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Random fields are ubiquitous mathematical structures in physics, with applications ranging from thermodynamics and statistical physics to quantum field theory and cosmology. Recent works on information geometry of Gaussian random fields…

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm…

We investigate the invariant metrics and complex geodesics in the universal Teichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk using Milin's coefficient inequalities. This technique allows us to establish that all…

In this paper, we leverage the properties of non-Euclidean Geometry to define the Geodesic distance (GD) on the space of statistical manifolds. The Geodesic distance is a real and intuitive similarity measure that is a good alternative to…

Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving…

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on…

Shape analysis and compuational anatomy both make use of sophisticated tools from infinite-dimensional differential manifolds and Riemannian geometry on spaces of functions. While comprehensive references for the mathematical foundations…

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete…

We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we…

In this article we describe applications of the numerical method of discrete differential forms in computational GR. In particular we consider the initial value problem for vacuum space-times that admit plane gravitational waves. As…

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes…

A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic…

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by…

In this paper we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The…

Optimization techniques are at the core of many scientific and engineering disciplines. The steepest descent methods play a foundational role in this area. In this paper we studied a generalized steepest descent method on Riemannian…

In this work, we propose to study the global geometrical properties of generative models. We introduce a new Riemannian metric to assess the similarity between any two data points. Importantly, our metric is agnostic to the parametrization…

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to…

Local and global illumination were recently defined in Riemannian manifolds to visualize classical Non-Euclidean spaces. This work focuses on Riemannian metric construction in $\mathbb{R}^3$ to explore special effects like warping, mirages,…

We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This…

Geometric data analysis and learning has emerged as a distinct and rapidly developing research area, increasingly recognized for its effectiveness across diverse applications. At the heart of this field lies curvature, a powerful and…