Reduces a complex hypersurface to a simplified equation with primary invariants.
problem Analyzing Levi degenerate CR manifolds in 5 dimensions.
method Applying Lie's theory, integrating and straightening chains, and using Poincaré-Moser reduction.
result Shows a convergent change of coordinates that simplifies the equation of the manifold.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
A new direct construction method for Cartan-Moser chains.
problem Detecting Cartan-Moser chains from advanced considerations.
method Inspection of Lie prolongations of infinitesimal automorphisms.
result Found a simple cubic degenerate orbit locus.
Study normal forms for a specific type of complex hypersurface.
problem Tackle the normal forms of a special class of rigid hypersurfaces in complex space.
method Apply Chen-Merker method to find relative invariants and construct a bridge between Poincaré and Cartan normal forms.
result Establish the Poincaré-Moser complete normal form for the hypersurface.
Paper develops a weighted linearization approach for vector fields.
problem Linearizability of vector fields under weighted conditions.
method Formal Moser trick applied to power series, addressing weighted non-resonance condition.
result Formal Moser trick works over any field of characteristic zero.
Reduces field theories using Poisson-Poincaré method.
problem Reduction of field theories using Poisson-Poincaré method.
method Poisson-Poincaré reduction for field theories.
result Reduction procedure for field theories.
Completes reduction scheme in Lagrange-Poincaré category.
problem Lagrangian reduction by stages in the whole category.
method Analyzes Noether theorem, Hamiltonian reduction, geometric aspects.
result Affirmative answer to open question of Lagrangian reduction.
On a doubling metric measure space endowed with a "carré du champ", we consider Lp estimates (Gp) of the gradient of the heat semigroup and scale-invariant Lp Poincaré inequalities (Pp). We show that the combination of (Gp) and (Pp) for p≥2 always implies two-sided Gaussian heat kernel bounds. Th…
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
Method for computing Khovanov homology of tangles.
problem Limited explicit computational studies of Khovanov homology for tangles.
method Arc reduction approach to compute Khovanov homology.
result Derived and computed Poincaré polynomials for simple and complex tangles.
This work extends reduction processes for nonholonomic discrete mechanical systems.
problem Nonholonomic discrete mechanical systems and their reductions.
method Introduces a category LDPd of discrete-time dynamical systems and a two-stage reduction process. result Two-stage reduction process produces systems isomorphic to one-stage reduction.
Paper compares Lagrangian reduction methods for rigid body systems.
problem Modeling and reduction of rigid body systems with rotors.
method Euler-Poincaré reduction by the whole group and reduction by stages.
result Equivalence of equations and conservation laws are tracked.
We show that the Spivak normal fibration of an orientable 4-dimensional Poincaré complex has a vector bundle reduction.
We discuss Poincaré duality complexes X and the question whether or not their Spivak normal fibration admits a reduction to a vector bundle in the case where the dimension of X is at most 4. We show that in dimensions less than 4 such a reduction always exists, and in dimension 4 such a reduction exists provided X is o…
Contradicts claims about Poincaré complexes and homology manifolds.
problem Claims about Poincaré complexes and homology manifolds are contradicted.
method Constructs a Poincaré complex with specific properties to contradict the claims.
result A Poincaré complex with vanishing periodic total surgery obstruction is not necessarily homotopy equivalent to a homology manifold.
Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.
problem Scaling symmetries in Hamiltonian systems.
method Contact reduction of symplectic Hamiltonian systems.
result Generically possible reduction to contact Hamiltonian systems, reducing inputs needed.
We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of…
Geometric derivation of quantum dynamics from Lie group actions.
problem Deriving quantum dynamics from geometric principles.
method Euler-Poincaré reduction on adjoint-coupled semidirect products.
result Reproduces the Lindblad equation from geometric reduction.
We present in modern language the contents of the famous note published by Henri Poincaré in 1901 "Sur une forme nouvelle des équations de la Mécanique", in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equa…
Study nonholonomic systems with collisions using variational principles.
problem Variational problems on nonholonomic systems with collisions.
method Extended variational principle, introduced connection on principal bundles, applied Lagrange–Poincaré–Pontryagin reduction.
result Implicit Lagrange–d'Alembert–Pontryagin equations for nonholonomic systems with collisions.
The paper derives the QGS equations using stochastic central extensions.
problem Deriving the viscous quasi-geostrophic equations on the torus.
method Central extensions of Lie groups and Lie algebras, stochastic Lagrangian formulation, and Euler-Poincaré reduction.
result Stochastic perturbations to the central extension lead to solutions of the QGS equations.
Let π:P→Mn be a principal G-bundle, and let L:J1P→Λn(M) be a G-invariant Lagrangian density. We obtain the Euler-Poincare equations for the reduced Lagrangian l defined on C(P), the bundle of connections on P.
Paper proves constants for Moser-Trudinger inequality on surfaces.
problem Establishing constants for Moser-Trudinger inequality on surfaces.
method Using systole, isoperimetric constant, and curvature as parameters.
result Constants can be chosen to depend on only 3 parameters.
