Derives properties of heat kernel for Rumin complex on Heisenberg groups.
problem Analyzing heat kernel properties for Rumin complex.
method Derives properties of heat equation with Hodge operator on Heisenberg groups.
result Constructs Calderón reproducing formula using heat kernel for Rumin forms.
Sobolev mappings preserve the Rumin complex on contact manifolds.
problem Preserving the Rumin complex under Sobolev mappings on contact manifolds.
method Using the Pullback Theorem, Pansu pullback is shown to induce chain mappings between Rumin complexes and de Rham complexes.
result The Rumin flat complex is bilipschitz invariant under Sobolev mappings between contact manifolds.
Researchers compute determinants and torsions of Rumin complex in specific Lie group representations.
problem Computing determinants and torsions of Rumin complex in specific Lie group representations.
method Analyzing Schrodinger and generic representations of the (2,3,5) nilpotent Lie group.
result Computed the spectrum and zeta regularized determinant of Rumin differentials in Schrodinger representations and evaluated their alternating product in generic representations.
New definition of Rumin complex for nilpotent Lie groups.
problem No new problem introduced.
method Alternative definition of Rumin complex on nilpotent Lie groups.
result Direct application of ℓq,p cohomology results to all nilpotent Lie groups. Alternative construction of Rumin complex on Lie groups.
problem Constructing Rumin complex on homogeneous nilpotent Lie groups.
method Using ideas from parabolic geometry, an alternative construction to the classical one on Carnot groups.
result Explicit computations for the Engel group using the new approach.
Harmonic forms and Rumin complex linked on Sasakian manifolds.
problem Relationship between harmonic forms and Rumin complex on Sasakian manifolds.
method Analytic torsion function and Rumin complex analysis.
result Kernel of Rumin Laplacian matches Hodge-de Rham Laplacian on compact Sasakian manifolds.
We explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the {Rumin complex} in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.
New CR invariant treatment of Rumin complex via differential forms.
problem CR invariant treatment of Rumin complex.
method New treatment via differential forms, identification of balanced A∞-structures, Hodge decomposition theorems. result Sharp upper bound on Kohn--Rossi groups and CR analogue of Frölicher inequalities.
The paper explores the Rumin complex and spectral sequence on Carnot groups.
problem Understanding the relationship between Rumin complex and spectral sequence on Carnot groups.
method Investigates the Rumin complex and spectral sequence on Carnot groups, focusing on the filtration by homogeneous weights.
result Provides a detailed insight into the relationship between the Rumin complex and the spectral sequence on Carnot groups.
Poincar{é} and Sobolev inequalities for differential forms on Heisenberg balls, involving Rumin's differentials, are given. Furthermore, a global homotopy of Rumin's complex which improves differentiability of Rumin forms is provided on any bounded geometry contact manifold.
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.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
Explicitly expresses torsion functions on lens spaces.
problem Calculating torsion on lens spaces.
method Expressed torsion functions in terms of Hurwitz zeta function and Ray-Singer torsion.
result Found that torsion functions vanish at the origin and determined their values.
Given a contact manifold $M_#$ together with a transversal infinitesimal automorphism ξ, we show that any local leaf space M for the foliation determined by ξ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on $M_#$ descends to a complex of differential operators on…
We propose a definition for analytic torsion of the Rumin complex on contact manifolds. This is given by the derivative at zero of a well-chosen combination of zeta functions of a fourth-order modified Rumin Laplacian. The regular value at zero (before differentiation) of this well-chosen combination of zeta functions …
The paper proves a decomposition theorem for forms on sub-Riemannian contact manifolds.
problem Developing a Lp-Hodge decomposition on sub-Riemannian contact manifolds. method Using a Sobolev approach and recent results from [4] and [6].
result Established an Lp-Hodge decomposition theorem for Rumin's forms on sub-Riemannian contact manifolds. Eta invariant of (2,3,5) nilmanifolds vanishes but eta function is nontrivial.
problem Eta invariant of (2,3,5) nilmanifolds.
method Study of eta function and invariant of a self-adjoint differential operator of Heisenberg order two.
result Formula expressing the eta function of (2,3,5) nilmanifolds in terms of elementary functions.
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Simple construction of Rumin algebra for contact manifolds.
problem Computing the de Rham cohomology algebra of contact manifolds.
method Using Markl's Homotopy Transfer Theorem for a simple explicit construction.
result Recovery of the Rumin algebra as a contact invariant C∞-algebra. The purpose of these notes is to explain parts of Gromov's survey of Carnot-Carathedory spaces, in the light of subsequent results of M. Rumin. Among the rich material provided by Gromov, most of which pertains to analysis on metric spaces, we choose to concentrate on the H{ö}lder equivalence problem for Carnot manifol…
Spectral sequence analysis for Sobolev mappings in Carnot groups.
problem Analyzing spectral sequences for Sobolev mappings in Carnot groups.
method Showed Pansu pullback induces a spectral sequence mapping.
result Pansu pullback induces a spectral sequence mapping.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
The paper compares three hypoelliptic Laplacians on a specific 5D Cartan group.
problem Defining suitable hypoelliptic Laplacians for sharp estimates on Carnot groups.
method Introducing and comparing three hypoelliptic Laplacians on a specific Carnot group.
result Sharp div-curl type inequalities for the three hypoelliptic Laplacians.
