Related papers: Closed geodesics with prescribed intersection numb…

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in…

Circular and radial geodesics are studied in the spacetime described by the $\gamma$ metric. Their behaviour is compared with the spherically symmetric situation, bringing out the sensitivity of the trajectories to deviations from spherical…

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If $G$ is a graph, then ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that one can assign a fixed geodesic to…

In this paper, we provide upper and lower bounds on the crossing numbers of dense graphs on surfaces, which match up to constant factors. First, we prove that if $G$ is a dense enough graph with $m$ edges and $\Sigma$ is a surface of genus…

Let $\gamma_0$ be a curve on a surface $\Sigma$ of genus $g$ and with $r$ boundary components and let $\pi_1(\Sigma)\curvearrowright X$ be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number…

The problem of map enumeration concerns counting connected spatial graphs, with a specified number $j$ of vertices, that can be embedded in a compact surface of genus $g$ in such a way that its complement yields a cellular decomposition of…

Given a compact orientable surface with finitely many punctures $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential unoriented simple closed curves in $\Sigma$. We determine a complete set of relations for a function…

The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e. of any curve obtained by continuous deformation. Given a curve $c$ represented by a closed walk of length at…

The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…

We study the asymptotics of large directed graphs, constrained to have certain densities of edges and/or outward $p$-stars. Our models are close cousins of exponential random graph models (ERGMs), in which edges and certain other subgraph…

We consider orientation-preserving actions of finite groups $G$ on pairs $(S^3, \Sigma)$, where $\Sigma$ denotes a compact connected surface embedded in $S^3$. In a previous paper, we considered the case of closed, necessarily orientable…

We extend the application of Hamiltonian Monte Carlo to allow for sampling from probability distributions defined over symmetric or Hermitian positive definite matrices. To do so, we exploit the Riemannian structure induced by Cartan's…

Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum…

We give a uniform bound of the bounded geodesic image theorem for the closed oriented surfaces. The proof utilizes the bicorn curves introduced by Przytycki and Sisto (see arXiv:1502.02176). With the uniformly bounded Hausdorff distance of…

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $L/k$, where $L$ is the length of the geodesic. We investigate the existence and behavior of these curves on doubled polygons and show…

The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a…

Resonance relations among periodic orbits on given energy hypersurfaces are very important for getting deeper understanding of the dynamics of the corresponding Hamiltonian systems. In this paper, we establish two new resonance identities…

Let X be a Hadamard manifold and $\Gamma$ a discrete group of isometries of X which contains an axial isometry without invariant flat half plane. We study the behavior of conformal densities on the geometric limit set of $\Gamma$ in order…

Given a compact orientable surface $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential simple loops on $\Sigma$. We determine a complete set of relations for a function from $\Cal S(\Sigma)$ to $\bold Z$ to be a…