arXiv research

Counting tripods on a flat torus using lattice point counting.

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

Estimates point counts in Teichmüller space for mapping class groups.

The paper counts geodesic loops on surfaces without conjugate points.

Formula counts rational curves with a specific singular point in projective space.

Invariant counts maximum stable umbilic splits.

We show how to define and count lattice points in the moduli space $\modm_{g,n}$ of genus g curves with n labeled points. This produces a polynomial with coefficients that include the Euler characteristic of the moduli space, and tautological intersection numbers on the compactified moduli space.

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary num…

Counting lattice points in moduli space of Klein surfaces.

The number of diagrams of stationary points free vector fields in the 2-disk $\mathbb{B}^{2}$ is counted in the article. It is shown that the number of such diagrams with $2k$ exceptional points on the boundary $\mathbb{S}^{1}$ equals $3^{k-2}(C_{k}+2C_{k-1})$, where $C_{k}$ is the corresponding Catalan number. An algo…

Flow Matching for count data improves sample quality and efficiency.

Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, t…

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

In this paper a neuro-robotics model capable of counting using gestures is introduced. The contribution of gestures to learning to count is tested with various model and training conditions. Two studies were presented in this article. In the first, we combine different modalities of the robot's neural network, in the s…

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top degree coefficients tautological intersection numbers on the compactified moduli spac…

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

Proposes a robust EM algorithm for analyzing incomplete panel count data.

The study counts geodesics on special manifolds without focusing points.

Regular integer lattices are characterized by k unit vectors that build up their generator matrices. These have rank k for D-lattices, and are rank-deficient for A-lattices, for E_6 and E_7. We count lattice points inside hypercubes centered at the origin for all three types, as if classified by maximum infinity norm i…

The study counts ideal points in 2-bridge knot complements using knot diagrams.

The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…

We study the combinatorial geometry of "lattice" Jenkins--Strebel differentials with simple zeroes and simple poles on $\mathbb{C}P^1$ and of the corresponding counting functions. Developing the results of M. Kontsevich we evaluate the leading term of the symmetric polynomial counting the number of such "lattice" Jenki…

We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows us to settle the existence and uniqueness problem. The machinery of dual polygons…

We present the first framework for Gaussian-process-modulated Poisson processes when the temporal data appear in the form of panel counts. Panel count data frequently arise when experimental subjects are observed only at discrete time points and only the numbers of occurrences of the events between subsequent observati…

Study on symplectic semi-characteristic using cohomology and vector fields.

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

In this article we consider a variant of Rabinowitz Floer homology in order to define a homological count of discriminant points for paths of contactomorphisms. The growth rate of this count can be seen as an analogue of Givental's nonlinear Maslov index. As an application we prove a Bott-Samelson type obstruction theo…

Estimates point counts on Riemannian varieties over finite fields.

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

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…

New phases identified in neural scaling laws with compute limits.

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…

We introduce a notion of complexity for Sefiert homology spheres by establishing a correspondence between lattice point counting in tethrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order in the set of Seif…

This paper was motivated by work of Arnold where he explains how to count "snakes", i.e. Morse functions on the real axis with prescribed behavior at infinity. This leads immediately to a count of excellent Morse functions on the circle, where following Thom's terminology, excellent means that no two critical points li…

We study the hyperbolic random geometric graph introduced in Krioukov et al. For a sequence $R_n \to \infty$, we define these graphs to have the vertex set as Poisson points distributed uniformly in balls $B(0,R_n) \subset B_d^α$, the $d$-dimensional Poincaré ball (unit d-ball with the Poincaré metric $d_α$ correspondi…

Study counts sub-chord diagrams to classify spherical curves.

The study counts critical points of Steklov eigenfunctions on manifolds.

The study counts critical points in knot cobordisms using abelian and metacyclic invariants.

ENTED efficiently decomposes binary and count tensors using nonparametric Gaussian processes.

Bound critical points for minimal Radó functions.

Firms miscount their customers who stop buying without saying goodbye.

The number of closed billiard trajectories in a rational-angled polygon grows quadratically in the length. This paper gives an analogue on K3 surfaces, by considering special Lagrangian tori. The analogue of the angle of a billiard trajectory is a point on a twistor sphere, and the number of directions admitting a spec…

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 …

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

We count meromorphic differentials with fixed residues and poles of fixed orders.

We obtain a growth estimate for the number of lattice points inside any Q-Gorenstein cone. Our proof uses the result of Futaki-Ono-Wang on Sasaki-Einstein metric for the toric Sasakian manifold associated to the cone, a Yau's inequality, and the Kawasaki-Riemann-Roch formula for orbifolds.

We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $\mathbb{C}$, and the space of unordered $n$-tuples of linearly independent lines in $\mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations give…

We count the conjugacy classes of maximal tori in the groups of symplectomorphisms of S^2 \times S^2 and of the blow-up of CP^2 at a point.