Lie group integrators improve global error estimates.
problem Global error estimates for Lie group integrators.
method Relate local error to global error, derive from bounds.
result Lie-Butcher theory proves global error estimates for Lie group integrators.
We solve extending local Lie groupoids to global ones, linking integrability and associativity.
problem Extending local Lie groupoids to global Lie groupoids.
method We characterize integrable local Lie groupoids and establish a relationship between their integrability and associativity failure.
result A precise relationship between integrability of Lie algebroids and associativity failure in local integration.
A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold M. A pseudoaction generates a pseudogroup of transformations of M in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the…
Global minimizers exist for Tonelli Lagrangians on half-Lie groups.
problem Existence and properties of minimizers for Lagrangians on infinite-dimensional spaces.
method Introduced Tonelli Lagrangians on half-Lie groups, proved existence of minimizers and flow lines.
result Global minimizers exist above certain energy thresholds.
Global analysis of Dixmier traces and Wodzicki residues on compact Lie groups.
problem Computing Dixmier traces and Wodzicki residues on compact Lie groups.
method Global quantisation approach, using global symbols and representation theory.
result Explicit formulae for Dixmier traces and Wodzicki residues on compact Lie groups.
In this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r≥3. The second method provides us with global solutio…
Local and global rigidity results for Lie group actions on pseudo-Riemannian manifolds.
problem Rigidity of isometric actions of simple Lie groups on pseudo-Riemannian manifolds.
method Analyzing normal bundles and using Lie group properties.
result Local and global rigidity theorems for specific Lie groups and manifolds.
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
problem Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
method Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
result Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.
We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these glo…
For a strict Lie 2-group, we develop a notion of Lie 2-algebra-valued differential forms on Lie groupoids, furnishing a differential graded-commutative Lie algebra equipped with an adjoint action of the Lie 2-group and a pullback operation along Morita equivalences between Lie groupoids. Using this notion, we define co…
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
A classification is given of the exceptional Z2×Z2-symmetric spaces G/K by A.Kollross, where G is an exceptional compact Lie group or Spin(8), and moreover the structure of K is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms…
Conserved quantities on multisymplectic manifolds extend symplectic geometry results.
problem Defining and analyzing conserved quantities on multisymplectic manifolds.
method Defining globally conserved quantities as Lie derivatives of exact forms, extending classical results in symplectic geometry.
result A family of globally conserved quantities can be obtained under certain symmetries and homotopy co-momentum maps.
We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.
Survey on Higgs bundle moduli spaces and their connected components.
problem Counting connected components of Higgs bundle moduli spaces.
method Analyzes moduli spaces for Higgs bundles associated with real Lie groups and closed Riemann surfaces.
result Explicit descriptions of some moduli space components are possible.
The study pinches the Ricci functional on solvmanifolds, identifying special metrics.
problem Analyzing the Ricci functional on solvmanifolds.
method Using properties of the beta operator and left-invariant metrics.
result Solvsolitons and almost-abelian Lie groups are global maxima of the Ricci pinching functional.
The paper constructs and analyzes globally exceptional Z3 x Z3-symmetric spaces for specific Lie groups.
problem Understanding and constructing symmetric spaces for exceptional Lie groups.
method Introduced and used the concept of Γ-symmetric spaces, constructed automorphisms, and determined group structures.
result Explicit constructions and structures of exceptional Z3 x Z3-symmetric spaces for G2, F4, and E6.
Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein …
The paper finds solutions for specific curvature conditions on 5D Lie groups.
problem Finding metrics with prescribed Ricci curvature on 5D nilpotent Lie groups.
method Applied Milnor-type theorem technique to prove global existence of (g, c).
result Global existence of (g, c) for prescribed Ricci curvature on 5D nilpotent Lie groups.
Develops a new framework for generalized Ricci flow on Lie groups.
problem Global existence and geometric properties of Ricci flow on Lie groups.
method Inspired by Lauret's bracket flow, studies generalized Ricci flow on discrete quotients of Lie groups.
result Establishes global existence on solvmanifolds in arbitrary dimensions.
We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as a…
Extends pseudo-differential operators theory to compact Lie groups.
problem Global pseudo-differential operators on compact Lie groups.
method Develops a subelliptic pseudo-differential calculus for compact Lie groups.
result Establishes subelliptic versions of Fefferman and Calderón-Vaillancourt theorems.
New framework for equivariant neural networks using Lie group decompositions.
problem Limitations of existing equivariant neural network methods for Lie groups.
method Lie group structure and geometry, decomposition into subgroups and submanifolds.
result Equivariant neural networks for affine transformations outperform previous methods.
This research bridges Killing vectors and Lie algebras through induced vector fields.
problem Understanding the relationship between Killing vector fields and Lie algebras.
method Defining and exploring induced vector fields to connect Killing vector fields with isometry Lie groups.
result Established a new connection between Killing vector fields and Lie algebras.
Lie groupoid equivariant neural networks are a new type of neural network.
problem Designing neural networks that respect the structure of Lie groupoids.
method Introducing Lie groupoid equivariant convolutions and layers, and showing their equivalence to Lie algebroid-equivariant networks.
result Lie groupoid equivariant neural networks are equivalent to certain Lie algebroid-equivariant networks.
