We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…
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48 results for “infinite-dimensional Frobenius theorem”
We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionle…
This paper reconstructs Lie groupoids from their bisections and applies it to prequantisation.
problem Reconstructing Lie groupoids from their bisections and applying it to prequantisation.
method Investigates the relation of bisections to Lie groupoids and constructs Lie groupoids from bisection Lie groups using recent progress in the infinite-dimensional Frobenius theorem.
result Characterizes an integrability constraint of closed Lie subalgebras to closed Lie subgroups in terms of modified discreteness conditions on periods.
Following the approach of Carlet et al.(2011)\cite{CDM}, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection betw…
We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobeniu…
A new algebra for Frobenius manifolds solves PDEs and constraints.
problem Understanding the algebraic structure of Frobenius manifolds.
method Constructing a Virasoro-like algebra and deriving PDEs and constraints.
result Solves a family of quadratic PDEs for the genus-zero free energy.
Motivated by the algebraic open-closed string models, we introduce and discuss an infinite-dimensional counterpart of the open-closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equ…
Study infinite-dimensional Toda manifold at irregular singularity, revealing non-uniqueness of formal solutions.
problem Non-uniqueness of formal solutions to the Dubrovin equation at irregular singularity.
method Revisited canonical coordinates, formal solutions analysis, Borel resummation, Stokes matrices computation.
result Infinite-dimensional Stokes matrices computed from resummed formal solutions.
Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.
problem Geometric proof of the Frobenius theorem on Riemannian manifolds.
method Introduces a new canonical connection and applies it to prove the Frobenius theorem.
result Geometric proof of the Frobenius theorem.
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
Frobenius theorem proven for a generalized supergeometry.
problem Integrable distributions on -supermanifolds.
method Proof of the Frobenius theorem for -supermanifolds.
result Local and global versions of the theorem are established.
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
Paper constructs solutions to WDVV equations for Frobenius manifolds.
problem Solving open WDVV equations for Frobenius manifolds.
method Explicit construction of flat F-manifolds and principal hierarchies.
result Recovery of polynomial solutions for A- and D-type singularities.
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
New findings on currents and Frobenius theorem properties.
problem Understanding geometric properties of currents and their relation to Frobenius theorem.
method Analyzing integral and normal currents, and their relation to Frobenius theorem.
result Integral currents cannot be tangent to nowhere involutive distributions of planes.
Generalizes Frobenius theorem to quasiconformal deformations.
problem Integrability of plane fields generated by quasiconformal deformations.
method Generalization of classical Frobenius theorem to plane fields.
result A.e. involutive plane fields are integrable.
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they ge…
With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre bundle sections) exist. In particular, loop supergroups can be realised globally in …
Extends involutivity to non-Lipschitz subbundles and proves the Frobenius Theorem.
problem Defining involutivity for non-Lipschitz subbundles and proving the Frobenius Theorem.
method Using generalized functions, the Frobenius Theorem is extended to log-Lipschitz subbundles with sharp regularity estimates.
result For log-Lipschitz involutive subbundles, there exists a homeomorphism with specific regularity properties.
We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.
We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing…
The paper proves a homogeneous Frobenius theorem for N-manifolds.
problem Homogeneous actions and N-manifolds.
method Homogeneity approach and fiber bundle structure.
result Simple proof of homogeneous Frobenius theorem.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
We formulate a notion of (uniform) asymptotic involutivity and show that it implies (unique) integrability of corank-1 continuous distributions in dimensions three or less. This generalizes and extends a classical theorem of Frobenius Theorem which says that an involutive C^1 distribution is uniquely integrable.
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
The paper extends Frobenius-type theorems to non-smooth settings with Hölder estimates.
problem Extending Frobenius-type theorems to non-Lipschitz subbundles and vector fields.
method Develops a singular version of the Frobenius theorem for log-Lipschitz vector fields and proves Hölder estimates.
result Sharp regularity results for log-Lipschitz vector fields and their parameterizations.
The abstract discusses extensions of Jacobi groups and their orbit space properties.
problem Understanding the properties of extended Jacobi groups and their orbit spaces.
method Proving an analogue of Chevalley Theorem and constructing a Frobenius structure.
result Construction of a Dubrovin Frobenius structure on the orbit space.
We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-…
New groups and manifolds from Weyl groups, with Frobenius structures.
problem Understanding new extended affine Weyl groups and their properties.
method Developed new Weyl groups and constructed Frobenius manifold structures.
result Existence of Frobenius manifold structures on orbit spaces of new Weyl groups.
This paper describes how, in the case of algebraic surfaces, the well-known theorem of Donaldson-Uhlenbeck-Yau can be proved in a framework of generalized 'multiplier ideal sheaves', following the ideas of Siu. The key concept is that the destabilizing sheaf satisfies a differential inclusion relation. This relation is…
We endow the de Rham cohomology of any Poisson or Jacobi manifold with a natural homotopy Frobenius manifold structure. This result relies on a minimal model theorem for multicomplexes and a new kind of a Hodge degeneration condition.
