arXiv research

Determines algebra structure of complex differential forms operators.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

Extends index theorem to uniformly elliptic operators on manifolds.

Graded Transformers embed algebraic structure in neural networks through graded transformations.

New perspective on APS indices preserves orientations and gradings through bordisms.

Study shows links with unbounded annular Khovanov gradings but bounded Floer gradings.

Constructs graded jet bundles for Z-graded manifolds and vector bundles.

Study graded coverings for supermanifolds, proving their universal properties.

It is shown that the new formula for the field theory Poisson brackets arise naturally in the extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect …

HOFLON automates process start-ups and grade-changes using offline RL and online optimization.

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the …

It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differ…

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formu…

Wavelet scattering method improves glioma grade prediction accuracy.

The abstract generalizes a construction for splitting supermanifolds and studies Lie supergroup cases.

Recently, for a family of ungraded Dirac operators over some space $B$ J. Lott constructed an index gerbe. In the present paper we show (in analogy to the holonomy formula for the determinant bundle in the graded case) that the holonomy of the index gerbe (a priori a hermitean line bundle with connection over the free …

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and releva…

We take advantage of different generalizations of the tangent manifold to the context of graded manifolds, together with the notion of super section along a morphism of graded manifolds, to obtain intrinsic definitions of the main objects in supermechanics such as, the vertical endomorphism, the canonical and the Carta…

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally,…

The paper studies how geometric transformations affect semi-classical operators on specific Lie groups.

The time evolution operator $K$ is introduced in the graded context and its main properties are discussed. In particular, the operator $K$ is used to analize the projectability of constraint functions arising in the Lagrangian formalism for singular Lagrangians.

The paper extends T-duality and Jacobi forms to Witten gerbe modules.

We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…

We extend the classical characterization of a finite-dimensional Lie algebra g in terms of its Maurer-Cartan algebra-the familiar differential graded algebra of alternating forms on g with values in the ground field, endowed with the standard Lie algebra cohomology operator-to sh Lie-Rinehart algebras. To this end, we …

MGDL refines deep neural networks by training grades sequentially, improving stability.

Exposes graded and microformal geometry, focusing on $Q$-manifolds.

We define an index of the fermionic signature operator on even-dimensional globally hyperbolic spin manifolds of finite lifetime. The invariance of the index under homotopies is studied. The definition is generalized to causal fermion systems with a chiral grading. We give examples of space-times and Dirac operators th…

Develops a chain-level model for Chas-Sullivan products using Morse theory with differential graded coefficients.

Let $S$ be a spinor bundle of a pseudo-Euclidean vector bundle $(E,\mathrm{g})$ of even rank. We introduce a new filtration on the algebra $\mathcal{D}(M,S)$ of differential operators on $S$. As main property, the associated graded algebra $\mathrm{gr}\mathcal{D}(M,S)$ is isomorphic to the algebra $\mathcal{O}(\mathcal…

Introduces a new operator generating higher Koszul brackets on differential forms.

New Morse theory for path homology with coefficients.

Short note observes quantum Hochschild homology as a composition of known operations.

We define a differential graded algebra for Legendrian graphs and tangles in the standard contact Euclidean three space. This invariant is defined combinatorially by using ideas from Legendrian contact homology. The construction is distinguished from other versions of Legendrian contact algebra by the vertices of Legen…

Defines a filtration on variational bicomplex for concise functional form conditions.

Study spectral functionals on manifolds with torsion.

The paper proves a homogeneous Frobenius theorem for N-manifolds.

In this paper, we compute the slice genus for many low-crossing virtual knots. For instance, we show that 1295 out of 92800 virtual knots with 6 or fewer crossings are slice, and that all but 248 of the rest are not slice. Key to these results are computations of Turaev's graded genus, which we show extends to give an …

New Bol operators identified on superstrings.

For a smooth family F of admissible elliptic pseudodifferential operators with differential form coefficients associated to a geometric fibration of manifolds M--> B we show that there is a natural zeta-form z(F,s) and zeta-determinant- form det(F) in the de-Rham algebra of smooth differential forms, generalizing the c…

This paper analyses non-regular $|2|$-graded geometries, and show that they share many of the properties of regular geometries -- the existence of a unique normal Cartan connection encoding the structure, the harmonic curvature as obstruction to flatness of the geometry, the existence of the first two BGG splitting ope…

The paper studies asymptotics and zeta functions on compact nilmanifolds.

New operations defined on moduli spaces for bundles with orientations.

The study explores Lie superalgebras constructed from Lie algebras using Schouten-like brackets.

We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed context and, in particular, enriches it with new natural operations. Applications wi…

Constructive approach to Lie algebra gradings, computing maximal and enumerating all gradings.

The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle co…

A garland based on a manifold $P$ is a finite set of manifolds homeomorphic to $P$ with some of them glued together at marked points. Fix a manifold $M$ and consider a space $\NN$ of all smooth mappings of garlands based on $P$ into $M$. We construct operations $\bullet$ and $[-,-]$ on the bordism groups $\bor_*(\NN)$ …

Three definitions of graded vector bundles are shown to be equivalent.