The paper integrates Rota-Baxter Lie algebras into Lie group structures and geometries.
problem Integrating Rota-Baxter operators into Lie group structures and geometries.
method Introducing Rota-Baxter operators on Lie groups, Lie algebroids, and groupoids.
result Geometrization of Rota-Baxter Lie algebras and groups.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
The paper extends a fibration theorem to all orbifolds, proving an isomorphism conjecture.
problem Proving the Farrell-Jones Isomorphism conjecture for orbifolds.
method Extending a fibration theorem to all orbifolds of genus ≥1. result Proves the Farrell-Jones Isomorphism conjecture for orbifolds.
Study Hofer's metric in compact Lie groups using group invariant convex geometry.
problem Hofer's metric in compact Lie groups and its geodesics.
method Group invariant convex geometry, functional and matrix analysis.
result Existence and characterization of geodesics in compact Lie groups.
Researchers prove solvmanifolds are global maxima for Ricci pinching functional in new cases.
problem Whether solvmanifolds are global maxima for Ricci pinching functional.
method Analyzing left-invariant metrics on solvable Lie groups, proving global maxima in specific cases.
result Proves solvmanifolds are global maxima for Ricci pinching functional in two new cases.
Local-to-global principle for Morse actions on symmetric spaces.
problem Recognizing Morse actions on symmetric spaces.
method Equivariant Morse quasiisometric embeddings of trees into symmetric spaces.
result Algorithmic recognizability of Morse actions and construction of Morse Schottky subgroups.
We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.
Proposes a new metric learning method using Lie group geodesics.
problem Improving distance metrics for k-NN classification.
method Geodesic interpolation on Lie transformation group to calculate velocities and produce a diffeomorphic global transformation.
result Effective in synthetic and real datasets, improving k-NN classification.
Study on simplicity of Lie skew braces, proving new results for compact cases.
problem Simplicity of Lie skew braces, focusing on compact connected cases.
method Reviewing correspondence, investigating ideals and rigidity, proving main result for compact Lie skew braces.
result Compact connected simple Lie skew braces are either trivial or have simple underlying Lie groups.
The study examines attractors in Cartan foliations and their properties.
problem Existence of attractors in Cartan foliations.
method Reduction of the problem to the action of Lie groups and holonomy groups.
result Conditions for the existence of attractors in reductive Cartan foliations.
Study on Einstein metrics on complex projective spaces with specific group actions.
problem Finding Einstein metrics invariant under cohomogeneity one Lie group actions.
method Analyzing Einstein equation for diagonal invariant metrics under five Takagi models.
result Nonexistence of smooth globally defined invariant Einstein metrics in four models, necessary condition in the fifth.
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{é}nez. As homogeneous manifolds, these spaces are of the G/H, where G is a connected compact simple Lie group with an automorphism γ~ of oder 4 and H is a fixed points subgroup Gγ of G. In the present article, for t…
The study proves stability of a flow on specific Lie groups.
problem Global stability of the Pluriclosed flow on compact Lie groups.
method Computation of cohomology, verification of flat metrics, and analysis of complex structures.
result Stability of the pluriclosed flow on compact Lie groups of rank two.
In this paper we determine the at least 4-dimensional affine reductive homogeneous manifolds for an at most 9-dimensional simple Lie group or an at most 6-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable …
Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
The paper consider the symmetric of Finsler spaces. We give some conditions about globally symmetric Finsler spaces. Then we prove that these spaces can be written as a coset space of Lie group with an invariant Finsler metric. Finally, we prove that such a space must be Berwaldian
This work characterizes global quotient stacks---smooth stacks associated to a finite group acting a manifold---among smooth quotient stacks [M/G], where M is a smooth manifold equipped with a smooth proper action by a Lie group G. The characterization is described in terms of the action of the connected componen…
This thesis explores symplectic foliations and local Lie groupoids, with applications in current theory.
problem Understanding calibratable symplectic foliations and local Lie groupoids.
method Applying de Rham's and Sullivan's theories to symplectic foliations and generalizing Mal'cev and Olver's theorems.
result Generalizations of theorems by Mal'cev and Olver for local Lie groupoids and algebroids.
Study proper actions of Lie groups on symmetric spaces, finding rigidity results and Hurwitz-Radon numbers.
problem Proper actions of non-compact semisimple Lie groups on pseudo-Riemannian symmetric spaces.
method Analysis of symmetric spaces and rigidity results.
result Any connected non-compact semisimple Lie group acting properly on these spaces must be globally isomorphic to Spin(n,1) up to compact factors. We treat the almost differentiable left A-loops as images of global differentiable sharply transitive sections σ:G/H→G for a Lie group G such that G/H is a reductive homogeneous manifold. In this paper we classify all 3-dimensional connected strongly left alternative almost differentiable left A-loops L, …
Study on deformation of affine structures on Lie groups using cohomology.
problem Understanding the gap between global topological invariants and left-invariant affine structures.
method Comparing De Rham and Koszul-Vinberg cohomology groups on Lie groups SO(2), H3(R), and SGal(3).
result Vanishing theorem for the second KV-cohomology group, proving structural rigidity of orbits.
The cohomology of affine structures on Lie groups is compared with that of Koszul-Vinberg algebras.
problem Comparing De Rham cohomology and KV-cohomology on Lie groups SO(2), H3(R), and Galilei group SGal(3).
method Constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups.
result Evaluating the algebraic quotient of the cohomology groups.