Related papers: Geometry of Deformed Cellular Spaces
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local…
Let $g_t$ be a smooth 1-parameter family of negatively curved metrics on a closed hyperbolic 3-manifold $M$ starting at the hyperbolic metric. We construct foliations of the Grassmann bundle $Gr_2(M)$ of tangent 2-planes whose leaves are…
We review recent results and ongoing investigations of the symplectic and Poisson geometry of derived moduli spaces, and describe applications to deformation quantization of such spaces.
The development of traction-force microscopy, in the past two decades, has created the unprecedented opportunity of performing direct mechanical measurements on living cells as they adhere or crawl on uniform or micro-patterned substrates.…
The geometry of generative models serves as the basis for interpolation, model inspection, and more. Unfortunately, most generative models lack a principal notion of geometry without restrictive assumptions on either the model or the data…
At the classical level, redefinitions of the field content of a Lagrangian allow to rewrite an interacting model on a flat target space, in the form of a free field model (no potential term) on a curved target space. In the present work we…
In some scientific fields, a scaling is able to modify the topology of an observed object. Our goal in the present work is to introduce a new formalism adapted to the mathematical representation of this kind of phenomenon. To this end, we…
The remarkable range of biological forms in and around us, such as the undulating shape of a leaf or flower in the garden, the coils in our gut, or the folds in our brain, raise a number of questions at the interface of biology, physics and…
We focus on studying, numerically, the scalar curvature tensor in a two-dimensional discrete space. The continuous metric of a two-sphere is transformed into that of a lattice using two possible slicings. In the first, we use two integers,…
In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…
Deployable structures, essential across various engineering applications ranging from umbrellas to satellites, are evolving to include soft, morphable designs where geometry drives transformation. However, a major challenge for soft…
Living cells respond to mechanical changes in the matrix surrounding them by applying contractile forces that are in turn transmitted to distant cells. We calculate the mechanical work that each cell performs in order to deform the matrix,…
We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings,…
The membrane curvature of cells and intracellular compartments continuously adapts to enable cells to perform vital functions, from cell division to signal trafficking. Understanding how membrane geometry affects these processes in vivo is…
Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation…
Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a class of ' cellular networks'…
The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The…
The Gromov-Hausdorff (GH) distance is traditionally used for measuring distances between metric spaces. It is defined as the minimal distortion of embedding one surface into the other, while the optimal correspondence can be described as…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…