Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
Graded Transformers embed algebraic structure in neural networks through graded transformations.
problem Efficiently modeling hierarchical and structured data in neural networks.
method Introduces Linearly Graded Transformer (LGT) and Exponentially Graded Transformer (EGT) with graded scaling operators.
result Establishes rigorous guarantees and improved efficiency for structured data.
Three definitions of graded vector bundles are shown to be equivalent.
problem Defining graded vector bundles in three different ways.
method Equivalence of categories among sheaves, graded modules, and locally trivial graded manifolds.
result All three approaches to graded vector bundles are equivalent.
The paper introduces Z-graded manifolds and their properties.
problem Defining functions and vector fields on Z-graded manifolds. method Introduced Z-graded manifolds from local models, defined functions and vector fields, and reviewed correspondences. result Properties and definitions of Z-graded manifolds. New connections adapted to graded structures defined for a class of manifolds.
problem Defining connections for graded bundles and their applications.
method Formalism of supermanifolds to describe Lie algebroids, defining weighted A-connections.
result Existence of adapted connections on graded bundles and double vector bundles.
The paper explores new algebraic structures and morphisms in graded settings.
problem Understanding new algebraic structures and morphisms in graded settings.
method Introducing and analyzing L∞-, P∞-, and S∞-algebras, and thick morphisms in a Z2imesZ-graded context. result Shifted S∞-thick morphisms induce L∞-morphisms of shifted S∞-structures. Normal forms for Q-structures on graded manifolds explained.
problem Understanding structures of Q-manifolds on graded manifolds.
method Local and global normal forms results for Q-structures.
result Structures are concentrated along the zero-locus of curvatures.
We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T^nQ playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose…
Study Lie superalgebroids from graded Poisson structures.
problem Construct Lie superalgebroids from graded Poisson structures.
method Construct Lie superalgebroids using graded Poisson structures.
result Lie superalgebroids constructed from graded Poisson structures.
Study bi-graded Lie algebras and their applications.
problem Properties of Z2imesZ2-graded Lie algebras. method Harish-Chandra pairs approach to Lie group-algebra correspondence.
result Examples of application in bi-graded setting.
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…
The paper defines and studies the category of Z-graded manifolds, including their intrinsic structure and formal properties.
problem Understanding the categorical properties and intrinsic structure of Z-graded manifolds.
method Describing local models, explaining formality, and formulating analogues of theorems.
result Proper definitions of objects and morphisms in the category of Z-graded manifolds, and formulation of Batchelor's theorem.
The paper investigates gradings of complex simple Lie algebras, focusing on ∣3∣-gradings and their algebraic structures.
problem Investigating the algebraic structure of ∣3∣-gradings of complex simple Lie algebras. method Completely determining the possible reductive algebras n0 and proving the uniqueness of a specific free nilpotent Lie algebra. result The only free nilpotent Lie algebra of step 3 that appears as the negative part of a ∣3∣-grading is the usual ∣3∣-grading of the exceptional Lie algebra g2. The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
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 …
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth…
New criteria for effective prolongations of graded Lie algebras.
problem Effective criteria for the finiteness of prolongations of graded Lie algebras.
method Explicit construction of matrices to check the rank for effective criteria.
result Results apply to geometries defined by structure algebras on contact distributions.
Constructs a functor for graded bundles to vector bundles, characterizing symmetric structures.
problem Characterizing and fully characterizing the image of a functor from graded bundles to vector bundles.
method Constructs a full linearisation functor that takes a graded bundle of degree k to a k-fold vector bundle, fully characterizing the image and discussing related cases.
result Obtains a subcategory of k-fold vector bundles consisting of symmetric k-fold vector bundles equipped with a family of morphisms indexed by the symmetric group Sk. Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle.…
The paper explores dual structures in graded manifolds.
problem Understanding dual objects in graded manifolds.
method Developed dual objects for graded bundles and applied them to double vector bundles and graded bundles of degree 2.
result Elegant characterizations of double vector bundles and graded bundles of degree 2.
Peer grading is the process of students reviewing each others' work, such as homework submissions, and has lately become a popular mechanism used in massive open online courses (MOOCs). Intrigued by this idea, we used it in a course on algorithms and data structures at the University of Hamburg. Throughout the whole se…
Geometric structures on NQ-manifolds, i.e.~non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher analogues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contac…
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
This paper studies graded manifolds of type Δ and their equivalence with n-fold vector bundles.
problem Understanding the relationship between graded manifolds and vector bundles.
method Geometrization process for Zr-graded manifolds of type Δ. result Established an equivalence between a subcategory of n-fold vector bundles and graded manifolds of type Δ.
Introduces homogeneity supermanifolds for studying graded structures.
problem Graded structures on supermanifolds.
method Homogeneity degrees and weight vector field.
result Proofs of homogeneous Poincaré Lemma, Frobenius Theorem, and Darboux Theorem.
