The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
ChebLieNet uses Lie groups to create invariant spectral graph networks.
problem Handling anisotropic data in graph neural networks.
method Develops anisotropic convolutional layers on Lie groups with Riemannian metrics.
result Demonstrates the effectiveness of balancing equivariance and invariance.
The paper proves a generalized Bour's theorem for invariant surfaces.
problem Proving a generalized Bour's theorem for invariant surfaces.
method Equivariant geometry techniques applied to three-manifolds.
result A generalized Bour's theorem holds for invariant surfaces.
Classifies 4D spaces with circle symmetry and positive curvature.
problem Classifying spaces with specific geometric properties.
method Equivariant homeomorphism classification, infinitesimal geometry analysis.
result Classification of 4D Alexandrov spaces and orbifolds with circle symmetry.
Introduces orbifolds from charts and various topological perspectives.
problem No specific problem stated; introduces new mathematical objects.
method Charts and classical topological perspectives.
result Orbifolds defined and properties from Algebraic, Differential, and Riemannian Geometries.
Reduces 3-body problem to shape and spherical geometry.
problem Investigates the motion of three bodies in a plane.
method Geometric reduction using equivariant Riemannian geometry.
result Time parametrization of moduli curve determined by shape curve and potential function.
Explains examples of Lagrangian flow with circle symmetry.
problem Understanding Lagrangian flow with symmetry.
method Examining specific examples in C2. result Shows various types of flow, including compact and non-compact.
The paper provides obstructions to positive scalar curvature for certain manifolds with group actions.
problem Obstructions to the existence of complete invariant metrics with positive scalar curvature.
method Callias-type index theorem applied to proper actions by locally compact groups.
result Obstructions to positive scalar curvature vanish for certain Lie group actions.
We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold M, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the Ck-topology, k=2,...,∞, in the set of metrics of …
Defines metric bundles for manifold geometries, unifying various types of metrics.
problem Unified framework for various types of metrics on manifolds.
method Formalizes metric bundles and defines open fiberwise cones for nondegenerate symmetric bilinear forms.
result Unified framework subsumes Riemannian and pseudo-Riemannian metrics, and extends to other structures.
Proposes eDNNs and iDNNs for deep learning on manifolds.
problem Deep learning on manifolds with geometric preservation and intrinsic geometry incorporation.
method Intrinsic and extrinsic deep neural networks (iDNNs and eDNNs) with geometric embeddings and maps.
result Empirical risk minimizers of eDNNs and iDNNs converge optimally.
Let a compact Lie group act isometrically on a non-collapsing sequence of compact Alexandrov spaces with fixed dimension and uniform lower curvature and upper diameter bounds. If the sequence of actions is equicontinuous and converges in the equivariant Gromov--Hausdorff topology, then the limit space is equivariantly …
Extends coarse index theory to locally compact groups for Callias operators.
problem Developing an equivariant coarse index theory for non-cocompact actions.
method Using admissible modules and localised K-theory of group C∗-algebras. result Equivariant index for Callias operators is a special case of the localised index.
In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries ("quantizations") that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and argume…
The paper studies hyperbolic three-manifolds and their geometric constraints.
problem Understanding the interaction between hyperbolic geometry and homology cobordism.
method Derived explicit bounds on relative grading and invariants in monopole Floer homology.
result Explicit bounds on numerical invariants and subgroup structure of homology cobordism.
Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space …
The paper studies geometric properties of group equivariant operators and their Riemannian structure.
problem Understanding the geometric structure of group equivariant operators.
method Endowing the space of group equivariant non-expansive operators with a Riemannian manifold structure and using gradient descent methods.
result Gradient descent methods can be applied to minimize cost functions on the space of group equivariant non-expansive operators.
This work proposes a geometric approach to equivariant message passing on Riemannian manifolds.
problem Efficiently processing data on Riemannian manifolds with equivariance.
method Geometric insight into equivariant message passing on Riemannian manifolds, using an equivariant embedding and diffusion process.
result A new class of equivariant GNNs on Riemannian manifolds.
Bound on equivariant index for min-max surfaces.
problem Bounding the index of equivariant min-max surfaces.
method Equivariant min-max procedure with group action.
result Equivariant index bound by number of parameters.
Study reveals fundamental group properties of manifolds with specific curvature and growth.
problem Understanding the fundamental groups of manifolds with nonnegative Ricci curvature and linear volume growth.
method Analysis of covering spaces and rigidity results for RCD spaces.
result Fundamental groups of manifolds contain subgroups of finite index or are finite.
Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
Formulae for Riemannian foliations cohomology.
problem Equivariant cohomology of Riemannian foliations.
method Localization and integration formulas for equivariant basic cohomology.
result Duistermaat-Heckman theorem for transversely symplectic foliations.
