arXiv research

Study on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds, proving finiteness or infiniteness of induced distance.

Study investigates induced geometry on surfaces in 3D contact manifolds.

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

Defines Lorentzian distance on contactomorphisms, proving continuity and finite conditions.

We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of smooth compact manifolds with smooth metrics, is tangen- tial to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of smooth manifolds with rough metri…

In this paper we study the metric geometry of the space $Σ$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $τ$. The trace induces an incomplete Riemannian metric $<x,y>_a=τ(ya^{-1}xa^{-1})$, and though the techniques involved are quite different,…

The paper improves conformal prediction by analyzing the beta law of conditional coverage.

Uniform foliations with Reeb components on 3-manifolds.

Study the tradeoff between signal distortion and human perception over finite channels.

We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm $||.||_{\dot{H}^{-1}(ν_q)}$, that is known to linearize the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic formulation of…

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teic…

We prove an effective version of a theorem relating curve complex distance to electric distance in hyperbolic 3-manifolds, up to errors that are polynomial in the complexity of the underlying surface. We use this to give an effective proof of a result regarding maps between curve complexes of surfaces induced by finite…

We study the geometry of the space of densities $\VolM$, which is the quotient space $\Diff(M)/\Diff_μ(M)$ of the diffeomorphism group of a compact manifold $M$ by the subgroup of volume-preserving diffemorphisms, endowed with a right-invariant homogeneous Sobolev $\dot{H}^1$-metric. We construct an explicit isometry f…

The Gromov-Hausdorff distance provides a metric on the set of isometry classes of compact metric spaces. Unfortunately, computing this metric directly is believed to be computationally intractable. Motivated by applications in shape matching and point-cloud comparison, we study a semidefinite programming relaxation of …

Researchers solve 3D cube complex boundary rigidity problem.

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial or…

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodes…

Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called because it induces an L^2 topology on each tangent space. It turns out that this top…

We prove that the L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L^2 metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponentia…

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the i…

We introduce a new distance metric for non-linear embeddings of Tempered Exponential Measures.

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

Finite mapping class groups for Heegaard splittings with distance ≥ 3, but not for distance 2.

In human cognition, the expansion of perceived between-category distances and compression of within-category distances is known as categorical perception (CP). There are several hypotheses about the causes of CP (e.g., language, learning, evolution) but no functional model. Whether CP is essential to categorisation or …

Proves CLT for Brownian paths on pinched negative curvature manifolds.

Estimates distances between fixed points of certain Teichmüller actions.

Estimates path-valued data using signature metrics and local kernels.

Bounds on geodesic distances on Stiefel manifold derived from new metrics.

Paper estimates Wasserstein distance for Ricci shrinkers.

This article is about a natural distance function induced by smooth cobordisms between links. We show that the cobordism distance of torus links is determined by the profiles of their signature functions, up to a constant factor.

Study uses equivariant topology to measure distances between G metric spaces.

Generates infinite-depth hierarchical clusters from few examples.

The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…

Study proves finiteness for distance functions on curved surfaces with controlled curvature.

Let $Γ$ be a finite graph and let $Γ^{\mathrm{e}}$ be its extension graph. We inductively define a sequence $\{Γ_i\}$ of finite induced subgraphs of $Γ^{\mathrm{e}}$ through successive applications of an operation called "doubling along a star". Then we show that every finite induced subgraph of $Γ^{\mathrm{e}}$ is iso…

Distance function to a finite set is a topological Morse function.

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the g…

Completes the space of vector-valued one-forms on manifolds.

The paper analyzes how the one-dimensional Wasserstein distance captures pointwise density differences in finite samples.

Finite Goeritz groups for links with long bridge decompositions.

This study approximates distances between Gaussian processes and covariance operators using RKHS.

A new invariant captures geometric features of circle embeddings.

This work extends stochastic localization to joint probability measures for data analysis.

The paper studies convergence of cosmological spacetimes using null distance.

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by thes…

Paper proves circle packings converge to Riemann mapping for Jordan domains.

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manag…