# Publications

Yangian-like algebras associated with current R-matrices different from the Yang ones are introduced. These algebras are of two types. The so-called braided Yangians are close to the Reflection Equation algebras, arising from involutive or Hecke symmetries. The Yangians of RTT type are close to the corresponding RTT algebras. Some properties of these two classes of the Yangian-like algebras are studied. Thus, evaluation morphisms for them are constructed, their bi-algebra structures are described, and quantum analogs of certain symmetric polynomials, in particular, the quantum determinant, are introduced. It is proved that in any braided Yangian this determinant is always central, whereas, in general, this is not true for the Yangians of RTT type. Analogs of the Cayley-Hamilton-Newton identities in the braided Yangians are exhibited. A bosonic realization of the braided Yangians is performed.

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

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.

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.

Strong interaction among charge carriers can make them move like viscous fluid. Here we explore alternating current (AC) effects in viscous electronics. In the Ohmic case, incompressible current distribution in a sample adjusts fast to a time-dependent voltage on the electrodes, while in the viscous case, momentum diffusion makes for retardation and for the possibility of propagating slow shear waves. We focus on specific geometries that showcase interesting aspects of such waves: current parallel to a one-dimensional defect and current applied across a long strip. We find that the phase velocity of the wave propagating along the strip respectively increases/decreases with the frequency for no-slip/no-stress boundary conditions. This is so because when the frequency or strip width goes to zero (alternatively, viscosity go to infinity), the wavelength of the current pattern tends to infinity in the no-stress case and to a finite value in a general case. We also show that for DC current across a strip with no-stress boundary, there only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.

Cactus group is the fundamental group of the real locus of the Deligne–Mumford moduli space of stable rational curves. This group appears naturally as an analog of the braid group in coboundary monoidal categories. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain corresponding to arbitrary semisimple Lie algebra g. Cactus group appears in our construction as a subgroup in the Galois group of Bethe Ansatz equations. Following the idea of Pavel Etingof, we conjecture that this action is isomorphic to the action of the cactus group on the tensor product of crystals coming from the general coboundary category formalism. We prove this conjecture in the case g=sl2 (in fact, for this case the conjecture almost immediately follows from the results of Varchenko on asymptotic solutions of the KZ equation and crystal bases). We also present some conjectures generalizing this result to Bethe vectors of shift of argument subalgebras and relating the cactus group with the Berenstein-Kirillov group of piecewise-linear symmetries of the Gelfand–Tsetlin polytope.

We discuss the relation between the cluster integrable systems and *q*-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.

We also define quantum *q*-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using *q*-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting t=−1. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a (k,n) torus knot as a certain sum over diagrams.

The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “a” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character k∕n. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when k=mn+1.

We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU(2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra.

We develop a recursive approach to computing Neveu-Schwarz conformal blocks associated with n-punctured Riemann surfaces. This work generalizes the results of [1] obtained recently for the Virasoro algebra. The method is based on the analysis of the analytic properties of the superconformal blocks considered as functions of the central charge *c*. It consists of two main ingredients: the study of the singular behavior of the conformal blocks and the analysis of their asymptotic properties when *c* tends to infinity. The proposed construction is applicable for computing multi-point blocks in different topologies. We consider some examples for genus zero and one with different numbers of punctures. As a by-product, we propose a new way to solve the recursion relations, which gives more efficient computational procedure and can be applied to SCFT case as well as to pure Virasoro blocks.

We study degenerations of Bethe subalgebras *B*(*C*) in the Yangian Y(gln), where *C* is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parameterizes all possible degenerations, is the Deligne–Mumford moduli space of stable rational curves M0,n+2¯. All subalgebras corresponding to the points of M0,n+2¯are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. The Deligne–Mumford spaceM0,n+2¯ generalizes to other root systems as some De Concini–Procesi resolution of some toric variety. We state a conjecture generalizing our results to Bethe subalgebras in the Yangian of arbitrary simple Lie algebra in terms of this De Concini–Procesi resolution.

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skewinvertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).

This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.

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 derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with *n* regular singular points on the Riemann sphere and generic monodromy in GL (*N*,ℂ). The corresponding operator acts in the direct sum of *N* (*n* − 3) copies of *L*2 *(S*1). Its kernel has a block integrable form and is expressed in terms of fundamental solutions of *n* − 2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant *n*-point system via a decomposition of the punctured sphere into pairs of pants. For *N* = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov–Okounkov partition function). Further specialization to *n* = 4 gives a series representation of the general solution to Painlevé VI equation.

In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra In this paper we analyze its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we exhibit an analog of the Cayley-Hamilton identityfor the generating matrix of such a braided Yangian. Besides, by passing to a limit of this braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev.

The classical matrix-tree theorem discovered by G.Kirchhoff in 1847 expresses the principal minor of the (n x n) Laplace matrix as a sum of monomials of matrix elements indexed by directed trees with n vertices. We prove, for any k >= n, a three-parameter family of identities between degree k polynomials of matrix elements of the Laplace matrix. For k=n and special values of the parameters the identity turns to be the matrix-tree theorem.

For the same values of parameters and arbitrary k >= n the left-hand side of the identity becomes a specific polynomial of the matrix elements called higher determinant of the matrix. We study properties of the higher determinants; in particular, they have an application (due to M.Polyak) in the topology of 3-manifolds.

Homological index of a holomorphic 1-form on a complex analytic variety with an isolated singular point is an analogue of the usual index of a 1-form on a non-singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1-forms on a (purely dimensional) complex analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex analytic varieties with isolated singular points related to "vanishing Chern numbers" at them.