Study geometric structures and their interactions under different metrics.
problem Understanding interactions between geometric structures under various metrics.
method Analyzing generalized polynomial structures and their behavior under different metrics on the generalized tangent bundle.
result Showed the commutation or anti-commutation of generalized polynomial structures forming triple structures.
Study of Hermitian structures on Lie groups with 2D commutator subgroups.
problem Classifying Hermitian structures on Lie groups with specific commutator subgroups.
method Explicit classification of Type I and Type II structures, computation of Bismut connections, and examples of Kahler structures.
result Classification of Kahler structures within Type I and Type II structures.
The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu struc…
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Study characteristic classes for TC structures on principal G-bundles.
problem Classifying principal G-bundles with TC structures.
method Algebraic-geometric construction using power maps on BcomG. result Construction of characteristic classes for TC structures on SU(n), U(n), and Sp(n) bundles. In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, RξφT=TRξφ, where T=A or T=S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing ρ-commutative manifolds, Q-manifolds, and modular classes. result Generalized modular classes for non-commutative spaces.
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
The paper defines constraints for commuting endomorphisms in generalized tangent bundles.
problem Identifying constraints for commuting endomorphisms in generalized tangent bundles.
method Using Gröbner basis techniques to construct and study tensors forming ideals.
result Explicit construction and study of tensors forming ideals of commuting endomorphisms.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.
problem Encoding superintegrable Hamiltonian systems using product structures.
method Introducing commutative and associative product structures on Euclidean spaces of dimension at least three, satisfying specific conditions.
result All abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise from these product structures.
New bialgebra structures for relative Poisson algebras are introduced.
problem Extending bialgebra structures from commutative differential algebras to relative Poisson algebras.
method Introducing new bialgebra structures (relative PCA bialgebras) and using commutative 2-cocycles.
result New bialgebra structures (relative PCA bialgebras) are equivalent to certain Manin triples.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …
Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.
problem Understanding algebraic structures on de Rham cohomology of Poisson and Jacobi manifolds.
method Using DG operads and quasi-isomorphisms, they show the de Rham cohomology structure is trivial.
result The de Rham cohomology of Poisson and Jacobi manifolds has no higher structure beyond commutativity.
Covariance is shown as a commutator in random variable calculus.
problem Expressing covariance as a commutator of operators.
method Demonstrated through commutator identities involving expectations and products of functions.
result Revealed the underlying Lie algebraic structure in efficient influence curve calculus.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
Presented an algebra structure for a specific geometric surface.
problem Understanding algebraic structures of geometric surfaces.
method Explicit presentation of Kauffman bracket skein algebra.
result Explicit algebraic structure for a 5-punctured sphere.
In this paper we study real hypersurfaces in the complex quadric space Qm whose structure Jacobi operator commutes with their structure tensor field. We show that the Reeb curvature α of such hypersurfaces is constant and if α is non-zero then the hypersurface is a tube around a totally geodesic submanifold $\ma…
Study differential and integral calculus on noncommutative C*-algebras.
problem Develop calculus on noncommutative spaces.
method Formal smooth structure on nonpure states of C*-algebras.
result Prove Stokes' theorem in both commutative and noncommutative settings.
In this paper, we prove a theorem that gives a simple criterion for generating commuting pairs of generalized almost complex structures on spaces that are the product of two generalized almost contact metric spaces. We examine the implications of this theorem with regard to the definition of generalized Sasakian and ge…
Derives Kerr metric from two commuting complex structures.
problem Deriving the Kerr metric from complex geometry.
method Observation of two commuting complex structures and two Killing vector fields leads to an ansatz for the metrics.
result Derives the Kerr metric using linear algebra.
The paper introduces statistical and geometric structures on anti-commutable pre-Leibniz algebroids.
problem Generalizing differential geometric structures to algebroids.
method Introducing statistical, conjugate connection, and Hessian structures on anti-commutable pre-Leibniz algebroids.
result Statistical and conjugate connection structures are equivalent for admissible connections.
Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.
problem Counting periodic geodesics with specific commutator structure.
method Reduction to counting critical realizations of trivalent graphs.
result Asymptotic count of geodesics with bounded length and commutator structure.
