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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., Comment: 7 pages

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., Comment: 25 pages, 23 figures. Fixed a gap in the argument, added a toroidal graph example

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., Comment: 9 pages, 1 figure

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 \[ N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2, \] where here $\Delta_g e_j = -\lambda_j^2 e_j$ 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 $N_x$., Comment: 31 pages, 1 figure. We add a section introducing a few more physical interpretations, including "Can you hear at which point a drum is struck?"

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 $\Gamma \subseteq \mathbb F_q^\times$ of bounded index., Comment: 22 pages, 3 figures. Accepted incorporating referee comments

The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(\lambda) := \sum_{\lambda_j \leq \lambda} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} \lambda^{\operatorname{codim} H} + O(\lambda^{\operatorname{codim} H - 1})$$ where $e_j$ constitute a Hilbert basis of Laplace-Beltrami eigenfunctions on a Riemannian manifold $M$ with $\Delta_g e_j = -\lambda_j^2 e_j$, and $H$ is an embedded submanifold. We show for some suitable definition of `$\sim$', $$ N(\lambda) \sim C_{H,M} \lambda^{\operatorname{codim} H} + Q(\lambda) \lambda^{\operatorname{codim} H - 1} + o(\lambda^{\operatorname{codim} H - 1}) $$ 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., Comment: 36 pages, 1 figure

Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_\lambda$ on a compact, boundary-less Riemannian manifold with $\Delta e_\lambda = -\lambda^2 e_\lambda$. We study eigenfunction triple products \[ \langle e_\lambda e_\mu, e_\nu \rangle = \int e_\lambda e_\mu \overline{e_\nu} \, dV. \] 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 $\lambda,\mu,$ and $\nu$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $\lambda, \mu,$ and $\nu$ fail to satisfy the triangle inequality. As a consequence, we improve a result by Lu, Sogge, and Steinerberger., Comment: 29 pages, 1 figure

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 $\{\psi_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle \gamma_H e_j, \psi_k \rangle_{L^2(H)} = \int_H e_j \overline \psi_k \, dV_H, \] of restrictions $\gamma_H e_j$ 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 $\{(\mu_k, \lambda_j)\}_{j,k - 0}^{\infty}$ of the (square roots of the) Laplacian $\Delta_M$ on $M$ and the Laplacian $\Delta_H$ on $H$ in a family of suitably `thick' regions in $\mathbb R^2$. Thick regions include (1) the truncated cone $\mu_k/\lambda_j \in [a,b] \subset (0,1)$ and $\lambda_j \leq \lambda$, and (2) the slowly thickening strip $|\mu_k - c\lambda_j| \leq w(\lambda)$ and $\lambda_j \leq \lambda$, where $w(\lambda)$ is monotonic and $1 \ll w(\lambda) \lesssim \lambda^{1 - 1/n}$. Key tools for obtaining these asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem., Comment: 36 pages. Referees' suggestions incorporated. To appear in Sci. China Math

We show that on compact Riemann surfaces of nonpositive curvature, the generalized periods, i.e. the $\nu$-th order Fourier coefficients of eigenfunctions $e_\lambda$ over a closed smooth curve $\gamma$ which satisfies a natural curvature condition, go to 0 at the rate of $O((\log\lambda)^{-1/2})$, if $0<|\nu|/\lambda<1-\delta$, for any fixed $0<\delta<1$. 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 $O((\log\lambda)^{-1/2})$. A direct corollary of our results and the QER theorem of Toth and Zelditch is that for a geodesic circle $\gamma$ on a compact hyperbolic surface, the restriction $e_{\lambda_j}|_\gamma$ of an orthonormal basis $\{e_{\lambda_j}\}$ has a full density subsequence that goes to zero in weak-$L^2(\gamma)$. One key step of our proof is a microlocal decomposition of the measure over $\gamma$ into tangential and transversal parts., Comment: 26 pages, 3 figures, 2 corollaries on weak $L^2$ convergence added. These results should be compared with arXiv:1711.09864 by the second author on the closed geodesic case

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., Comment: 33 pages

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

Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $\gamma$ be a closed geodesic in $M$. And let $e_\lambda$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $\Delta_g$ with $-\Delta_g e_\lambda = \lambda^2 e_\lambda$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_\gamma e_\lambda \, ds = O((\log\lambda)^{-1/2}),$$ 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 $\gamma$ whose geodesic curvature avoids, pointwise, the geodesic curvature of circles of infinite radius tangent to $\gamma$., Comment: 23 pages, 3 figures, arXiv admin note: text overlap with arXiv:1702.03552

Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $\Delta_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_\lambda$ be $L^2$-normalized eigenfunctions of $\Delta_g$ with eigenvalue $\lambda$, i.e. \[ -\Delta_g e_\lambda = \lambda^2 e_\lambda. \] We prove \[ \left| \int_{\mathbb R} b(t) e_\lambda (\gamma(t)) \, dt \right| = o(1) \quad \text{ as } \lambda \to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $\gamma$ is a curve parametrized by arc-length whose geodesic curvature $\kappa(\gamma(t))$ avoids two critical curvatures $\mathbf k(\gamma'^\perp(t))$ and $\mathbf k(-\gamma'^{\perp}(t))$ 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 $-v$.

This study investigated the effect of three specific hands-on teaching strategies on the attainment and alteration of preservice elementary teachers' conceptions about earth-sun-moon relationships. The subjects (n=76) were enrolled in an elementary school science methods course. The descriptive nature of this study explored: (1) the effect of two instructional strategies--those using models versus those not using models--on preservice teachers' abilities to develop models to explain the occurrence of the lunar phases; and (2) whether spatial skills and reasoning levels interact with the ability to develop explanatory models. The results of the study suggest that models do have an effect. The group using only mental models did not show any significant change from pretest to posttest on the retention test. The group using physical models had significant categorical shifts from pretest to posttest. No significant interaction between spatial ability and model development was found. Results also suggest that concrete manipulated models appear to work best for novel situations, but that the type of models used in teaching abstract phenomena should be carefully determined. The paper concludes with several recommendations for further study. Contains 26 references. (PR)

A significant problem for many schools in the United States is the shortage of qualified mathematics and science teachers entering the profession; this, in turn, is having an adverse effect on the quality of education. The purpose of this study was to survey the 320 mathematics and science education graduates (40% responded) from Kansas State University (KSU), from the years 1975-1988, in order to determine the factors that may have contributed to their leaving or never entering the teaching profession. In addition, the survey provided information for the College of Education at KSU to consider in its efforts to improve the teacher education program. Included is an introduction, methodology, conclusion, recommendations to KSU, and recommendations for further studies. (The document includes 16 tables and 15 references.) (KR)