arXiv research

The squashed 7-sphere $S^{7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{R}^{8} - \{ 0 \}$ has full holonomy ${\rm Spin}(7)$. There is a canonical calibrating 4-form $Φ$ on $\mathbb{R}^{8} - \{ 0 \}$. A minimal 3-submanifold in $S^{7}$ is called associative if its cone …

The study constructs associative 3-folds in squashed 3-Sasakian manifolds.

New solutions found for G2 system on squashed manifolds.

Researchers describe $G_2$-structures on $7$-sphere, finding harmonic representatives.

Study calculates deformations of instantons on a specific $Spin(7)$-manifold.

New Sasaki-Einstein 7-spheres found via Berglund-Hübsch transpose.

MPNNs over-squash distant node information, study shows.

We construct a new infinite family of models of exotic 7-spheres. These models are direct generalizations of the Gromoll-Meyer sphere. From their symmetries, geodesics and submanifolds half of them are closer to the standard 7-sphere than any other known model for an exotic 7-sphere.

Graph rewiring method alleviates over-squashing in GNNs.

In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit…

New Riemannian GNNs reduce over-squashing in graphs with negative curvature.

Graphs can be smoothed or squashed too, study finds.

Study $\mathrm{G}_2$-instantons on 7-sphere from 4-sphere instantons.

gLSTM improves graph neural networks by increasing storage capacity to prevent over-squashing.

Graph pruning improves neural network performance by addressing squashing and smoothing issues.

Graph neural networks struggle to propagate long-range information, causing over-squashing.

Classifies two families of simply connected 7-manifolds with minimal homological complexity.

Advocates against over-smoothing and over-squashing in GNNs, suggesting they are less critical than previously thought.

A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface $Σ$ so that the surfa…

New method uses curvature to improve graph neural networks.

Study on curvature properties of exotic 7-sphere using quaternionic geometry.

New analysis shows over-squashing limits GNNs' power.

The paper studies $G_2$-Poisson equations on 7-spheres and classifies invariant solutions.

Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class $W^{1,p}$, $1 < p < \infty$. If $f$ almost minimizes its $p$ Dirichlet energy then $f$ is Hölder continuous. If $p=2$ and $f$ is squeeze and squash stationary then $f$ is in VMO.

A physicist explores exotic 7-spheres using fibre bundles and instantons.

We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.

Study local moduli of Sasaki-Einstein metrics on specific polynomial links.

Unified treatment of gauge theories and Yang-Mills theory duality.

New Sasaki-Einstein 7-manifolds found, including rational homology 7-spheres and connected sums.

A new method boosts graph neural networks by preventing over-smoothing and over-squashing.

Abstract: Investigates octonion product deformations and related geometries.

We construct a co-dimension $3$ completely non-holonomic sub-bundle on the Gromoll-Meyer exotic $7$ sphere based on its realization as a base space of a Sp(2)-principal bundle with the structure group Sp(1). The same method is valid for constructing a co-dimension 3 completely non-holonomic sub-bundle on the standard 7…

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient.

Nearly $G_2$-structures are unstable under a modified $G_2$-Laplacian co-flow.

Here we generalize the Gromoll-Meyer construction of an exotic 7-sphere by producing geometric models of exotic 8, 10 and Kervaire spheres as quotients of sphere bundles over spheres by free isometric actions. We give a geometric application at the end.

In this paper, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza--Klein black hole spacetime in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space-time metric using Lie point symmetries are presented and then it the $n$-dimensional optima…

We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing $G$-equivariance on the homogeneous space $G/H=\mathrm{SU}(4)/\mathrm{SU}(3)$ endowed with its Sasaki-Einstein structure, and $G/H=\mathrm{Sp}(2)/\mathrm{Sp}(1)$ as a 3-Sasakian manifold. In both cases …

The paper identifies conditions for free circle actions on specific 7-manifolds.

A metric with positive sectional curvature on the Gromoll-Meyer exotic 7-sphere is constructed explicitly. The proof relies on a 2-parameter family of left invariant metrics on Sp(2) and a one-parameter family of conformal deformations via an isoparametric function F on it. One byproduct is a metric with positive secti…

Gromoll and Meyer have represented a certain exotic 7-sphere $Σ^7$ as a biquotient of the Lie group $G = Sp(2)$. We show for a 2-parameter family of left invariant metrics on $G$ that the induced metric on $Σ^7$ has strictly positive sectional curvature at all points outside four subvarieties of codimension $\geq 1$ wh…

DRew dynamically rewires message passing to improve long-range tasks.

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the near…

Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS_4 x X^7 and preserving at least half of the supersymmetry have been recently classified. Requiring that amount of supersymmetry forces X to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely-act…

For each nonnegative integer we find an open (4m+9)-dimensional simply-connected manifold admitting complete nonnegatively curved metrics whose souls are non-diffeomorphic, homeomorphic, and have codimension 2. We give a diffeomorphism classification of the pairs (N, soul) when N is a nontrivial complex line bundle ove…

A new graph neural network (NBA-GNN) avoids revisiting nodes to improve accuracy.

Study 3-Sasakian and G2 structures on manifolds.

In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…

New formalism solves kinematical constraints in curved backgrounds and non-trivial states.