We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-Müter metric. We then consider a suitable third order perturbation of this Cheeger-Müter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.

It’s usually the task of the modeler to make their assumptions fit the real world as closely as is practical. It was now our task to make a real-world drum that conformed to the modeling assumptions.

Hi, I am Paul Schwahn, a postdoctoral researcher at Unicamp.

A classification theorem in Lean of solvable Lie algebras of dimension zero to three - LieLean/LowDimSolvClassification

A new semi-supervised machine learning package is introduced which successfully solves the Euclidean vacuum Einstein equations with a cosmological constant, without any symmetry assumptions. The model architecture contains subnetworks for each patch in the manifold-defining atlas. Each subnetwork predicts the components of a metric in that patch, with the associated Einstein conditions, of the form $R_{μν} - λg_{μν} = 0$, being used as independent loss components; in our conventions, $μ,ν= 1, 2, \cdots, n$, where $n$ is the dimension of the Riemannian manifold and $λ\in \{+1, 0, -1\}$. To ensure the consistency of the global structure of the manifold, another loss component is introduced across the patch subnetworks which enforces the coordinate transformation between the patches, $ g' = J g J^T$, for an appropriate analytically known Jacobian $J$. We test our method for the case of spheres represented with 2 patches in dimensions $2,3,4,5$; in dimensions $2, 3$ the geometries have been fully classified, however it is unknown whether a Ricci-flat metric can be put on spheres in dimensions $4, 5$, which we provide numerical evidence against.