arXiv research

Gromov boundary described for ray graph action.

Infinite clique of rays in plane minus Cantor set.

New rays on infinite type surfaces help understand their boundaries.

The ray graph of a surface with a Cantor set has infinite diameter and is hyperbolic.

Novel semi-supervised method for X-ray classification with minimal labels.

We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).

First-passage percolation affects graph properties like curvature and geodesics.

GraphXCOVID uses deep semi-supervised learning to identify COVID-19 from chest X-rays with minimal labels.

Study irrational rotations and construct 2-filling rays on infinite type surfaces.

New proof of harmonic map existence from punctured surfaces to crowned hyperbolic targets.

The study combines graph-minors and metric spaces, answering some questions and conjectures.

Study abelian factors in Lie algebras from graph edge labels.

The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-…

The paper shows how sublinearly Morse boundaries can be understood through combinatorial methods.

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singular…

Study extends geodesic ray transform results to orientable surfaces.

Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.

Study of rays and co-rays in Wasserstein space, introducing Busemann functions.

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. Firs…

The mapping class group $Γ$ of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of $Γ$ on this graph to find an explicit non tri…

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p \ge 0$ punctures, and let $ω(g,p)= 3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ in the curve graph satisfying the following optimal intersection property: for any natural number $k$, the endpoints …

We give a detailed microlocal study of X-ray transforms over geodesics-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation as…

Ray-Singer torsion is a mathematical concept with applications in physics.

We show that any grafting ray in Teichmüller space is (strongly) asymptotic to some Teichmüller geodesic ray. As an intermediate step we introduce surfaces that arise as limits of these degenerating Riemann surfaces. Given a grafting ray, the proof involves a Teichmüller ray with a conformally equivalent limit, and bui…

Develops geometric foundations for sublinear Morse boundaries in mapping class groups and Teichmüller spaces.

Survey on broken ray transforms and open problems.

We reduce the broken ray transform on some Riemannian manifolds (with corners) to the geodesic ray transform on another manifold, which is obtained from the original one by reflection. We give examples of this idea and present injectivity results for the broken ray transform using corresponding earlier results for the …

Three disjoint rays in euclidean 3-space form Borromean rays provided their union is knotted, but the union of any two components is unknotted. We construct infinitely many Borromean rays, uncountably many of which are pairwise inequivalent. We obtain uncountably many Borromean hyperplanes.

Study of ray transforms on spherically symmetric manifolds with low regularity metrics.

Study light ray transform in pseudo-Euclidean space, derive inversion formula, and prove stability.

Proves symplectomorphism of noncompact manifolds with embedded rays.

Study characterizes kernel of mixed ray transform on simple surfaces.

Unique geodesics selected by energy minimization in Teichmüller space.

Lecture notes on analysis tools for X-ray tomography.

The Shannon theorem is extended to locally compact groups.

The article recovers tensor fields from partial data using weighted divergent ray transforms.

Sharp mapping properties and regularization for X-ray transform on disks of constant curvature.

CNNs trained on one hospital's x-rays perform poorly on x-rays from other hospitals.

Regularity results for geodesic X-ray transform on nonsmooth manifolds

Study light ray transform on Lorentzian manifolds without conjugate points.

Paper shows invertibility of tensor X-ray transform on certain manifolds.

Local X-ray transform works well near boundaries in hyperbolic spaces.

Enhanced Graph Convolutional Network predicts diseases faster and more accurately.

Machine learning predicts x-ray pulse properties from XFEL parameters.

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

We consider the broken ray transform on Riemann surfaces in the presence of an obstacle, following earlier work of Mukhometov. If the surface has nonpositive curvature and the obstacle is strictly convex, we show that a function is determined by its integrals over broken geodesic rays that reflect on the boundary of th…

Study characterizes X-ray transform kernel for periodic slabs and related manifolds.

Model learns to focus on key areas of chest X-rays.