New differential complexes on symplectic manifolds.
problem Developing calculus on symplectic manifolds.
method Coupling a symplectic manifold to a vector bundle with a constrained curvature.
result Construction of new differential complexes.
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48 results for “differential complex”
Survey explores cohomology's roles in applied math and sciences.
problem Understanding cohomology's role in solving differential equations.
method Examining differential complexes and structure-preserving discretizations.
result Various fundamental concepts in mechanics are formulated using differential complexes.
On the algebra generated by math.DG
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.
The paper constructs a complex of differential operators on symplectic manifolds with metaplectic structures.
problem Handling differential complexes and PDEs on Hilbert bundles.
method Construction of a complex of differential operators acting on exterior forms with values in the dual of the Kostant's symplectic spinor bundle.
result The cohomology groups of the constructed complex are finitely generated projective Hilbert -modules.
Formula for complex SVD backpropagation developed.
problem No specific problem stated; focuses on complex SVD.
method Back propagation formula for complex SVD developed.
result Back propagation formula for complex SVD created.
Local index theory and the Riemann-Roch-Grothendieck theorem for complex flat vector bundles IImath.DG
Proves a theorem for complex flat vector bundles using differential forms.
problem No specific problem stated; focuses on proving a theorem.
method Uses differential forms to prove the Riemann-Roch-Grothendieck theorem.
result Proves the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles.
We obtain a new differentiable sphere theorem for compact Lagrangian submanifolds in complex Euclidean space and complex projective space.
The paper constructs Levi flat structures using structure sheaves and differential complexes.
problem Global solvability and regularity of Levi flat structures.
method Employing formal integrability and differential complexes, the paper constructs a resolution for the structure sheaf.
result Global exactness and Sobolev regularity of the differential complex for Levi flat structures.
The paper uses complex-valued functions to simplify plane differential geometry and kinematics.
problem Simplifying complex problems in plane differential geometry and kinematics.
method Consistent use of complex-valued functions of a real variable.
result Derives results in a particularly simple, uniform, and transparent way.
Proves exponential sample complexity separations in local differential privacy.
problem Sample complexity in locally private protocols.
method Connection between communication complexity and sample complexity, using specific lower bounds for two problems.
result Exponential separations between differentially private protocols.
Poincaré and Sobolev inequalities for differential forms on Heisenberg balls are derived.
problem Establishing inequalities for differential forms on Heisenberg balls.
method Using Rumin's differentials and a global homotopy of Rumin's complex.
result Global homotopy improves differentiability of Rumin forms on bounded geometry contact manifolds.
On manifold-like polyfolds as differential geometrical objects with applications in complex geometrymath.DG
Advances M-polyfolds for complex geometry applications.
problem Complex geometry challenges in differential geometry.
method Introduces and proves geometric structures within M-polyfolds.
result Establishes M-polyfolds as useful differential geometric objects.
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 differential structure on a noncommutative algebra defined in terms of a differential g…
Quantum complexity lowerbound proved using differential geometry.
problem Proving lower bounds on quantum complexity.
method Applied the Bishop-Gromov bound to Nielsen's complexity geometry.
result Lower bounds on quantum complexity are exponentially large.
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as detour complexes. This elaborates on a lecture given at the IMA Summer Programme ``S…
Differentiating groupoidsmath.DG
Article proves tangent complex structure of Lie n-groupoid.
problem Differentiating Lie n-groupoids.
method Proves representability of presheaf by tangent complex.
result Tangent complex of Lie n-groupoid carries Lie n-algebroid structure.
Smooth complex surfaces with triple intersections using differential geometry.
problem Smooth complex surfaces with trivial canonical bundle and triple intersections.
method Explicit construction of local smoothings and solutions to nonlinear elliptic PDEs.
result Existence of smoothings for -semistable SNC complex surfaces with trivial canonical bundle.
Reformulates elasticity complex with new differential and Hodge star operators.
problem Elasticity complex and compatibility condition reformulation.
method Generalized differential complex of Dubois-Violette-Henneaux.
result Integrating formula to recover displacement from strain.
We construct a versal family of deformations of CR structures in five dimensions, using a differential complex closely related to the differential form complex introduced by Rumin for contact manifolds.
The paper explores quadratic differentials in spherical CR geometry and their properties.
problem Understanding quadratic differentials in spherical CR geometry.
method Analyzing the Rumin complex, defining differential operators, and studying quasiconformal maps.
result Definition and properties of quadratic differentials in spherical CR geometry.
Survey of geometry developments, including complex structures on surfaces.
problem Enumerative geometry and complex structures on surfaces.
method Differential and algebraic geometry, nonlinear elliptic PDEs.
result Extensions to 4-manifolds and complex structures on surfaces of general type.
We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…
New path integrals for elasticity derived from differential complex theory.
problem Deriving path integrals for elasticity equations.
method Using Bernstein-Gelfand-Gelfand (BGG) construction and properties of the de Rham complex, derived path integral operators for elasticity.
result Path integral operators for elasticity satisfying and .
