We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see Gaiotto et al., and Kapranov et al.): from a manifold M with the pair of gradient-like commuting vector fields, subject to some general position conditions we construct an L∞-algebra and Maurer–Cartan element in it. We also provide Morse-theoretic examples for the algebra of the infrared data.

Miquel dynamics is a discrete-time dynamical system on the space of square-grid circle patterns. For biperiodic circle patterns with both periods equal to two, we show that the dynamics corresponds to translation on an elliptic curve, thus providing the first integrability result for this dynamics. The main tool is a geometric interpretation of the addition law on the normalization of binodal quartic curves.

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.

We establish a correspondence between Young diagrams and differential operators of infinitely many variables. These operators form a commutative associative algebra isomorphic to the algebra of the conjugated classes of finite permutations of the set of natural numbers. The Schur functions form a complete system of common eigenfunctions of these differential operators, and their eigenvalues are expressed through the characters of symmetric groups. These operators generate differential equations for partition functions of Hurwitz numbers.

S.S. Chern conjectured that the Euler characteristic of every closed affine manifold has to vanish. We present an analog of this conjecture stating that the Euler-Satake characteristic of any compact affine orbifold is equal to zero. We prove that Chern's conjecture is equivalent to its analog for the Euler-Satake characteristic of compact affine orbifolds, and orbifolds may be ineffective. This fact allowed us to extend to orbifolds the known results of B.~Klingler and also results of B.~Kostant and D.~Sullivan on sufficient conditions to fulfill Chern's conjecture. Thus we prove that if an $n$-dimensional compact affine orbifold $\mathcal N$ is complete or if its holonomy group belongs to the special linear group $SL(n,\mathbb R),$ then the Euler-Satake characteristic of $\mathcal N$ has to vanish. An application to pseudo-Riemannian orbifolds is considered. Examples of orbifolds belonging to the investigated class are given. In particular, we construct an example of a compact incomplete affine orbifold with the vanishing Euler characteristic, the holonomy group of which does not belong to $SL(n,\mathbb R).$

We study movable singularities of Garnier systems using the connection of the latter with Schlesinger isomonodromic deformations of Fuchsian systems. Questions of the existence of solutions of some inverse monodromy problems are also considered.

We propose a general scheme of constructing braided differential algebras via algebras of ‘‘quantum exponentiated vector fields’’ and those of ‘‘quantum functions’’. We treat a reflection equation algebra as a quantum analog of the algebra of vector fields. The role of a quantum function algebra is played by a general quantum matrix algebra. As an example we mention the so-called RTT algebra of quantized functions on the linear matrix group GL(m). In this case our construction essentially coincides with the quantum differential algebra introduced by S. Woronowicz. If the role of a quantum function algebra is played by another copy of the reflection equation algebra we get two different braided differential algebras. One of them is defined via a quantum analog of (co)adjoint vector fields, the other algebra is defined via a quantum analog of right-invariant vector fields. We show that the former algebra can be identified with a subalgebra of the latter one. Also, we show that‘‘quantum adjoint vector fields’’ can be restricted to the so-called ‘‘braided orbits’’ which are counterparts of generic GL(m)-orbits in gl∗(m). Such braided orbits endowed with these restricted vector fields constitute a new class of braided differential algebras.

On any reflection equation algebra corresponding to a skew-invertible Hecke symmetry (i.e., a special type solution of the Quantum Yang-Baxter equation) we define analogs of the partial derivatives. Together with elements of the initial reflection equation algebra they generate a "braided analog" of the Weyl algebra. When q→1, the braided Weyl algebra corresponding to the Quantum Group U q(sl(2)) goes to the Weyl algebra defined on the algebra Sym(u(2)) or U(u(2)) depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra U(u(2)), find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra U(u(2)). Eventual applications of our approach are discussed.

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.

Motivated by the algebraic open–closed string models, we introduce and discuss an infinite-dimensional counterpart of the open–closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equipped Cardy–Frobenius algebra, with the closed and open sectors being represented by the conjugation classes of permutations and of pairs of permutations, i.e. by the algebras of Young diagrams and of bipartite graphs respectively.

We define contravariant forms on diagonal reduction algebras, algebras of h-deformed differential operators and on standard modules over these algebras. We study properties of these forms and their specializations. We show that the specializations of the forms on the spaces of h-commuting variables present zero singular vectors iff they are in the kernel of the specialized form. As an application we compute norms of highest weight vectors in the tensor product of an irreducible finite dimensional representation of the Lie algebra gln with a symmetric or wedge tensor power of its fundamental representation.

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.

This paper proposes an axiomatic form for cyclic foam topological field theories, that is topological field theories corresponding to string theories, where particles are arbitrary graphs. World surfaces in this case are 2-manifolds with one-dimensional singularities. I prove that cyclic foam topological field theories are in one-to-one correspondence with graph-Cardy-Frobenius algebras. Examples of cyclic foam topological field theories and graph-Cardy-Frobenius algebras are constructed.

Let M be a hyperkähler manifold, and η a closed, positive (1, 1)-form with rkη<dimM. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kähler form under a Lagrangian fibration L, all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of L. Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kähler forms in the hyperkähler triple goes to η.

We obtain, in local coordinates, the explicit form of the two-dimensional, superintegrable systems of Matveev and Shevchishin involving linear and cubic integrals. This enables us to determine for which values of the parameters these systems are indeed globally defined on S^2.

We review the formalism of free fermions used for construction of tau-functions of classical integrable hierarchies and give a detailed derivation of group-like properties of the normally ordered exponents, transformations between different normal orderings, the bilinear relations, the generalized Wick theorem and the bosonization rules. We also consider various examples of tau-functions and give their fermionic realization.

The Dotsenko-Fateev integral is an analytical function of one complex variable expressing the amplitude in the 4-point correlator of the 2D conformal field theory. Dotsenko-Fateev found ODE of third order with Fuchsian singularities satisfied by their integral. In the present paper, this work is extended to generalized Dotsenko-Fateev integrals, in particular those associated to arbitrary multipoint correlators, and Pfaff systems of PDE of Fuchsian type are constructed for them. The ubiquity of the Fuchsian systems is in that they permit to obtain local expansions of solutions in the neighborhoods of singularities of the system.