arXiv research

Constructs a function to count closed geodesics on Riemannian manifolds.

Study geodesic paths on flat surfaces, comparing length and singularity counts.

Study counts geodesics on modular surface, linking to necklace counting.

The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.

We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously …

Counts arcs in surfaces, proving convergence of geodesic currents.

Counting hyperbolic multi-geodesics with individual component lengths.

Geodesics count exponentially between triangulations of surfaces with enough topology.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

Study counts geodesic surfaces in knot complements, finding unique ones for small knots.

The paper counts geodesic loops on surfaces without conjugate points.

Research extends geodesic length function study to three holed sphere.

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-…

Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; an…

Study counts geodesics on hyperbolic 3-manifolds, proving prime theorems.

The Poincaré series for surfaces with boundary extends to the complex plane.

Counting periodic geodesics of bounded length and commutator structure on hyperbolic surfaces.

For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overlineΣ$, and for every compactly supported filling current $α$, we compute as $L \to \infty$, the number of mapping classes $φ$ so that $f(φ(α))\leq L$. As an application, when the surface in question is close…

Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.

The study counts Salem numbers linked to arithmetic hyperbolic orbifolds.

Study counts and equidistributes geodesic orbits on curved spaces.

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.

The study counts geodesics on special manifolds without focusing points.

Extends curve functions to geodesic currents with a simple criterion.

For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the projective indefinite special-orthogonal group of signature $(p,q)$. We study counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic …

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

Mirzakhani's thesis counts geodesics on hyperbolic surfaces, finding a specific asymptotic formula.

Study counts ergodic measures in surface lamination strata.

Counting subgroups of a surface using convex core lengths.

Study geodesics entering a fixed cusp neighborhood multiple times.

The mapping class group of a surface $§$ acts on the set of closed geodesics on $§$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits t…

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.

We investigate the number of geodesics between two points $p$ and $q$ on a contact sub-Riemannian manifold M. We show that the count of geodesics on $M$ is controlled by the count on its nilpotent approximation at $p$ (a contact Carnot group). For contact Carnot groups we make the count explicit in exponential coordina…

Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.

Given integers $g,n \geq 0$ satisfying $2-2g-n < 0$, let $\mathcal{M}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B \colon \mathcal{M}_{g,n} \to \mathbf{R}_{\geq 0}$ which assigns to $X…

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

Study on geodesics and dihedral groups in lattices.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

We compute the asymptotic growth rate of the number N(C, R) of closed geodesics of length less than R in a connected component C of a stratum of quadratic differentials. We prove that for any 0 < θ< 1, the number of closed geodesics of length at most R that spend at least θ-fraction of time outside of a compact subset …

We compute the asymptotics, as R tends to infinity, of the number of closed geodesics in Moduli space of length at most R, or equivalently the number of pseudo-Anosov elements of the mapping class group of translation length at most R.

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

We investigate the geometry of word metrics on fundamental groups of manifolds associated with the generating sets consisting of elements represented by closed geodesics. We ask whether the diameter of such a metric is finite or infinite. The first answer we interpret as an abundance of closed geodesics, while the seco…

The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.

Study shows central limit theorem for counting measures in non-smooth spaces.

Let $X$ be a compact, geodesically complete, locally CAT(0) space such that the universal cover admits a rank one axis. Assume $X$ is not homothetic to a metric graph with integer edge lengths. Let $P_t$ be the number of parallel classes of oriented closed geodesics of length $\le t$; then $\lim\limits_{t \to \infty} P…

Estimates the number of closed curves on surfaces with power-saving error terms.

We show that the number of square-tiled surfaces of genus $g$, with $n$ marked points, with one or both of its horizontal and vertical foliations belonging to fixed mapping class group orbits, and having at most $L$ squares, is asymptotic to $L^{6g-6+2n}$ times a product of constants appearing in Mirzakhani's count of …