of the Lax equation specified on a singular curve and the classical r-matrix depending on the spectral parameter. Generalized Toda systems have also been considered. Their phase portraits have been described.

Let $W_G(q_1,q_2,\ldots)$ be a weighted symmetric chromatic polynomial of a graph $G$. S. Chmutov, M. Kazarian and S. Lando in the paper arXiv:1803.09800v2 proved that the generating function $\mathcal{W}(G)$ for the polynomials $W_G(q_1,q_2,\ldots)$ is a $\tau$-function of the Kadomtsev--Petviashvili integrable hierarchy. We proved that the function $\mathcal{W}(G)$ itself is a solution of a linear integrable hierarchy. Also we described the initial conditions for the general formal $\tau$-function of the KP-hierarchy which guarantee that the $\tau$-function is a solution of a linear integrable hierarchy.

In a previous paper we associated to each invertible constant pseudo difference operator of degree one, two integrable hierarchies in the algebra of pseudo difference operators Ps, the so-called dKP() hierarchy and its strict version. We show here first that both hierarchies can be described as the compatibility conditions for a proper linearization. Next we present a geometric framework for the construction of solutions of the hierarchies, i.e. we associate to each hierarchy an infinite dimensional variety such that to each point of the variety one can construct a solution of the corresponding hierarchy. This yields a Segal–Wilson type framework for all these integrable hierarchies.