Toric actions in cosymplectic geometry linked to symplectomorphisms.
problem Understanding toric actions in cosymplectic geometry.
method Showed that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
result Compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
problem Finding generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
method Weyl asymptotic law for G-equivariant volume spectrum, generic density result. result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
Study compares weak and homotopy moment maps in multisymplectic geometry.
problem Existence and equivariance of moment maps in multisymplectic geometry.
method Comparison of weak and homotopy moment maps.
result Analysis of existence and equivariance phenomena.
The paper defines conditions for a Riemannian structure on a symplectic quotient.
problem Existence of Riemannian structures on symplectic quotients.
method Analyzes conditions for existence given a Lie group action with equivariant momentum mapping.
result Determines conditions under which an induced Riemannian structure exists.
We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We appl…
The paper computes metrics and Einstein tensors on even-dimensional manifolds.
problem Computing metrics and Einstein tensors on even-dimensional Riemannian manifolds.
method Development of spectral Einstein functionals and equivariant Bismut Laplacian.
result Explicit computation of the equivariant noncommutative residue density.
Generalizes Molino's theory for Riemannian foliations.
problem Studying Riemannian foliations and their properties.
method Generalization of Molino's theory with discussion of projections and equivariant basic Â-genus characters.
result Equivariant basic cohomological isomorphism for Killing foliation.
Two constructions of Chern character for equivariant vector bundles in noncommutative geometry.
problem Chern character for equivariant vector bundles in noncommutative geometry.
method Two constructions using cyclic cohomology of crossed product algebras.
result Equivalence of two constructions under proper action.
Generalizes skyrmion theory to gauged maps with G-action.
problem Finding non-trivial minimizers of gauged skyrmion energy.
method Introduces a gauged energy functional and studies BPS equations.
result Classifies solutions for G=mU(1) and G=mSU(2). This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes-Chern character of an equivariant Dirac spectral triple. The formula that we obtain for which was used in the first paper of the series. The computation has two main ste…
In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for the volumes. We also find applications to the theor…
This paper proves the existence of a smooth embedding for symmetrical manifolds.
problem Embedding symmetrical Riemannian manifolds isometrically in Euclidean spaces.
method Proves the existence of a C∞ equivariant isometric embedding. result Existence of a C∞ equivariant isometric embedding of M in $\RR^q$. Selects points from Jordan domains on Riemannian surfaces.
problem Selecting points from Jordan domains in Riemannian surfaces.
method Fiber bundle theory and conformal mappings.
result Space of Jordan domains retracts onto round disks.
Paper proves equivariant Fried conjecture for specific flows.
problem Equivariant Fried conjecture for suspension flows.
method Analyzes suspension flows of equivariant isometries.
result Proves conjecture for various groups and cases.
Study on Finsler spaces with unique prime geodesics.
problem Estimating the number of orbits of prime closed geodesics in Finsler manifolds.
method Generalizing works on two prime closed geodesics to equivariant situation, studying homogeneous Finsler geometry.
result Closed Finsler manifold with only one orbit of prime closed geodesic is a compact rank-one Riemannian symmetric space when dimension is even or metric is reversible.
The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--C…
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
Classifies 4D Alexandrov spaces with torus actions.
problem Classifying Alexandrov spaces with torus actions.
method Equivariant classification and homeomorphism analysis.
result Alexandrov spaces are homeomorphic to Riemannian orbifolds.
The paper proves the existence of a center in disks within Riemannian manifolds.
problem Proving the existence of a center in disks within Riemannian manifolds.
method Analyzing the existence of a center in C1 embedded k-disks in Riemannian n-manifolds. result Existence and equivariance of centers in certain conditions, non-existence in others.
In this paper, we establish an infinitesimal equivariant index formula in the noncommutative geometry framework using Greiner's approach to heat kernel asymptotics. An infinitesimal equivariant index formula for odd dimensional manifolds is also given. We define infinitesimal equivariant eta cochains, prove their regul…
Study stabilizes Morse-Bott cohomology for equivariant manifolds.
problem Equivariant cohomology of manifolds with group actions.
method Stabilization technique to construct Morse-Bott functions.
result Realization of equivariant transversality and orientability.
Infinite G-invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G-homology class admits infinitely many distinct realizations by embedded minimal G-hypersurfaces. Constructs equivariant spectral flow for Dirac-type operators on manifolds.
problem Calculating spectral flow for Dirac-type operators on manifolds with group actions.
method Equivariant spectral flow construction for paths of Dirac-type operators on manifolds.
result Relates delocalised η-invariants and ρ-invariants for different positive scalar curvature metrics.
New method generates equilibrium glass configurations efficiently.
problem Sampling equilibrium configurations of amorphous materials is slow and difficult.
method Riemannian stochastic interpolation framework combining Riemannian stochastic interpolant and equivariant flow matching.
result Enforcing geometric and symmetry constraints significantly improves generative performance.
Survey on GKM theory in low dimensions, highlighting combinatorics-geometry interplay.
problem Generalizing classical ideas from quasi-toric manifolds to torus actions.
method GKM theory applied to low-dimensional cases.
result Particularly fruitful interaction between geometry and combinatorics in low dimensions.