arXiv research

Counting tripods on a flat torus using lattice point counting.

Estimates point counts in Teichmüller space for mapping class groups.

Counting hyperbolic multi-geodesics with individual component lengths.

Calegari, Marques, and Neves count minimal surfaces in hyperbolic manifolds.

Uniform counting formulas for orthogeodesics in Kleinian groups converge.

We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.

The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.

Anosov groups study matrix coefficients and orbit counting in symmetric spaces.

The paper counts geodesic loops on surfaces without conjugate points.

Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Hu…

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

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

In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that thi…

Counting subgroups of a surface using convex core lengths.

Study counts minimal surfaces in curved 3D spaces, finding hyperbolic space minimizes area.

New test detects differences in heterogeneous datasets.

The paper counts conjugacy classes of pseudo-Anosov homeomorphisms in Teichmüller space.

Counting essential surfaces in 3-manifolds yields concise formulae and detailed asymptotics.

In this article, associated with each lattice $T\subseteq \mathbb{Z}^n$ the concept of a harmonic-counting measure $ν_T$ on a sphere $S^{n-1}$ is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of …

Study a market with uncertain informed traders, finding price impact depends on both asset value and informed trader count distribution.

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 …

The paper improves count data regression models for overdispersed data.

Study counts orbits of mapping class group in shearing coordinates.

Study on counting orbits and Poincaré series for specific hyperbolic metrics.

We study the spectral theory of asymptotically hyperbolic manifolds with ends of warped product type. Our main result is an upper bound on the resonance counting function with a geometric constant expressed in terms of the respective Weyl constants for the core of the manifold and the base manifold defining the ends.

Estimates surface count with prescribed foliations.

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

Let $D^-$ and $D^+$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\to+\infty$ for the number of common perpendiculars of length at most $t$ from $D^-$ to $D^+$, count…

We prove that there is a true asymptotic formula for the number of one sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic structure, and it is noninteger, in contrast to counting results of Mirzakhani for…

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

Counting spheres in hyperbolic space with effective methods.

The study counts geodesics on special manifolds without focusing points.

Novel Bayesian method for high-dimensional count data prediction.

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 …

Study counts surface subgroups in curved 3D manifolds.

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

New exact tests detect changepoints in binary and count data, especially when normal approximations fail.

We consider an elliptic self-adjoint first order differential operator L acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of the operator L is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Gi…

We extend asymptotic formulas for saddle connections on translation surfaces.

The paper counts mapping classes by Nielsen-Thurston type, finding growth rates for different subsets.

Graphical estimation of count time series dependencies.

This study analyzes LTS in sparse models with finite sample error bounds.

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, thi…

We compute explicitly the absolute contribution of square-tiled surfaces having a single horizontal cylinder to the Masur-Veech volume of any ambient stratum of Abelian differentials. The resulting count is particularly simple and efficient in the large genus asymptotics. Using the recent results of Aggarwal and of Che…

We compute the volumes of the eigenform loci in the moduli space of genus two Abelian differentials. From this, we obtain asymptotic formulas for counting closed billiards paths in certain L-shaped polygons with barriers.

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which, if $M$ is compact, is defined for $w \geq 0$ and $T > 0$ by $$N_{M,w}(T) = \sum\…

Efficient tensor decomposition for count data models achieves near-optimal multiway analysis.

Square-tiled surfaces can be classified by their number of squares and their cylinder diagrams (also called realizable separatrix diagrams). For the case of $n$ squares and two cone points with angle $4 π$ each, we set up and parametrize the classification into four diagrams. Our main result is to provide formulae for …