arXiv research

The paper finds bounds on shortest dense curves on surfaces.

Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

Algorithms find second and third shortest non-trivial closed walks on surfaces.

A curve around a sphere must be at least 4π long.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

Study bounds the length of shortest periodic geodesics on certain curved spaces.

A curve of minimum length to enclose a unit sphere in 3D is at least 4π.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.

New proof confirms surfaces can be divided into polygons.

Sharp bounds found on shortest geodesic on punctured spheres.

We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.

New bounds on shortest geodesic loops on a sphere.

Optimizes the first eigenvalues of Riemann surfaces for large genus.

Spirals are not shortest paths in certain sub-Riemannian geometries.

We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic length of compressible curves.

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

The distortion of a curve measures the maximum arc/chord length ratio. Gromov showed any closed curve has distortion at least pi/2 and asked about the distortion of knots. Here, we prove that any nontrivial tame knot has distortion at least 5pi/3; examples show that distortion under 7.16 suffices to build a trefoil kno…

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use t…

This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.

Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.

Geodesics in 3D space with 1-2 analytic obstacles, proving geodesic independence.

Paper proves curves can be smoothed to reduce self-intersection by exactly 1.

What length of rope (of given diameter) is required to tie a particular knot? To answer this question, we define some new notions of thickness for a space curve, one based on Gromov's distortion, and another generalizing the thickness of Litherland, Simon et al. We prove a basic inequality between these thickness measu…

If a variational problem comes with no boundary conditions prescribed beforehand, and yet these arise as a consequence of the variation process itself, we speak of a free boundary values variational problem. Such is, for instance, the problem of finding the shortest curve whose endpoints can slide along two prescribed …

Relatively extremal knots are the relative minima of the ropelength functional in C^1 topology. On the set curves of fixed length, they are the relative maxima of thickness (normal injectivity radius) functional, including the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has thickness equal to h…

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $δ$-hyperbolic for some $δ=δ(g,p)$. In this paper, we show that there exists some $δ>0$ independent of $g,p$ such that the cu…

The study of random surfaces reveals asymptotic lengths of separating geodesics.

Study geodesics and shortest arcs on Lie groups with specific metrics.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

Study geodesics and shortest arcs on Lie groups with specific metrics.

It is a classical theorem of Loewner that the systole of a Riemannian torus can be bounded in terms of its area. We answer a question of a similar flavor of Robert Young showing that if $T$ is a Riemannian 2-torus with boundary in $\mathbb R ^n$, such that the boundary curve is a standard unit circle, then the length o…

Generative Flow Networks solve shortest path problems in graphs.

Deep learning approximates shortest path distances in large graphs.

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

Origami edge-paths connect coherent curves on surfaces.

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

Study on shortest arcs on hyperbolic surfaces with boundary.

Study finds bounds for systole length on arithmetic punctured spheres.

We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call "homotopic curve shortening" (HCS), starts with a closed curve (which might self-intersect) in the presence of a set $P\subset \mathbb R^2$ of point obstacles, and evolves in discrete…

3D spheres can't be swept by short curves, complicating geodesic length estimates.

Geodesics connect model modes in neural network loss landscapes.

Algorithms compute length spectra of torus graphs efficiently.