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1. Diffusion Curvature for Estimating Local Curvature in High Dimensional Data

   Bhaskar, Dhananjay; MacDonald, Kincaid; Fasina, Oluwadamilola; Thomas, Dawson; Rieck, Bastian; Adelstein, Ian; Krishnaswamy, Smita (January 2022, Advances in Neural Information Processing Systems 35 (NeurIPS 2022))

   We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local curvature based on the laziness of a random walk starting at a point or region of the data. We show that this laziness directly relates to volume comparison results from Riemannian geometry. We then extend this scalar curvature notion to an entire quadratic form using neural network estimations based on the diffusion map of point-cloud data. We show applications of both estimations on toy data, single-cell data and on estimating local Hessian matrices of neural network loss landscapes. 

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2. Simulated surface diffusion in nanoporous gold and its dependence on surface curvature

   https://doi.org/10.1016/j.commatsci.2023.112430  

   Winkeljohn, Conner Marie; Shahriar, Sadi; Seker, Erkin; Mason, Jeremy K (October 2023, Computational Materials Science)

   The morphological evolution of nanoporous gold is generally believed to be governed by surface diffusion. This work specifically explores the dependence of mass transport by surface diffusion on the curvature of a gold surface. The surface diffusivity is estimated by molecular dynamics simulations for a variety of surfaces of constant mean curvature, eliminating any chemical potential gradients and allowing the possible dependence of the surface diffusivity on mean curvature to be isolated. The apparent surface diffusivity is found to have an activation energy of ~0.74 eV with a weak dependence on curvature, but is consistent with the values reported in the literature. The apparent concentration of mobile surface atoms is found to be highly variable, having an Arrhenius dependence on temperature with an activation energy that also has a weak curvature dependence. These activation energies depend on curvature in such a way that the rate of mass transport by surface diffusion is nearly independent of curvature, but with a higher activation energy of ~1.01 eV. The curvature dependencies of the apparent surface diffusivity and concentration of mobile surface atoms is believed to be related to the expected lifetime of a mobile surface atom, and has the practical consequence that a simulation study that does not account for this finite lifetime could underestimate the activation energy for mass transport via surface diffusion by ~0.27 eV. 

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3. A Review of and Some Results for Ollivier–Ricci Network Curvature

   https://doi.org/10.3390/math8091416  

   Azarhooshang, Nazanin; Sengupta, Prithviraj; DasGupta, Bhaskar (September 2020, Mathematics)

   null (Ed.)

   Characterizing topological properties and anomalous behaviors of higher-dimensional topological spaces via notions of curvatures is by now quite common in mainstream physics and mathematics, and it is therefore natural to try to extend these notions from the non-network domains in a suitable way to the network science domain. In this article we discuss one such extension, namely Ollivier’s discretization of Ricci curvature. We first motivate, define and illustrate the Ollivier–Ricci Curvature. In the next section we provide some “not-previously-published” bounds on the exact and approximate computation of the curvature measure. In the penultimate section we review a method based on the linear sketching technique for efficient approximate computation of the Ollivier–Ricci network curvature. Finally in the last section we provide concluding remarks with pointers for further reading. 

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4. Decay of scalar curvature on uniformly contractible manifolds with finite asymptotic dimension

   https://doi.org/10.1002/cpa.22128  

   Wang, Jinmin; Xie, Zhizhang; Yu, Guoliang (August 2023, Communications on Pure and Applied Mathematics)

   Abstract Gromov proved a quadratic decay inequality of scalar curvature for a class of complete manifolds. In this paper, we prove that for any uniformly contractible manifold with finite asymptotic dimension, its scalar curvature decays to zero at a rate depending only on the contractibility radius of the manifold and the diameter control of the asymptotic dimension. We construct examples of uniformly contractible manifolds with finite asymptotic dimension whose scalar curvature functions decay arbitrarily slowly. This shows that our result is the best possible. We prove our result by studying the index pairing between Dirac operators and compactly supported vector bundles with Lipschitz control. A key technical ingredient for the proof of our main result is a Lipschitz control for the topologicalK‐theory of finite dimensional simplicial complexes. 

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5. The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature

   https://doi.org/10.1090/proc/15240  

   Lim, Alice (August 2021, Proceedings of the American Mathematical Society)

   In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative N N -Bakry Émery Ricci curvature. We study the Splitting Theorem and a property called the geodesic loops to infinity property in relation to spaces with nonnegative N N -Bakry Émery Ricci curvature. In addition, we show that if M n M^n is a complete, noncompact Riemannian manifold with nonnegative N N -Bakry Émery Ricci curvature where N > n N>n , then H n − 1 ( M , Z ) H_{n-1}(M,\mathbb {Z}) is 0 0 . 

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