arXiv research

In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick applicatio…

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

Improved bounds for eigenfunctions on hyperbolic surfaces found.

We study the asymptotic properties of the conormal cycle of nodal sets associated to a random superposition of eigenfunctions of the Laplacian on a smooth compact Riemannian manifold without boundary. In the case where the dimension is odd, we show that the expectation of the corresponding current of integration equidi…

Partial coverings of hyperbolic surfaces equidistribute with geodesics.

Study equidistribution for flows on geometrically finite convergence group actions.

Study shows how to count and equidistribute cusped Hitchin representations with entropy gaps.

The paper shows equidistribution of geodesics and nets on manifolds.

The paper shows measures equidistribute on affine submanifolds with a rate.

Polynomial error equidistribution for SL2 groups.

Proves polynomial error rate for equidistribution of unipotent flows.

Geodesics in curved spaces spread evenly over time.

Hyperbolic groups' infinite orbits spread evenly in spaces.

The paper describes how Hodge loci are typically equidistributed in complex varieties.

Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.

Paper shows how to evenly distribute intersections in hyperbolic spaces.

Paper proves geodesics are evenly distributed on surfaces.

The paper equidistributes zeros of random polynomials and sections on manifolds.

New findings on geometric flows and equidistribution in Hilbert geometry.

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

Common perpendiculars equidistribute in negatively curved spaces.

Study on geodesics and dihedral groups in lattices.

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…

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quat…

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

Study shows twist tori equidistribute in moduli space, with other families having singular distributions.

Study counts and equidistributes geodesic orbits on curved spaces.

In this paper we study the equidistribution of expanding horospheres in infinite volume geometrically finite rank one locally symmetric manifolds and apply it to the orbital counting problem in apollonian sphere packing.

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

Geodesics on hyperbolic surfaces become evenly spread over time.

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 …

Asymptotics for equidistribution of circles on hyperbolic surfaces.

We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R)…

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

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

For convex co-compact hyperbolic manifolds $Γ\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $δ_Γ< n/2$, we show that the high-frequency Eisenstein series associated to a point $ξ$ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …

Let $M$ be a finite volume hyperbolic manifold, we show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal n…

Effective estimates for lattice orbits in homogeneous spaces.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

Study shows automorphisms of Markov surfaces share periodic points if they share a common iterate.

New method for minimal submanifolds in spheres using eigenfunctions.

Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.

Study critical points of Laplace eigenfunctions in polygons.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

Given a simple closed curve $γ$ on a connected, oriented, closed surface $S$ of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on $S$ having a simple closed geodesic of length $L$ of the same topological type as $γ$ equidistributes with respect to a …

The study counts critical points of Steklov eigenfunctions on manifolds.

Study heat profiles and eigenfunctions using Brownian motion.

Eigenfunction gradients on curved spaces imply rigid structure.