arXiv research

Develops a new spectrum for annular links, recovering a transverse invariant at extreme gradings.

Upper bounds for volume spectrum depend on volume, dimension, and a conformal invariant.

Upper bounds on Laplacian eigenvalues on manifolds with non-negative curvature.

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…

Motivated by the work of Abreu and Freitas, we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over $\mathbb{p}^1$.

The Hodge spectra help distinguish orbifolds from manifolds with singularities.

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…

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \cite{E} is equivalent to the problem of recovering, up to a conformal equivalence, a positive function $a\in C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{iθ}\}$ from the eigenvalue spectrum of t…

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show …

In this note we prove that toric Kähler metrics on complex projective space which are also $U(n)$-invariant are determined by their equivariant spectrum i.e. the list of eigenvalues of the Laplacian together with weights of the torus representation on the eigenspaces.

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also …

We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as well as their lengths, are invariants of the Steklov spectrum. The proofs are based on pseudodifferential techniques for the Dirichlet-to-Neuma…

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of total…

We introduce the \emph{metric spectrum}, which measures the exponential rate of approximation to an isolated invariant set of points starting in its stable set, and relate it to the Lyapunov spectrum. We determine the metric spectrum of each Morse component of the finest Morse decomposition of a linear induced flow on …

New invariant extends curvature estimates to noncompact manifolds.

We analyze the frequency spectrum of quantum neural networks using algebraic methods and prove maximality results.

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…

We show how to assign to any immersed torus in $\R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the toru…

Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.

Intuition drawn from quantum mechanics and geometric optics raises the following long-standing question: can the length spectrum of a closed Riemannian manifold be recovered from its Laplace spectrum? The Poisson relation states that for any closed Riemannian manifold $(M,g)$ the singular support of the trace of its wa…

The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.

New theorem shows metrics of certain groups are close if their lengths are identical.

Study magnetic potentials on Anosov manifolds using spectral data.

In a previous paper we have constructed an invariant of four-dimensional manifolds with boundary in the form of an element in the stable homotopy group of the Seiberg-Witten Floer spectrum of the boundary. Here we prove that when one glues two four-manifolds along their boundaries, the Bauer-Furuta invariant of the res…

Given a compact boundaryless Riemannian manifold $Y$ on which a compact Lie group $G$ acts, there is always a metric on $Y$ such that the action is by isometries. Assuming $Y$ is equipped with such a metric, recall that the $G$-invariant Laplacian is the restriction of the ordinary Laplacian to the space of functions w…

We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL$(2,\mathbb{R})$-invariant submanifold is completely degenerate, i.e. $λ_2 = \cdots = λ_g = 0$, then the submanifold must be an arithmetic Teichmueller curve in the moduli space of Abelian differentials over surfaces of genus three…

Study calculates geometric invariants from Navier-Lamé spectrum.

The paper studies properties of optimal metrics associated to curves on surfaces.

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta …

New field invariant refines real spectrum and relates to absolute Galois group.

A new graph signature invariant to graph automorphisms.

Spatial refinement of Bar-Natan homology constructed.

Take a riemanniann nilmanifold, lift its metric on its universal cover. In that way one obtains a metric invariant under the action of some co-compact subgroup. We use it to define metric balls and then study the spectrum of the laplacian for the dirichlet problem on them. We describe the asymptotic behaviour of the sp…

We introduce a spectrum of monotone coarse invariants for metric measure spaces called Poincaré profiles. The two extremes of this spectrum determine the growth of the space, and the separation profile as defined by Benjamini--Schramm--Timár. In this paper we focus on properties of the Poincaré profiles of groups with …

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a measure of the complexity of a 3-manifold using the number of triple points of min…

Classifies GL(2,R)-invariant subvarieties in complex geometry.

Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.

New mathematical tools for studying knots and links.

We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley …

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac op…

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/λ_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum λ_{i}^{-2}(Ω)$ where $Ω$ is an extrinsic ball of a proper m…

This paper relates the spectrum of the scalar Laplacian of an asymptotically hyperbolic Einstein metric to the conformal geometry of its ``ideal boundary'' at infinity. It follows from work of R. Mazzeo that the essential spectrum of such a metric on an $(n+1)$-dimensional manifold is the ray $[n^2/4,\infty)$, with no …

For a given bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $…

Corners can be identified by a drum's sound spectrum.

Spectral flow connects manifold geometry to rigidity criteria.

New invariant distinguishes lens spaces via categorified homotopy E_3-algebra.

Constructs a spectrum for knot Floer homology without holomorphic geometry.

New CR manifolds found with same Kohn Laplacian spectra.