arXiv research

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a sp…

We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada's method. We also observe that if a good orbifold $\mathcal{O}$ and a smooth manifold $M$ are isospectral, then they cannot admit non-trivial finite Riemannian covers $M_1 \to \mathcal{O}$ …

We show that certain families of iso-length spectral hyperbolic surfaces obtained via the Sunada construction are not generally simple iso-length spectral.

The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.

New hyperbolic 3-manifolds with multiple cusps are found that sound the same but look different.

In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multip…

Constructs non-isometric iso-length-spectral surfaces.

The authors exhibit pairs of infinite-volume, hyperbolic three-manifolds that have the same scattering poles and conformally equivalent boundaries, but which are not isometric. The examples are constructed using Schottky groups and the Sunada construction.

In this paper we describe recent results on explicit construction of lens spaces that are not strongly isospectral, yet they are isospectral on $p$-forms for every $p$. Such examples cannot be obtained by the Sunada method. We also discuss related results, emphasizing on significant classical work of Ikeda on isospectr…

In this paper we construct, for n >= 2, arbitrarily large families of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we …

We generalize Sunada's method to produce new examples of closed, locally non-isometric manifolds which are isospectral. In particular, we produce pairs of isospectral, simply-connected, locally non-isometric normal homogeneous spaces. These pairs also allow us to see that in general group actions with discrete spectra …

We construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's Theorem due to DeTurck and Gordon.

The relationship between the Chern-Simons invariant and eta-invariant of a 3-manifold is shown to lead to an obstruction to a group being the fundamental group of a closed oriented 3-manifold. The proof uses Sunada's construction of isospectral manifolds as covering spaces over a common base space.

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, ``can one hear the shape of a drum?'' In order to construct simply connected examples, we exploit the observation that an orbifold whose unde…

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada's method in the cusped setting, and so in addition our pairs are fin…

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichlet-to-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin …

The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked…

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, t…

We give a systematic way to construct almost conjugate pairs of finite subgroups of $Spin(2n+1)$ and $Pin(n)$ for $n\in \mathbb{N}$ sufficiently large. As a geometric application, we give an infinite family of pairs $M_1^{d_n}$ and $M_2^{d_n}$ of nearly Kähler manifolds that are isospectral for the Dirac and Laplace op…

The Ray-Singer isospectral theorem (1971) is applied to a general spectral function for Laplacians of twisted p-forms (say) on homogeneous Clifford-Klein factors of the three-sphere. The inducing formulae necessary to express any spectral quantity for any twisting in terms of those for cyclic subgroups of the tetrahedr…

We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric…

Paper shows surfaces can't be heard to be orientable.

The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimens…

In this note we prove that a generic Riemannian manifold of dimension $\geq 3$ does not admit any nontrivial local conformal diffeomorphisms. This is a conformal analog of a result of Sunada concerning local isometries, and makes precise the principle that generic manifolds in high dimensions do not have conformal symm…

We associate a two-step nilpotent Lie algebra to an arbitrary Schreier graph. We then use properties of the Schreier graph to determine necessary and sufficient conditions for this Lie algebra to extend to a three-step nilpotent Lie algebra. As an application, if we start with pairs of non-isomorphic Schreier graphs co…

For any $n\geq 7$, $k\geq 3$, we give pairs of compact flat $n$-manifolds $M, M'$ with holonomy groups $\mathbb Z_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is Kähler while $M'$ is not. Furthermore, with…

We introduce the Γ-extension of the spectrum of the Laplacian of a Riemannian orbifold, where Γis a finitely generated discrete group. This extension, called the Γ-spectrum, is the union of the Laplace spectra of the Γ-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the…

Consider a finite connected graph possibly with multiple edges and loops. In discrete geometric analysis, Kotani and Sunada constructed the crystal associated to the graph as a standard realization of the maximal abelian covering of the graph. As an application of what the author showed in an earlier paper with Seshadr…

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which als…

We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…

In this paper we construct arbitrarily large families of smooth projective varieties and closed Riemannian manifolds that share many algebraic and analytic invariants. For instance, every non-arithmetic, closed hyperbolic $3$--manifold admits arbitrarily large collections of non-isometric finite covers which are strong…

Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate …

We continue our exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold and find that if $(M,g)$ and $(N,h)$ are a pair of locally homogeneous, locally non-isometric isospectral three-manifolds, where $M$ is an elliptic three-manifold, then $(1)$ $N$ is also an…

The flat trace of geodesic Koopman operators varies with negatively curved surfaces.

Improved iterative methods for risk parity portfolio weights.

We describe a novel optimization method for finite sums (such as empirical risk minimization problems) building on the recently introduced SAGA method. Our method achieves an accelerated convergence rate on strongly convex smooth problems. Our method has only one parameter (a step size), and is radically simpler than o…

A new method combines Laplace and Variational Bayes for scalable inference.

Unified framework for model explanation methods based on feature removal.

This work reviews and evaluates methods for predicting prediction intervals in regression problems.

Derives kernel PCA with Nyström method for scalability.

In this paper, the author considers the numerical computation of CVA for large systems by Mote Carlo methods. He introduces two types of stochastic mesh methods for the computations of CVA. In the first method, stochastic mesh method is used to obtain the future value of the derivative contracts. In the second method, …

New method combines spectral and sparse methods for Gaussian processes.

A comprehensive benchmark of 15 scRNA-seq imputation methods across various datasets and analyses.

New methods using natural gradient for structured optimization.

Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of t…

We investigate methods for pricing American options under the variance gamma model. The variance gamma process is a pure jump process which is constructed by replacing the calendar time by the gamma time in a Brownian motion with drift, which makes it a time-changed Brownian motion. In general, the finite difference me…

A new method speeds up deep neural network training.

We propose a new stochastic dual coordinate ascent technique that can be applied to a wide range of regularized learning problems. Our method is based on Alternating Direction Multiplier Method (ADMM) to deal with complex regularization functions such as structured regularizations. Although the original ADMM is a batch…