We establish a version of the complex Frobenius theorem in the context of a complex subbundle S of the complexified tangent bundle of a manifold, having minimal regularity. If the subbundle S defines the structure of a Levi-flat CR-manifold, it suffices that S be Lipschitz for our results to apply. A principal tool in …
The paper extends Frobenius' Theorem to non-involutive surfaces below threshold.
problem Generalizing Frobenius' Theorem to surfaces with lower regularity.
method Combining fractional Sobolev estimates, Stokes-type theorem, and Lusin's Theorem.
result Sharp exponents for the tangency set's null measure.
For each Frobenius algebra there is defined a skein module of surfaces embedded in a given 3-manifold and bounding a prescribed curve system in the boundary. The skein relations are local and generate the kernel of a certain natural extension of the corresponding topological quantum field theory. In particular the skei…
The abstract discusses transversality for infinite dimensional manifolds.
problem Transversality for infinite dimensional Fréchet manifolds.
method Constructing degree of nonlinear Fredholm mappings using transversality results.
result Proof of rank theorem, invariance of domain theorem, and Bursuk-Ulam type theorem.
Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentialsmath.DG
The paper proves convergence of WDVV potentials and semisimplicity of Frobenius manifolds.
problem Convergence of WDVV potentials and semisimplicity of Frobenius manifolds.
method Analytical proof of integrable deformations of meromorphic connections and application to Frobenius manifolds.
result Convergence of semisimple formal Frobenius manifolds to analytic manifolds.
Lectures on Symplectic Geometry, Poisson Geometry, Deformation Quantization and Quantum Field Theorymath-ph
Lectures on symplectic and Poisson geometry, quantization, and quantum field theory.
problem Exploring symplectic and Poisson structures and their applications in quantum field theory.
method Introduction to differential geometry, symplectic geometry, Poisson geometry, and deformation quantization.
result Detailed understanding of symplectic and Poisson structures and their quantization.
Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be .
Checks difference flatness via involutive distributions.
problem Nonlinear discrete-time system flatness testing.
method Computing a sequence of involutive distributions.
result Efficient implementation in computer algebra.
Proves convergence of normal forms for infinite-dimensional Lie pseudo-group actions.
problem Analyzing convergence of normal forms for complex manifolds.
method Equivariant moving frame method and Cartan-Kähler Theorem.
result Proves convergence of normal form power series for infinite-dimensional Lie pseudo-group actions.
Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a "one-leaf" analogue of the classical Frobenius integrability theorem…
Study index theory for infinite-dimensional manifolds with LT actions.
problem Index theory for infinite-dimensional manifolds with LT actions.
method Introduce LT-equivariant KK-theory and construct three KK-elements: index, Clifford symbol, and Dirac elements.
result Satisfy a relation called the (KK-theoretical) index theorem or KK-theoretical Poincaré duality.
Study how large-scale flows align small-scale vortices in 3D Euler equations.
problem Understanding how large-scale flows align small-scale vortices in 3D Euler equations.
method Constructing a Lagrangian coordinate to identify when the Lie bracket is zero and investigating the locality of the pressure term.
result Clarified conditions under which small-scale vortices are aligned by large-scale flows.
Research covers geometry, analysis, and integration on infinite-dimensional spaces.
problem Exploring geometric and analytical structures in infinite-dimensional settings.
method Analyzes numerical schemes, Lie groups, connections, and integration theory.
result Developed new methods for integration and analysis on infinite-dimensional manifolds.
Universal approximation theorem for differentiable maps on infinite-dimensional manifolds
problem Approximation of differentiable maps on infinite-dimensional manifolds
method Weighted universal approximation theorem
result Universal approximation theorem for differentiable maps
Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carathéodory spacesmath.DG
New insights into integrability and rectifiability in sub-Riemannian geometry.
problem Understanding rectifiability in sub-Riemannian spaces.
method Refined Frobenius Theorem for non-involutive distributions, new metric space class.
result Carnot-Carathéodory spaces are extremal in rectifiability.
The study shows how discrete graphs can resemble hypercube structures under certain curvature conditions.
problem Understanding the structure of graphs with specific curvature conditions.
method Analyzing weighted graphs with lower Ricci curvature bounds and eigenvalue closeness to establish structural similarity.
result Discrete graphs with specific curvature conditions are close to hypercube structures in terms of Frobenius distance and eigenfunctions.
Abstract reviews distributions and subbundles in differential geometry.
problem Understanding distributions and subbundles in differential geometry.
method Systematic review of distributions and subbundles, including sheaves and differentiability cases.
result Detailed consideration of Orbit Theorem and its applications.