The paper connects higher-dimensional mechanics to Lie n-algebroids.
problem Understanding interactions in higher-dimensional gauge systems.
method Comparing BV/BRST formalism with Lie n-algebroids and defining polytorsion.
result Relates topological n-branes to differential geometry on Lie n-algebroids.
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of mo…
The paper studies Lie n-algebroids and their representations up to homotopy.
problem Understanding Lie n-algebroids and their representations.
method Analyzes differential graded modules and representations up to homotopy, describes adjoint and coadjoint modules, and computes the Weil algebra.
result Alternative characterisation of non-degeneracy of higher Poisson structures.
This thesis generalizes structures on Q-manifolds and Lie n-algebroids.
problem Representation theory and linear structures of Q-manifolds and Lie n-algebroids. method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie n-algebroids. result Establishes an equivalence between VB-Lie n-algebroids and (n+1)-term representations up to homotopy of Lie n-algebroids. Characterizes Filippov n-algebroids using connections and formulas.
problem Generalizing Lie algebroids to Filippov n-algebroids.
method Introducing Filippov connections and transforming the Jacobi identity into the Bianchi-Filippov identity.
result Expressed the n-ary bracket using a torsion-free formula.
Article proves tangent complex structure of Lie n-groupoid.
problem Differentiating Lie n-groupoids.
method Proves representability of presheaf by tangent complex.
result Tangent complex of Lie n-groupoid carries Lie n-algebroid structure.
Uniform criteria for stability of fixed points in various geometric structures.
problem Stability of fixed points in Poisson geometry and higher Lie theory.
method Uniform approach to criteria for stability, using differential graded Lie algebras and cohomology.
result Vanishing of a finite-dimensional cohomology group implies stability of fixed points.
In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of…
We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic funct…
Characterizes Anosov representations and strongly convex cocompact groups with eigenvalue gaps.
problem Understanding Anosov representations and their properties.
method Characterizations via equivariant limit maps, Cartan property, and uniform gap summation.
result Characterizations of Anosov representations and strongly convex cocompact subgroups.
We study the simplicial {\ell} q,p cohomology of Carnot groups G. We show vanishing and non-vanishing results depending of the range of the (p, q) gap with respect to the weight gaps in the Lie algebra cohomology of G.
Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.
problem Finding free semigroups with critical exponents arbitrarily close to a subgroup's in dense subgroups of Lie groups.
method Analyzing Zariski dense discrete subgroups of Lie groups, showing the existence of free semigroups with critical exponents arbitrarily close to the subgroup's.
result The existence of free semigroups with critical exponents arbitrarily close to the subgroup's in dense subgroups of Lie groups.
We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian manifolds. We classify all structures with such symmetry dimensions. Geometric propertie…
We extend an L2 energy gap result due independently to Min-Oo and Parker (1982) for Yang-Mills connections on principal G-bundles, P, over closed, connected, four-dimensional, oriented, smooth manifolds, X, from the case of positive Riemannian metrics to the more general case of good Riemannian metrics, includ…
New quantum models unify Alexander and generalized Alexander polynomials for AC links.
problem Defining and distinguishing AC links and virtual knots.
method Generalizing AC links to virtual tangles and using quantum supergroups.
result Generalized Alexander polynomials are distinct from Alexander polynomials for AC links.
Unique submaximal symmetry found for certain parabolic geometries.
problem Determining the next realizable symmetry dimension in parabolic geometries.
method Analyzing submaximally symmetric structures of type (G,P) for specific Lie groups. result Local uniqueness of submaximally symmetric structures established.
A quasi-Lie scheme is a geometric structure that provides t-dependent changes of variables transforming members of an associated family of systems of first-order differential equations into members of the same family. In this note we introduce two quasi-Lie schemes for studying second-order Gambier equations in a geome…
The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a un…
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.
Unified framework for critical scaling of inverse temperature in self-attention.
problem Conflicting inverse-temperature laws for long-context self-attention.
method Counting gaps and defining an upper-tail accumulation scale.
result Critical inverse-temperature scale determined by gap-counting function.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
Researchers classify cmc surfaces using Jacobi elliptic functions.
problem Classifying rotational cmc surfaces in non-Euclidean space forms.
method Lie sphere geometric description of rotational linear Weingarten surfaces.
result Explicit parametrizations of cmc surfaces in hyperbolic space.