Paper constructs subcomplexes from filtered Riemannian manifolds.
problem Understanding subcomplexes on filtered Riemannian manifolds.
method General construction of subcomplexes from two distinct complexes.
result Aligns with Rumin complex on regular subRiemannian manifolds.
Researchers prove inequalities for differential forms in Heisenberg groups, extending Euclidean results.
problem Quantitative formulations of topological problems in stratified Lie groups.
method Use of Rumin's complex and Poincaré/Sobolev inequalities for differential forms.
result Extension of L∞-inequalities to Heisenberg groups for forms of degree at least 2. Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the L2-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…
Compact currents and charges in Carnot groups proved.
problem Compactness of normal currents in Carnot groups.
method Dual compactness argument for Rumin forms using pseudo-differential calculus.
result Compactness of normal currents in Carnot groups in flat topology.
Analytic torsion matches Ray-Singer for specific nilmanifolds.
problem Matching analytic torsion with Ray-Singer in specific nilmanifolds.
method Examined rank two distributions on 5D nilmanifolds, proving torsion equality.
result Analytic torsion equals Ray-Singer torsion in these specific nilmanifolds.
New complexes refine multicomplexes for subRiemannian geometry.
problem Analyzing subRiemannian geometry on Carnot groups.
method Spectral complexes from truncated multicomplexes.
result Retains cohomology of multicomplexes and refines Rumin complex.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Let G be a semisimple Lie group with finite center, K⊂G a maximal compact subgroup, and P⊂G a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use G-invariant differential forms on G/K×G/P to construct G-equivariant Poisson transforms mapping differential forms on G/P to …
The purpose of this study is to analyse two related topics: the Rumin cohomology and the H-orientability in the Heisenberg group Hn. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator D, giving examples in th…
This paper constructs Poisson transforms and analyzes their properties on complex hyperbolic spaces.
problem Understanding discrete series representations of SU(n+1,1) using differential forms.
method Constructing Poisson transforms and analyzing their boundary asymptotics and intertwining properties with the Rumin complex.
result The constructed transforms realize the direct sum of all discrete series representations of SU(n+1,1).
In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the general…
New complexes derived from any filtered cochain complex compute the same cohomology.
problem Constructing cohomologically equivalent subcomplexes from filtered cochain complexes.
method Presenting a general construction that produces subcomplexes from any filtered cochain complex of finite depth.
result The construction of subcomplexes depends only on the filtration up to isomorphism.
The paper proves Rademacher's theorem for Heisenberg groups.
problem Proving Rademacher's theorem for Heisenberg groups.
method New definition of intrinsic Lipschitz graphs, extension and approximation theorems, use of Heisenberg currents and Rumin's complex.
result Rademacher's theorem for intrinsic Lipschitz graphs in Heisenberg groups.
We introduce a training method for both better word representation and performance, which we call GROVER (Gradual Rumination On the Vector with maskERs). The method is to gradually and iteratively add random noises to word embeddings while training a model. GROVER first starts from conventional training process, and th…
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…
We characterize locally Lipschitz mappings and existence of Lipschitz extensions through a first order nonlinear system of PDEs. We extend this study to graded group-valued Lipschitz mappings defined on compact Riemannian manifolds. Through a simple application, we emphasize the connection between these PDEs and the Ru…
We relate Lq,p-cohomology of bounded geometry Riemannian manifolds to a purely metric space notion of ℓq,p-cohomology, packing cohomology. This implies quasi-isometry invariance of Lq,p-cohomology together with its multiplicative structure. The result partially extends to the Rumin Lq,p-cohomolog…
Paper proves inequalities for forms on sub-Riemannian manifolds.
problem Establishing inequalities for differential forms on sub-Riemannian contact manifolds.
method Using structure of Rumin's complex, Sobolev-Gaffney inequality for Heisenberg groups, and geometric properties.
result Gaffney type inequality in Sobolev spaces for differential forms on sub-Riemannian contact manifolds with bounded geometry.
In this paper, we prove interior Poincar{é} and Sobolev inequalities in Euclidean spaces and in Heisenberg groups, in the limiting case where the exterior (resp. Rumin) differential of a differential form is measured in L 1 norm. Unlike for L p , p > 1, the estimates are doomed to fail in top degree. The singular integ…
The Alesker-Poincare pairing for smooth valuations on manifolds is expressed in terms of the Rumin differential operator acting on the cosphere-bundle. It is shown that the derivation operator, the signature operator and the Laplace operator acting on smooth valuations are formally self-adjoint with respect to this pai…
This paper aims to define and study currents and slices of currents in the Heisenberg group Hn. Currents, depending on their integration properties and on those of their boundaries, can be classified into subspaces and, assuming their support to be compact, we can work with currents of finite mass, define t…
The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universa…
Machine learning models predict depression risk based on various factors.
problem Identifying individuals at greatest risk for depression.
method Random Effects/Expectation Maximization (RE-EM) trees and Mixed Effects Random Forest (MERF) algorithms.
result Machine learning models accurately predict depression severity and identify key predictors.
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.