arXiv subjects

Emmett L. Wyman

Publications and source records attributed to Emmett L. Wyman.

15 recordsLinked to original sources

arXiv2026

Polygons and multi-product of eigenfunctions

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j\}_j$, which satisfy $Δ_g e_j = -λ_j^2 e_j$. We study the following inner product of eigenfunctions \[ \langle e_{i_1} e_{i_2} \ldots e_{i_k}, e_{i_{k+1}} \rangle = \int e_{i_1} e_{i_2}\ldots e_{i_k} \overline{e_{i_{k+1}}} \, dV. \] We show that, after a mild averaging in the frequency variables, the main $\ell^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.

math.AP↗

arXiv2025

Kuznecov remainders and generic metrics

We obtain improved remainder estimates in the Kuznecov sum formula for period integrals of Laplace eigenfunctions on a Riemannian manifold $M$. Building upon the two-term asymptotic expansion established in [arXiv:2204.13525], we prove that for a Baire-generic class of metrics, the oscillatory second term in the Kuznecov formula can be eliminated, yielding an improved remainder estimate.

math.AP↗

arXiv2024

The VC-dimension of quadratic residues in finite fields

We study the Vapnik-Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, $\mathbb F_q$, when considered as a subset of the additive group. We conjecture that as $q \to \infty$, the squares have the maximum possible VC-dimension, viz. $(1+o(1))\log_2 q$. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is $\geq (\frac{1}{2} + o(1))\log_2 q$. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups $Γ\subseteq \mathbb F_q^\times$ of bounded index.

math.CO↗

arXiv2024

Hearing the shape of a drum by knocking around

We study a variation of Kac's question, "Can one hear the shape of a drum?" if we allow ourselves access to some additional information. In particular, we allow ourselves to ``hear" the local Weyl counting function at each point on the manifold and ask if this is enough to uniquely recover the Riemannian metric. This is physically equivalent to asking whether one can determine the shape of a drum if one is allowed to knock at any place on the drum. We show that the answer to this question is ``yes" provided the Laplace-Beltrami spectrum of the drum is simple. We also provide a counterexample illustrating why this hypothesis is necessary.

math.AP↗

arXiv2024

Cospectral vertices, walk-regular planar graphs and the echolocation problem

We study cospectral vertices on finite graphs in relation to the echolocation problem on Riemannian manifolds. First, We prove a computationally simple criterion to determine whether two vertices are cospectral. Then, we use this criterion in conjunction with a computer search to find minimal examples of various types of graphs on which cospectral but non-similar vertices exist, including minimal walk-regular non-vertex-transitive graphs, which turn out to be non-planar. Moreover, as our main result, we classify all finite 3-connected walk-regular planar graphs, proving that such graphs must be vertex-transitive.

math.CO↗

arXiv2023

Can you hear your location on a manifold?

We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point $x$ on the manifold, up to symmetry, from its pointwise counting function \[ N_x(λ) = \sum_{λ_j \leq λ} |e_j(x)|^2, \] where here $Δ_g e_j = -λ_j^2 e_j$ and $e_j$ form an orthonormal basis for $L^2(M)$. This problem has several natural physical interpretations, two of which are acoustic: 1. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echoes and reverberations? 2. If a drum of a known shape is struck at some unknown point, can you determine this point by listening to the quality of the sound the drum produces? The main result of this paper provides an affirmative answer to this question for a generic class of metrics. We also probe the problem with a variety of simple examples, highlighting along the way helpful geometric invariants that can be pulled out of the pointwise counting function $N_x$.

math.AP↗

arXiv2023

Surfaces in which every point sounds the same

We address a maximally structured case of the question, "Can you hear your location on a manifold," posed in arXiv:2304.04659 for dimension $2$. In short, we show that if a compact surface without boundary sounds the same at every point, then the surface has a transitive action by the isometry group. In the process, we show that you can hear your location on Klein bottles and that you can hear the lengths and multiplicities of looping geodesics on compact hyperbolic quotients.

math.AP↗

arXiv2023

A two term Kuznecov sum formula

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(λ) := \sum_{λ_j \leq λ} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} λ^{\operatorname{codim} H} + O(λ^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $Δ_g e_j = -λ_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(λ) \sim C_{H,M} λ^{\operatorname{codim} H} + Q(λ) λ^{\operatorname{codim} H - 1} + o(λ^{\operatorname{codim} H - 1}) $$ where $Q$ is a bounded oscillating term and is expressed in terms of the geodesics which depart and arrive $H$ in the normal directions. In work by Canzani, Galkowski, and Toth, they establish (as a corollary to a stronger result involving defect measures) that if the set of recurrent directions of geodesics normal to $H$ has measure zero, then we obtain improved bounds on the individual terms in the sum -- the period integrals. We are able to give a dynamical condition such that $Q$ is uniformly continuous and `$\sim$' can be replaced with `$=$'. This implies improved bounds on period integrals, and this condition is weaker than the recurrent directions having measure zero. Moreover, our result implies improved bounds for period integrals if there is no $L^1$ measure on $SN^*H$ that is invariant under the first return map. This generalizes a theorem of Sogge and Zelditch and of Galkowski.