The paper extends statistical estimation techniques under differential privacy.
problem Establishing sample complexity bounds for estimation tasks under differential privacy.
method Proposes analogues of Le Cam's method, Fano's inequality, and Assouad's lemma under central differential privacy.
result Optimal sample complexity bounds for discrete distribution estimation under total variation and distances.
New Spencer complexes for Lie groupoids developed.
problem Developing Spencer complexes for Lie groupoids.
method Extending Malgrange's diagonal calculus to .
result Construction of non-linear and linear Spencer complexes.
New Poisson transforms link differential forms on homogeneous spaces to Riemannian symmetric spaces.
problem Linking differential forms on homogeneous spaces to Riemannian symmetric spaces.
method Construction of Poisson transforms using finite dimensional representations of reductive Lie groups.
result Explicit design of Poisson transforms compatible with BGG-complex for real hyperbolic space.
Introduces multiplicative differential forms on Lie groupoids with VB-groupoids values.
problem Describing multiplicative differential forms on Lie groupoids with VB-groupoids values.
method Introduces multiplicative differential forms on Lie groupoids with values in VB-groupoids, presents a Lie theory for differential forms on Lie groupoids with values in 2-term representations up to homotopy, defines a differential complex whose 1-cocycles are multiplicative forms with values in VB-groupoids.
result Complete description of multiplicative differential forms on Lie groupoids with values in VB-groupoids.
Differentiable ABMs face challenges in inference and optimisation.
problem Challenges in parameter inference and optimisation for differentiable ABMs.
method Discussion and experiments highlighting challenges.
result Challenges remain in constructing differentiable ABMs.
Harmonic Complex Structuresmath.DG
In this paper, we introduce a new concept so called harmonic complex structure by using harmonic theory for vector bundle-valued differential forms. It is a new structure intermediates between complex structure and Kähler structure. From differential geometric viewpoint, it is a natural generalization of Kähler structu…
Introduces differential forms to study inequalities between eigenvalues.
problem Eigenvalue inequalities for Laplacian and other operators.
method Uses differential forms and the de Rham complex.
result Shows differential forms are central to Rohleder's work.
The study characterizes complex structures using calculus of variations.
problem Variational characterization of complex structures.
method Calculus of variations for real vector bundle valued differential forms.
result Obtains variational characterization of complex structures.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
The study characterizes a complex curve of residueless meromorphic differentials on elliptic curves.
problem Characterizing the locus of residueless meromorphic differentials on elliptic curves.
method Multi-scale compactification of strata, formulas for genus and degree of maps, distinguishing components.
result Complete classification of connected components of residueless loci in exceptional strata.
This paper classifies components of meromorphic differential strata.
problem Understanding the boundary of multi-scale compactification of meromorphic differentials.
method Classifying connected components of residueless meromorphic differentials.
result Classification of connected components of strata of residueless meromorphic differentials.
Maps complex plane polynomials to light-like polygons in Einstein Universe.
problem Mapping between complex plane polynomials and light-like polygons.
method Constructs geometric homeomorphism between moduli spaces.
result Found minimal Lagrangian maps between ideal polygons.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
PINNs solve differential geometry problems in complex shapes.
problem Solving differential geometry problems in complex shapes.
method Training neural networks with loss functions inspired by differential conditions.
result PINNs are effective for differential geometry problems.
Computes cohomologies of blow-ups and projective bundles.
problem Cohomologies of blow-ups and projective bundles.
method Computes double complex of differential forms on projective bundles and blow-ups.
result Formulas for all cohomologies associated with the complex.
This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras such that the differential Gerstenhaber a…
We study variuos homological structures associated with Poisson algebra, the canonical differential complex for singular Poisson structure and the analogue of the star operator for such manifolds. Give the interpretation of the classical Koszul differential of exterior forms, as the supercommutator with some second ord…
Study differentiable maps on hypersurface links, finding fold maps with circle singular value sets.
problem Understanding differentiable maps on hypersurface links.
method Restricting holomorphic functions to hypersurface links and analyzing the resulting maps.
result Found fold maps with concentric circle singular value sets.
We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Why Diffeology?math.DG
Diffeology extends differential geometry to complex spaces.
problem Handling singular and infinite-dimensional settings in differential geometry.
method Introduces diffeology as a new framework.
result Diffeology provides a natural and effective framework for complex spaces.
Paper introduces differentiable sorting and ranking with time complexity.
problem Non-differentiability of sorting and ranking operations in machine learning.
method Differentiable proxies constructed as projections onto the permutahedron and reduction to isotonic optimization.
result First differentiable sorting and ranking operators with time and space complexity.
Given a generic Lagrangian system, its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condi…
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
Lecture notes on BGG complexes using Lie groups and algebras.
problem Constructing BGG complexes on open domains.
method Representation theory of semisimple Lie groups and Lie algebras.
result Introduction of BGG complexes with Lie group and algebra insights.