This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra to define, in particular, the notion of a differential graded algebra in the wo…
MGDL refines deep neural networks by training grades sequentially, improving stability.
problem Training deep neural networks is challenging due to nonconvex optimization landscapes.
method MGDL trains deep networks grade by grade, freezing previously learned grades and training new ones to fit residuals.
result MGDL guarantees vanishing error in a fixed-width multigrade ReLU architecture.
Given a unital associatve graded algebra we construct the graded q-differential algebra by means of a graded q-commutator, where q is a primitive N-th root of unity. The N-th power (N>1) of the differential of this graded q-differential algebra is equal to zero. We use our approach to construct the graded q-differentia…
Extends AKSZ to poly-symplectic structures for more complex spaces.
problem Developing a generalized AKSZ formulation for complex target spaces.
method Generalized AKSZ formulation and graded poly-symplectic structures.
result Recovering action functional and poly-symplectic structure of reduced phase space.
Heegaard Floer homology, first introduced by P. Ozsvath and Z. Szabo, associates to a 3-manifold Y a family of relatively graded Abelian groups HF(Y,t), indexed by Spin^c structures t on Y. In the case that Y is a rational homology sphere, Ozsvath and Szabo lift the relative Z-grading to an absolute Q-grading. This ind…
Abstract: Generalized reduction methods for symmetries in graded geometry.
problem Generalized reduction of symmetries in graded geometry.
method Graded symplectic reduction for Courant, Dirac, and generalized complex structures.
result Systematic recovery of reduction schemes for exact cases.
Constructs mixed Hodge structures on Kähler manifolds.
problem Real variations of mixed Hodge structures over compact Kähler manifolds.
method Using Sullivan's 1-minimal models of differential graded algebras associated with real variations of Hodge structures.
result Constructs real variations of mixed Hodge structures.
Classifies low-dimensional 3-Lie superalgebras.
problem Classifying low-dimensional 3-Lie superalgebras.
method Using induced Lie superalgebras and Clifford algebras with Z_2-graded structures.
result Explicit description of 3-Lie superalgebras by ternary commutators.
This work explores symplectic structures on graded manifolds and higher Lie groupoids.
problem Understanding symplectic structures on graded manifolds and their global counterparts.
method Introduction and study of graded manifolds, symplectic Q-manifolds, higher Lie groupoids, and their symplectic structures.
result Developed a graded analogue of Weinstein's tubular neighborhood theorem and explored its applications.
Study curvature and torsion in Courant algebroids using graded geometry.
problem Defining curvature and torsion in Courant algebroids.
method Graded geometric approach, introducing K-curvature and K-torsion.
result Natural graded geometric definition of Courant algebroid curvature and torsion.
The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases of knots with additional structure. The source of our additional grading may be…
Link cobordism maps are graded and determine surface genus.
problem Understanding graded structures in link Floer homology.
method Grading change formula and cobordism maps.
result Link cobordism maps are determined by surface genus.
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,…
New approach connects supergravity, string actions, and generalized geometry via graded Poisson structures.
problem Connecting supergravity and string effective actions with generalized geometry.
method Investigates deformations of graded Poisson structures to derive gravity actions.
result Derives gravity actions from natural deformations of a 2-graded symplectic manifold.
In this work, differential geometry of the Z3-graded quantum superplane is constructed. The corresponding quantum Lie superalgebra and its Hopf algebra structure are obtained.
New algebra pong algebra computed for knot Floer homology.
problem Computing A-infinity structure on knot Floer homology.
method Introduced differential graded algebra, pong algebra.
result Computed A-infinity structure on pong algebra's homology.
New equivalences found between graded supermanifolds and vector bundles.
problem Understanding equivalences between graded supermanifolds and vector bundles.
method Explicit geometric constructions using supergeometry tools.
result Desuperization equivalence functor as a composition of canonical equivalences.
This thesis generalizes structures on Q-manifolds and Lie n-algebroids.
problem Representation theory and linear structures of Q-manifolds and Lie n-algebroids. method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie n-algebroids. result Establishes an equivalence between VB-Lie n-algebroids and (n+1)-term representations up to homotopy of Lie n-algebroids. We define \textit{graded manifolds} as a version of supermanifolds endowed with an additional Z-grading in the structure sheaf, called \textit{weight} (not linked with parity). Examples are ordinary supermanifolds, vector bundles over supermanifolds, double vector bundles, iterated constructions like TTM, e…
The super or Z_2-graded Schouten-Nijenhuis bracket is introduced. Using it, new generalized super-Poisson structures are found which are given in terms of certain graded-skew-symmetric contravariant tensors Λof even order. The corresponding super `Jacobi identities' are expressed by stating that these tensors have zero…
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact str…
Study prolongations of nilpotent Lie algebras with specific structural subalgebras.
problem Understanding prolongations of nilpotent Lie algebras with specific structural subalgebras.
method Analyzing finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, focusing on those with decomposable reductive structural subalgebras.
result Obtained Levi-Malčev and Levi-Chevalley decompositions and precise properties of prolongations.