Related papers: The KP hierarchy, branched covers, and triangulati…
We use ideas of the geometry of bihamiltonian manifolds, developed by Gel'fand and Zakharevich, to study the KP equations. In this approach they have the form of local conservation laws, and can be traded for a system of ordinary…
We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of…
Many natural optimization problems derived from $\sf NP$ admit bilevel and multilevel extensions in which decisions are made sequentially by multiple players with conflicting objectives, as in interdiction, adversarial selection, and…
Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we…
A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in…
We first describe, over a field K of characteristic different from 2, the orbits for the adjoint actions of the Lie groups PGL(2, K) and PSL(2, K) on their Lie algebra sl(2, K). While the former are well known, the latter lead to the…
A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…
The usual dispersionless limit of the KP hierarchy does not work in the case where the dependent variable has values in a noncommutative (e.g. matrix) algebra. Passing over to the potential KP hierarchy, there is a corresponding scaling…
The KP equation is a nonlinear dispersive wave equation which provides an excellent model for resonant interactions of shallow-water waves. It is well known that regular soliton solutions of the KP equation may be constructed from points in…
We extend the old formalism of cut-and-join operators in the theory of Hurwitz $\tau$-functions to description of a wide family of KP-integrable {\it skew} Hurwitz $\tau$-functions, which include, in particular, the newly discovered…
Two new diagrammatic techniques on $3d\;\mathcal N=4$ quiver gauge theories, termed chain and cyclic quiver polymerisation are introduced. These gauge a diagonal $\mathrm{SU}/\mathrm{U}(k)$ subgroup of the Coulomb branch global symmetry of…
The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers.…
The description of number of dual (quasy)-exactly solvable models with its hidden symmetry algebra has been given at different levels of analysis within the framework of generalized Kustaanheimo-Stiefel (KS)-transformations. It's shown that…
We prove that the factorization of Appell's generalized hypergeometric series satisfying the so-called quadric property into a product of two Gauss' hypergeometric functions has a geometric origin: we first construct a generalized Kummer…
A simple description of the KP hierarchy and its multi-hamiltonian structure is given in terms of two Bose currents. A deformation scheme connecting various W-infinity algebras and relation between two fundamental nonlinear structures are…
We present a bridge between the KP soliton equations and the Calogero-Moser many-body systems through noncommutative algebraic geometry. The Calogero-Moser systems have a natural geometric interpretation as flows on spaces of spectral…
We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction…
For the first time we show that the quasiclassical limit of the symmetry constraint of the KP hierarchy leads to the generalized Zakharov reduction of the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry…
We establish a new class of integrable {\it systems of Kowalevski type}, associated with discriminantly separable polynomials of degree two in each of three variables. Defining property of such polynomials, that all discriminants as…
In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are…