arXiv research

The aim of this work is studying translating graphs by mean curvature flow in $\Real^3$. We prove non-existence of complete translating graphs over bounded domains in $\Real^2$. Furthermore, we show that there are only three types of complete translating graphs in $\Real^3$; entire graphs, graphs between two vertical p…

Inspired by the tremendous success of deep generative models on generating continuous data like image and audio, in the most recent year, few deep graph generative models have been proposed to generate discrete data such as graphs. They are typically unconditioned generative models which has no control on modes of the …

Study on stable translation lengths of surface homeomorphisms and their approximations.

This paper tackles graph translation challenges by predicting both node and edge attributes simultaneously.

We view molecular optimization as a graph-to-graph translation problem. The goal is to learn to map from one molecular graph to another with better properties based on an available corpus of paired molecules. Since molecules can be optimized in different ways, there are multiple viable translations for each input graph…

The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …

The paper explores new translating solitons and their relation to $Δ$-wings.

The paper proves a stability result for translating space-like graphs in Lorentz manifolds.

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…

In this paper we provide a full classification of complete translating graphs in $\mathbf{R}^3$. We also construct two $(n-1)$-parameter families of new examples of translating graphs in $\mathbf{R}^{n+1}$.

The study examines translation lengths of pseudo-Anosov maps on curve graphs.

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.

New bound for group action length without diameter restriction.

Circle packings on translation surfaces are consistent across different surfaces.

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian cur…

The purpose of these notes is to provide an introduction to those who want to learn more about translating solitons for the mean curvature flow in $\mathbb{R}^3$, particularly those which are complete graphs over domains in $\mathbb{R}^2$. In this paper we describe a full classification of complete translating graphs i…

Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.

Study on finiteness property of right-angled Artin groups actions on extension graphs.

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$.

The study classifies and investigates translators invariant under hyperpolar actions on symmetric spaces.

In this paper, we show that the minimal asymptotic translation length of the Torelli group $\mathcal{I}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group $Mod(S_g)$, which behaves like $1/g^2$. We also show that the minimal asymptotic translat…

The paper proves stability of certain graph types in Euclidean space with specific densities.

The present paper is a review of the current state of Graph-Link Theory (graph-links are also closely related to homotopy classes of looped interlacement graphs), dealing with a generalisation of knots obtained by translating the Reidemeister moves for links into the language of intersection graphs of chord diagrams. I…

G2Gs transforms target molecules into reactants without templates, improving accuracy.

The study proves stability of various graphical translators in mean curvature flow.

The paper classifies shapes of translating solitons for a specific flow.

In this article we prove two non-existence results for translating solitons of the mean curvature flow (translators for short) in $\mathbb{R}^{m+1}$. We also obtain an upper bound to the maximum height that a compact embedded translator in $\mathbb{R}^{3}$ can achieve. On the other hand, we study graphical perturbation…

The paper classifies shapes of translating solitons from isoparametric graphs.

Graphs from van der Corput sequence embed into Chamanara surface.

The problem of accelerating drug discovery relies heavily on automatic tools to optimize precursor molecules to afford them with better biochemical properties. Our work in this paper substantially extends prior state-of-the-art on graph-to-graph translation methods for molecular optimization. In particular, we realize …

Graph-to-Tree Neural Networks improve structured input-output translation in tasks like semantic parsing and math word problems.

Proposes local coordinate frames for improving model performance in complex dynamical systems.

The study explores normal generators for mapping class groups and their properties.

In this note we discuss graphs over a domain $Ω\subset N^2$ in the product manifold $N^2\times \mathbb{R}$. Here $N^2$ is a complete Riemannian surface and $Ω$ has peice-wise smooth boundary. Let $γ\subset\partialΩ$ be a smooth connected arc and $Σ$ be a complete graph in $N^2\times \mathbb{R}$ over $Ω$. We show that i…

The paper proves properties of surfaces with finite curvature in 3D space.

We recall the notion of (vertical) translating solitons in a product of a semi-Riemannian manifold $(M,g)$ and the real line. Mainly, we restrict our attention to those which are the graph of a smooth function. When dealing with submersions, we show a criteria to lift (or project) translating solitons from the base man…

Uniqueness of circle packings on certain translation surfaces is proven.

Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.

Scattering transforms are non-trainable deep convolutional architectures that exploit the multi-scale resolution of a wavelet filter bank to obtain an appropriate representation of data. More importantly, they are proven invariant to translations, and stable to perturbations that are close to translations. This stabili…

We construct a one-parameter family of singly periodic translating solutions to mean curvature flow that converge as the period tends to $0$ to the union of a grim reaper surface and a plane that bisects it lengthwise. The surfaces are semigraphical: they are properly embedded, and, after removing a discrete collection…

Classifies graph configuration spaces homeomorphic to manifolds.

Given $φ$ a pseudo-Anosov map, let $\ell_\mathcal{T}(φ)$ denote the translation length of $φ$ in the Teichmüller space, and let $\ell_\mathcal{C}(φ)$ denote the stable translation length of $φ$ in the curve graph. Gadre--Hironaka--Kent--Leininger showed that, as a function of Euler characteristic $χ(S)$, the minimal po…

Study on Monge-Ampère equations with polynomial growth rates.

SE(3)-Transformers maintain equivariance for 3D data under rotations and translations.

We prove that any complete immersed two-sided mean convex translating soliton $Σ\subset \mathbb{R}^3$ for the mean curvature flow is convex. As a corollary it follows that an entire mean convex graphical translating soliton in $\mathbb{R}^3$ is the axisymmetric "bowl soliton". We also show that if the mean curvature of…

Study shows mapping class groups are one-ended for surfaces with at least one end.

We consider the mean curvature flow of entire Lagrangian graphs with Lipschitz continuous initial data. Assuming only a certain bound on the Lipschitz norm of an initial entire Lagrangian graph in $\R^{2n}$, we show that the parabolic equation \eqref{PMA} for the Lagrangian potential has a longtime solution which is sm…