New framework for noncommutative Carrollian geometry using Lie-Rinehart pairs.
problem Developing a geometric framework for ultra-relativistic physics in noncommutative settings.
method Using ρ-Lie-Rinehart pairs to generalize Carrollian Lie algebroids to almost commutative geometry.
result Foundational principles of Carrollian geometry hold in almost commutative geometry.
New proof shows lower bound for commutator length in RAAGs.
problem Finding lower bounds for commutator length in RAAGs.
method Using letter-quasimorphisms to create negatively curved angle structures.
result Establishes a sharp lower bound of 1/2 for stable commutator length in RAAGs.
The shape operator and induced almost contact structure of nearly Kahler hypersurfaces do not commute.
problem Characterizing hypersurfaces in nearly Kahler spaces.
method Construction of twistor spaces and analysis of shape operators and induced structures.
result The shape operator and induced almost contact structure of nearly Kahler hypersurfaces do not commute.
New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.
problem Characterizing Hopf real hypersurfaces with commuting Jacobi operators.
method Investigating the commuting property between normal and structure Jacobi operators.
result A remarkable classification of Hopf real hypersurfaces in the complex quadric with commuting Jacobi operators.
We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered R-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theo…
We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
The paper explores SKT, balanced, and generalized Kähler structures on specific Lie groups.
problem Investigating invariant SKT, balanced, and generalized Kähler structures on compact quotients of almost nilpotent Lie groups.
method Characterization and classification of Hermitian almost nilpotent Lie algebras, study of structures under flows, and non-existence results.
result Construction of new compact SKT manifolds and examples of non-split generalized Kähler structures.
Infinitesimal calculations link fundamental groups to Lie algebras.
problem Calculating logarithm maps in fundamental groups.
method Hopf invariants defined by Harrison cohomology of commutative cochains.
result Zeroth Harrison cohomology is a universal dual to Malcev Lie algebra.
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
The paper examines conditions for commuting conjugates of finite-order mapping classes.
problem Conditions for commuting conjugates of finite-order mapping classes.
method Derived necessary and sufficient conditions for commuting conjugates.
result Conditions for the primitivity of finite-order mapping classes.
This paper extends Riemannian geometry concepts to Hom-ρ-commutative algebras.
problem Extending Riemannian geometry concepts to Hom-ρ-commutative algebras. method Recalling Hom-ρ-commutative algebras, developing metric, connection, torsion, curvature, and differential operators. result Established differential calculus and symplectic/Poisson structures on Hom-ρ-commutative algebras. The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give…
Let D be a homogeneous bounded domain of Cn and A a set of (anti--Wick) symbols that defines a commutative algebra of Toeplitz operators on every weighted Bergman space of D. We prove that if A is rich enough, then it has an underlying geometric structure given by a Lagrangian fo…
New approach predicts commuters' flow with 90.4% accuracy.
problem Limited ability to predict and reconstruct commuters' networks.
method Machine learning and 22 urban indicators.
result Predictions with 90.4% accuracy and 77.6% variance explained.
We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in t…
Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins--Singer model…
In this paper the author determines necessary and sufficient conditions for existence of the Ehresmann connection on a manifold foliated by locally free action of the commutative Lie group. Also here we describe structure of C0(M)∣L for a leaf L⊂M in case such a connection exists. Finally we give some res…
We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature, and derive its algebra structure. Our construction is based on the Riemannian tractor calculus, allowing to construct a prolongation of the differential system for symmetric Killing tensors. We a…
In this summary of Habilitation Thesis, it is outlined author's 18 years research activity on mathematical physics, geometric methods in particle physics and gravity, modifications and applications (after defending his PhD thesis in 1994). Ten most relevant publications are structured conventionally into three "strateg…
We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metr…
This paper explores how neural network width and depth behave as they approach infinity.
problem Understanding the behavior of neural functions as width and depth go to infinity.
method Formal definition of commutativity framework, study of neural covariance kernel, novel proof techniques.
result Taking width and depth to infinity in a deep neural network with skip connections results in the same covariance structure, regardless of the order of taking limits.
We study generalized Kaehler manifolds for which the corresponding complex structures commute and classify completely the compact generalized Kaehler four-manifolds for which the induced complex structures yield opposite orientations.
Maps commuting with sub-Laplacians on Carnot groups are conformal.
problem Characterizing maps preserving sub-Laplacians on sub-Riemannian Lie groups.
method Analyzing smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians.
result Sub-Laplacian determines the sub-Riemannian structure in Carnot groups.