arXiv research

The study examines symmetries in spaces with positive or non-negative curvature.

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Gr…

Study shows instability of certain MOTSs with continuous symmetry.

In this paper, by the method of moving planes, we establish the monotonicity and symmetry properties of convex solutions for Monge-Ampere systems on bounded smooth planar domains.

Study entropy bounds and finiteness for symmetric self-shrinkers.

This work extends PAC-Bayesian learning guarantees to non-compact symmetries and non-invariant data.

Study symmetric linear bandits with hidden symmetry, achieving improved regret bounds.

We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

This paper bounds min-entropy leakage for Blowfish privacy using graph symmetries.

The study examines how equivariance in networks affects generalization error using PAC-Bayesian bounds.

Researchers set entropy limits for specific types of self-shrinkers.

Study controls curvature in Ricci flows using necks.

Study proves symmetry of bounded domains in Riemannian manifolds.

Develops a theory for equivariant networks with partial domain symmetry.

We bound the symmetry algebra of a vector distribution, possibly equipped with an additional structure, by the corresponding Tanaka algebra. The main tool is the theory of weighted jets.

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

The study examines the stretch factors of outer automorphisms and their latent symmetry.

We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian s…

Proposes learning invariances in neural networks using a weight-space approach.

The paper investigates how symmetry in models affects their performance and generalization.

Gradient descent struggles to learn equivariant neural networks, even with symmetries.

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 invariant Lipschitz bandits, improving regret bounds.

Classifies knots that bound equivariant surfaces with free symmetries.

The equivariant Gromov--Hausdorff convergence of metric spaces is studied. Where all isometry groups under consideration are compact Lie, it is shown that an upper bound on the dimension of the group guarantees that the convergence is by Lie homomorphisms. Additional lower bounds on curvature and volume strengthen this…

Ancient Ricci flows with bounded girth found in 3D and higher.

We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symm…

Investigates CR structures in 7D, showing 8 is max symmetry dimension.

Study of sub-Riemannian problem on specific Lie groups, revealing symmetries and bounds.

New Lipschitz bound for ReLU networks resists weight rescaling.

New rigidity results for manifolds with maximal symmetry rank and positive intermediate Ricci curvature.

New method discovers symmetries in differential equations from data.

SymPE breaks symmetries in equivariant networks, improving performance across various tasks.

Develops tests for conditional symmetry under group actions.

We give a uniform and elementary treatment of many classical and new triply periodic minimal surfaces in Euclidean space, based on a Schwarz-Christoffel formula for periodic polygons in the plane. Our surfaces share the property that vertical symmetry planes cut them into simply connected pieces.

Study finds maximal symmetry groups for CR structures with specific properties.

New lower bounds on embedding dimensions for neural network architectures.

Study on CR structures in 7D, proving maximal symmetry dimension.

We prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub-Riemannian manifolds with tranverse symmetries. This representation is obtained from the study of Bochner-Weitzenbock type formulas for sub-Laplacians on 1-forms. As a consequence, we prove new hypoelliptic heat…

An extended Kleinian group whose orientation-preserving half is a Schottky group is called an extended Schottky group. These groups correspond to the real points in the Schottky space. Their geometric structures is well known and it permits to provide information on the locus of fixed points of symmetries of handlebodi…

Bayesian framework detects symmetries in chaotic dynamical systems.

The study finds polynomial upper bounds for singularities in Einstein-scalar field system.

Combining M-algebra and hyperbolic involutory algebra extends exceptional tangent spaces to 11 dimensions.

The study explores Hesse manifolds and their symmetries in multifield cosmological models.

Defines discrete symmetry of manifolds and proves bounds on its value.

Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.

Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. W…

We show the following symmetry property of a bounded Reinhardt domain $Ω$ in $\mathbb{C}^{n+1}$: let $M=\partialΩ$ be the smooth boundary of $Ω$ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In Appendix we sta…