math.AP↗

arXiv2023

Fourier coefficients of restrictions of eigenfunctions

Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $Δ_M$ on $M$ and the Laplacian $Δ_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $μ_k/λ_j \in [a,b] \subset (0,1)$ and $λ_j \leq λ$, and (2) the slowly thickening strip $|μ_k - cλ_j| \leq w(λ)$ and $λ_j \leq λ$, where $w(λ)$ is monotonic and $1 \ll w(λ) \lesssim λ^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.

math.AP↗

arXiv2021

Triangles and triple products of Laplace eigenfunctions

Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_λ$ on a compact, boundary-less Riemannian manifold with $Δe_λ= -λ^2 e_λ$. We study eigenfunction triple products \[ \langle e_λe_μ, e_ν\rangle = \int e_λe_μ\overline{e_ν} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $λ,μ,$ and $ν$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $λ, μ,$ and $ν$ fail to satisfy the triangle inequality. As a consequence, we improve a result by Lu, Sogge, and Steinerberger.

math.AP↗

arXiv2018

Improved Generalized Periods Estimates Over Curves on Riemannian Surfaces with Nonpositive Curvature

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $ν$-th order Fourier coefficients of eigenfunctions $e_λ$ over a closed smooth curve $γ$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\logλ)^{-1/2})$, if $0<|ν|/λ<1-δ$, for any fixed $0<δ<1$. Our result implies, for instance, the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of $O((\logλ)^{-1/2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesic circle $γ$ on a compact hyperbolic surface, the restriction $e_{λ_j}|_γ$ of an orthonormal basis $\{e_{λ_j}\}$ has a full density subsequence that goes to zero in weak-$L^2(γ)$. One key step of our proof is a microlocal decomposition of the measure over $γ$ into tangential and transversal parts.

math.AP↗

arXiv2018

Period integrals in nonpositively curved manifolds

We provide an improvement of a half power of log to standard bounds on integrals of Laplace eigenfunctions over submanifolds of codimension 2 or more, where the ambient space is a compact Riemannian manifold with negative sectional curvature. We provide the same improvement for hypersurfaces whose second fundamental form differs sufficiently from that of spheres of infinite radius. This result extends previous ones obtained in the 2-dimensional setting by Chen and Sogge; Sogge, Xi, and Zhang; and the author.

math.AP↗

arXiv2018

Explicit bounds on integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $γ$ be a closed geodesic in $M$. And let $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $Δ_g$ with $-Δ_g e_λ= λ^2 e_λ$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_γe_λ\, ds = O((\logλ)^{-1/2}),$$ an improvement over the general $O(1)$ bound. We show this integral enjoys the same decay for a wide variety of curves, where $M$ has nonpositive sectional curvature. These are the curves $γ$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $γ$.

math.AP↗

arXiv2017

Looping directions and integrals of eigenfunctions over submanifolds

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -Δ_g e_λ= λ^2 e_λ\qquad \text{ and } \qquad \| e_λ\|_{L^2(M)} = 1. \] Let $Σ$ be a $d$-dimensional submanifold and $dμ$ a smooth, compactly supported measure on $Σ$. It is well-known (e.g. proved by Zelditch in far greater generality) that \[ \int_Σe_λ\, dμ= O(λ^\frac{n-d-1}{2}). \] We show this bound improves to $o(λ^\frac{n-d-1}{2})$ provided the set of looping directions, \[ \mathcal{L}_Σ = \{ (x,ξ) \in SN^*Σ: Φ_t(x,ξ) \in SN^*Σ\text{ for some } t > 0 \} \] has measure zero as a subset of $SN^*Σ$, where here $Φ_t$ is the geodesic flow on the cosphere bundle $S^*M$ and $SN^*Σ$ is the unit conormal bundle over $Σ$.

math.AP↗

arXiv2017

Integrals of eigenfunctions over curves in surfaces of nonpositive curvature

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $Δ_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_λ$ be $L^2$-normalized eigenfunctions of $Δ_g$ with eigenvalue $λ$, i.e. \[ -Δ_g e_λ= λ^2 e_λ. \] We prove \[ \left| \int_{\mathbb R} b(t) e_λ(γ(t)) \, dt \right| = o(1) \quad \text{ as } λ\to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $γ$ is a curve parametrized by arc-length whose geodesic curvature $κ(γ(t))$ avoids two critical curvatures $\mathbf k(γ'^\perp(t))$ and $\mathbf k(-γ'^{\perp}(t))$ for each $t \in \operatorname{supp} b$. $\mathbf k(v)$ denotes the curvature of a circle with center taken to infinity along the geodesic ray in direction $-v$.

math.AP↗