Given a metric measure space (X,d,m) that satisfies the Riemannian Curvature Dimension condition, RCD∗(K,N), and a compact subgroup of isometries G≤Iso(X) we prove that there exists a G−invariant measure, mG, equivalent to m such that (X,d,mG) is still a…
In the scientific literature there are basically two schools of formulating Lagrangian (or Hamiltonian) mechanics in the (Lie) algebroid setting: in terms of prolongations and in terms of Tulczyjew triples. Despite the fact that in both approaches we describe the same phenomena, so far no comparison between prolongatio…
We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the…
C-projective structures are analogues of projective structures in the complex setting. The maximal dimension of the Lie algebra of c-projective symmetries of a complex connection on an almost complex manifold of C-dimension n>1 is classically known to be 2n2+4n. We prove that the submaximal dimension is equal to $…
A reverse Riesz estimate and spectral gap imply a Poincaré inequality.
problem Establishing a Poincaré inequality using a reverse Riesz estimate and spectral gap.
method Combining a reverse Riesz estimate and spectral gap condition to prove a Poincaré inequality.
result A Poincaré inequality is derived from a reverse Riesz estimate and spectral gap condition.
Study 1-Lefschetz contact solvmanifolds, proving their characterization.
problem Characterize 1-Lefschetz contact solvmanifolds.
method Prove the 1-Lefschetz condition on Lie algebras via 1-dimensional central extensions and explicit cohomology relations.
result Unimodular symplectic Lie algebras are 1-Lefschetz if and only if their contactizations are.
In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present…
The paper proves rigidity results for certain supergravity backgrounds in 11 dimensions.
problem Supersymmetry gap problem for supergravity backgrounds in 11 dimensions.
method Study of curvature restrictions and use of bijective correspondence between filtered deformations of Lie superalgebras and highly supersymmetric backgrounds.
result Rigidity results for specific supergravity backgrounds with low rank 4-form and high Killing spinor dimension.
New geometries defined for string models, filling gaps in the literature.
problem Developing mathematical structures for string models.
method Defining E-metric-connection geometries and locality structures.
result Unified framework for metric-affine and generalized geometries.
New method for studying t-dependent Hamilton equations on cosymplectic manifolds.
problem Existence and stability of solutions of t-dependent Hamilton equations. method Develops a cosymplectic energy-momentum method for Hamilton equations with more types of symmetries.
result Provides a more general framework for studying t-dependent Hamilton equations. The paper explores gaps in curvature-related metrics and rigidity.
problem Understanding gaps in curvature-related metrics and rigidity.
method Analyzes three types of gaps: spectral, metric-rigidity, and topological-rigidity.
result Proposes open problems in the field.
For convex domains with C1,ε boundary we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different closed complex faces of the boundary, then the automorphism group has finitely many components and the connected component of the identity is th…
Study gap-dependent regret bounds for risk-sensitive RL.
problem Risk-sensitive reinforcement learning with entropic risk measure.
method Propose cascaded gaps to adapt to problem structures, derive regret bounds.
result Exponential improvement over existing bounds in appropriate settings.
One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combinatio…
Paper analyzes origami slope gaps and their distribution, finding a unique pattern.
problem Analyzing slope gaps in origami surfaces.
method Derived slope gap distribution of a specific origami by considering return times under the horocycle flow.
result Found a unique distribution of origami slope gaps, not a sum of scaled Hall distributions.
Random hyperbolic surfaces have nearly optimal spectral gaps.
problem Proving the nearly optimal spectral gap conjecture for random Belyi surfaces.
method Using the Brooks-Makover model, the authors show a spectral gap greater than 1/4 - c/log(n).
result A random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than 1/4 - c/log(n).
Paper improves volume gap between minimal submanifolds and unit spheres.
problem Volume gap between minimal submanifolds and unit spheres.
method Modified Cheng-Li-Yau coefficients and applied Cheng-Yang eigenvalue estimate for Laplacian.
result Enhanced volume gap between minimal submanifolds and unit spheres.
Generative operators solve many convex problems with minimal parameters.
problem Worst-case parameter bounds limit the practical use of neural operators.
method Developed generative equilibrium operators (GEOs) using realizable finite-dimensional layers.
result GEOs can uniformly approximate solutions to convex optimization problems with logarithmic growth in parameters.
The article proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
problem Proving a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
method Establishing conformal log-concavity estimates for the first eigenfunction.
result Proves a conjecture about the fundamental gap for horoconvex domains in hyperbolic space.
Kahler-Einstein metrics linked to eigenvalue gaps on Fano manifolds.
problem Existence of Kahler-Einstein metrics on Fano manifolds.
method Characterization via eigenvalue gaps of Cauchy-Riemann and Hamiltonian vector fields.
result Existence of Kahler-Einstein metrics linked to eigenvalue gaps.
Local gaps in Ricci shrinkers depend only on dimension.
problem Understanding local properties of Ricci shrinkers.
method Proved local versions of Ricci curvature and entropy gap theorems.
result Local gaps depend only on dimension, not global entropy.
Study shows gaps in Bitcoin order book are linked to returns but only in the short term.
problem Understanding the relationship between gaps and returns in Bitcoin order books.
method Examined the dynamics of gaps and returns in a Bitcoin order book without considering long-term causation.
result The causal relationship between gaps and returns is limited to instantaneous causation.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.