Related papers: Differential-geometric and topological structure o…
We consider families of reductive complexes related by level-raising operators and originating from an associative algebra. In the main theorem it is shown that the multiple cohomology of that complexes is given by the factor space of…
We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules.We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors…
Purely real space versions of the differential equations describing the kinematics of a dislocated crystalline medium are considered. The differential geometric structures associated with them are revealed.
Inspired by Gromov's partial differential relations, we introduce a notion of differential transmutation, which allows to transfer some local properties of solutions of a PDE to solutions of another PDE, in particular local solvability,…
We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an…
We explain in some detail the geometric structure of spheres in any dimension. Our approach may be helpful for other homogeneous spaces (with other signatures) such as the de Sitter and anti-de Sitter spaces. We apply the procedure to the…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and…
This text can be considered as a non-technical and arithmetically motivated introduction to the definition of the limiting mixed Hodge structure. We state several assertions in terms natural to the classical theory of ordinary differential…
These notes provide an introduction to the algebra and geometry of differential operators and jet bundles. Their point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalisations of Lie…
The relation between differential geometry of surfaces and some Heisenberg ferromagnet models is considered.
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the…
In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of…
The closure conditions of the inexact exterior differential form and dual form (an equality to zero of differentials of these forms) can be treated as a definition of some differential-geometrical structure. Such a connection discloses the…
We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.
We consider various $A_{\infty}$-algebras of differential (super)forms, which are related to gauge theories and demonstrate explicitly how certain reformulations of gauge theories lead to the transfer of the corresponding…
We construct left, right and bilateral fundamental solutions for generalized steady Stokes' operators $S$ with smooth coefficients coefficients, associated with the de Rham complex of differentials on differential forms over a domain $X$ in…
We define convex-geometric counterparts of divided difference (or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of…
Hopf algebra structure on the differential algebra of the extended $q$-plane is defined. An algebra of forms which is obtained from the generators of the extended $q$-plane is introduced and its Hopf algebra structure is given.
As is well-known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plucker relations, Desnanot--Jacobi identities and their generalizations. We present a construction…