Related papers: Basics of jet modules and algebraic linear differe…
The theory of modular deformations is generalized for the category of complex analytic polyhedra which includes germs of complex space as well as any compact complex analytic space. The objective of the theory is a construction of fine…
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.
The article surveys published and not yet published results about moduli spaces of algebraic surfaces.
Generalizing differential geometry of smooth vector bundles formulated in algebraic terms of the ring of smooth functions, its derivations and the Koszul connection, one can define differential operators, differential calculus and…
We consider, for each smooth manifold $M$, the set $\mathbb{M}$ comprised by all the primary ideals of $\mathcal{C}^\infty(M)$ which are closed and whose radical is maximal. The classical Lie theory of jets (jets of submanifolds) must be…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…
We develop a theory of toroidal vertex algebras and their modules, and we give a conceptual construction of toroidal vertex algebras and their modules. As an application, we associate toroidal vertex algebras and their modules to toroidal…
Following the definitions of the algebras of differential operators, $\beta$-differential operators, and the quantum differential operators on a noncommutative (graded) algebra given in \cite{LR}, we describe these operators on the free…
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of…
Embedding symmetries in the architectures of deep neural networks can improve classification and network convergence in the context of jet substructure. These results hint at the existence of symmetries in jet energy depositions, such as…
We propose a categorical and algebraic study of quantale modules. The results and constructions presented are also applied to abstract algebraic logic and to image processing tasks.
A short explanation of the complexity of the structure of Demailly-semple invariant jet differentials was given
We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale…
A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a series of papers which show that such a…
We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.
We define a method which produces explicit cellular bases for algebras obtained via a Jones basic construction. For the class of algebras in question, our method gives formulas for generic Murphy--type cellular bases indexed by paths on…
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with…
Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials…
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a…
A formal definition of the graded algebra $\mathcal{R}$ of modular linear differential operators is given and its properties are studied. An algebraic structure of the solutions to modular linear differential equations (MLDEs) is shown. It…