Proves a theorem similar to Moser's using a normalization method.
problem Proving a theorem similar to Moser's in a specific context.
method Iterative normalization procedure based on Generalized Fischer Decompositions.
result An analogue of the Theorem of Moser proven.
Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.
problem Smooth triviality of Lie subalgebras and Lie ideals
method Establishing necessary and sufficient conditions and proving Moser's trick for foliations
result Direct proof of Moser's trick for foliations
Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyag…
Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.
problem Contact mechanical systems on Lie groups with symmetries.
method Reduction process using Lie group actions and symmetries.
result Euler-Poincaré-Herglotz equations on the reduced phase space.
Sharp inequalities on curved spaces with bounded curvature.
problem Establishing inequalities on curved spaces with curvature constraints.
method Using Sobolev and Moser-Trudinger inequalities on noncompact Riemannian manifolds with Ricci curvature bounded below.
result Best constants for inequalities on curved spaces with curvature constraints.
This study explores Kaluza-Klein reductions of new maximally supersymmetric backgrounds.
problem Exploring new maximally supersymmetric backgrounds in five dimensions.
method Classifying Kaluza-Klein reductions to four dimensions and determining preserved supersymmetry.
result Discovery of novel non-homogeneous four-dimensional Lorentzian spacetimes with N=1 supersymmetry. There are proven few analogues of the Theorem of Moser using The Approximation Theorem of Artin.
The paper proves a Moser-Trudinger inequality on metric spaces with curvature-dimension conditions.
problem Proving a Moser-Trudinger inequality on metric measure spaces.
method Rearrangement of functions on CD(k,n)-spaces satisfying a Polya-Szegö type inequality.
result Characterization of manifolds with lower bounded Ricci curvature admitting a Moser-Trudinger inequality.
Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…
Kuranishi's proof of complex deformation theory revisited
problem Existence of complex deformations on compact complex manifolds
method Hamilton-Nash-Moser implicit function theorem
result Revisits classical proof with modern tools
Proposes a new category of bundles for Lagrangian reduction in field theory.
problem Lagrangian reduction in field theory.
method Introduces a category of bundles to perform Lagrangian reduction by stages in covariant Field Theory.
result Formulates the Noether theorem in this new theoretical framework.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
Develops higher-order Euler-Poincaré field equations for principal G-bundles.
problem Formulating field equations for higher-order jet bundles of principal G-bundles.
method Reduction theory applied to G-invariant Lagrangian field theories on jet bundles, transferring Hamilton's principle to reduced configuration bundles. result Higher-order Euler-Poincaré field equations are equivalent to conservation of Noether current.
We prove a stability result for volume forms on fiber bundles with compact base and noncompact fibers. This generalizes the classical results of Moser and Greene--Shiohama, and recent work by the authors.
New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
The abstract proves a Moser-like theorem for C-symplectic structures and applies it to complex manifolds.
problem Analyzing the isotopy of C-symplectic structures and their applications.
method Proves an analogue of Moser's isotopy theorem for families of C-symplectic structures.
result Locally trivial degenerate twistorial deformation over the base of holomorphic Lagrangian fibrations.
We propose two constructions extending the Chern-Moser normal form to non-integrable Levi-nondegenerate (hypersurface type) almost CR structures. One of them translates the Chern-Moser normalization into pure intrinsic setting, whereas the other directly extends the (extrinsic) Chern-Moser normal form by allowing non-C…
Study applies inverse scattering to BKM systems, linking spectra and integrable systems.
problem Applying inverse scattering to BKM systems.
method Developed methods for BKM systems, relating Schrödinger-Hill operators, Neumann system, and KdV equations.
result Initial observations indicate potential for applying inverse scattering to BKM systems.
In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…
Theorems adapted for prequantum systems, showing symplectomorphism and gauge transformation.
problem Adapting classical theorems to prequantum systems.
method Establishing analogs of the Darboux, Moser, and Weinstein theorems.
result Prequantum systems with vanishing first cohomology are equivalent up to symplectomorphism and gauge transformation.
Unified product Lie groups and their quotient spaces are analyzed for dynamics.
problem Analyzing dynamics over homogeneous spaces using Lie group theory.
method Reduction and extension of Lie group structures to quotient spaces, formulation of Euler-Lagrange, Hamilton, and Euler-Poincaré equations.
result Unified product Lie groups and their quotient spaces provide a framework for formulating dynamics equations.
We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.
We study the Euler-Lagrange equations for a parameter dependent G-invariant Lagrangian on a homogeneous G-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group G, emphasizing the special invariance properties of the associated Euler-Poincaré equations with advected parameters.
New inequality criterion for a mean field equation on spheres.
problem Finding uniqueness in a mean field equation on spheres.
method Established a new Moser-Trudinger-Onofri inequality with a constraint on moments deviation.
result A threshold for deviation is a uniqueness criterion for the mean field equation.
Variational reduction simplifies Lagrangian systems with scaling symmetries.
problem Simplifying Lagrangian systems with scaling symmetries.
method Defining a variational reduction procedure for homogenous Lagrangian systems.
result Reconstructing trajectories from critical points of reduced variational principle.