arXiv research

Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.

Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)

We study the eta invariants of Dirac operators and the regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner M"uller and use relative traces to define these spectral invariants. We show the regularity of eta and zeta functions at s=0. The Selberg trace formula and the detai…

Three counterexamples show higher eigenvalue multiplicities than conjectured.

The paper calculates the full asymptotics of analytic torsions for compact orbifolds.

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace sp…

Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…

Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.

New methods derive a generalized Frenkel trace formula for Lie groups.

The goal of the course was a review of results mainly due to M. Olbrich and the first author. We consider a discrete cocompact subgroup $Γ$ of a semisimple Lie group $G$. We relate the group cohomology of $Γ$ with coefficients in the maximal globalization of a representation of $G$ with the multiplicities of unitary re…

Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.

Study Floer theory of hyperbolic three-manifolds using Dirac spectral flow.

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for …

This is an expanded version of [arXiv:1107.4836v1 [math.DS]]. Using techniques from [Chapter XI, The Selberg Trace Formula, in Eigenvalues in Riemannian Geometry, by Isaac Chavel], in which a differential-geometrically intrinsic treatment of counterparts of classical electrostatics was introduced, it is shown that on s…

We obtain an estimate from below for the remainder in Weyl's law on negatively curved surfaces. In the constant curvature case, such a bound was proved independently by Hejhal and Randol in 1976 using the Selberg zeta function techniques. Our approach works in arbitrary negative curvature, and is based on wave trace as…

Using the Arthur-Selberg trace formula we express the index of a Dirac operator on an arithmetic quotient over a totally real field with at least two real embeddings as the integral over the index form plus a sum of orbital integrals. For the Euler operator these orbital integrals are shown to vanish for products of ra…

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact th…

We derive a formula for the regularized trace of operators with compact spectrum which act on the space of square integrable functions on the quotient of a semisimple Liegroup of real rank one by a convex-cocompact subgroup. The sum of normalized orbital integrals associated to the hyperbolic conjugacy classes of this …

A derivation of the Cesàro-Fedorov relation from the Selberg trace formula on an orbifolded 2-sphere is elaborated and extended to higher dimensions using the known heat-kernel coefficients for manifolds with piecewise-linear boundaries. Several results are obtained that relate the coefficients, $b_i$, in the Shephard-…

Method constrains spectral gaps of hyperbolic spin surfaces using identities and semidefinite programming.

Study on Selberg's modified metric in symmetric spaces.

We introduce a pair of isospectral but non-isometric compact flat 3-manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, Didi has four half-twisting geodesics. Nevertheless, these spa…

Study asymptotics of Selberg zeta function on spin moduli space.

We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact $1$-forms $λ_1^*$ on rational homology spheres…

Using the Selberg trace formula, we show that for a hyperbolic 2-orbifold, the spectrum of the Laplacian acting on functions determines, and is determined by, the following data: the volume; the total length of the mirror boundary; the number of conepoints of each order, counting a mirror corner as half a conepoint; an…

Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.

Chern-Simons invariants of closed oriented Riemannian $3$-manifolds are introduced and studied from the basics. Their first-order variation is the Cotton tensor. The properties of the Cotton tensor: symmetry, conformal covariance, trace- and divergence-freedom, are recovered as corollaries of the Chern-Simons invariant…

Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.

We prove Patterson's conjecture about the singularities of the Selberg zeta function associated to a convex-cocompact, torsion free group acting on a hyperbolic space.

We discuss questions of isospectrality for hyperbolic orbisurfaces, examining the relationship between the geometry of an orbisurface and its Laplace spectrum. We show that certain hyperbolic orbisurfaces cannot be isospectral, where the obstructions involve the number of singular points and genera of our orbisurfaces.…

For a class of even dimensional conformally compact manifolds (X,g), we define a generalized Krein spectral function by applying a renormalized trace functional to the spectral measure of the Laplacian. We then show that this is the phase of the Kontsevich-Vishik determinant det S(s) of the scattering operator S(s) of …

Zeta functions for non-unitary twists are shown to have analytic continuation.

We propose a new method for studying $n$- and $Γ$-cohomology of globalizations of Harish-Chandra modules, where $G=KAN$ is a rank one semisimple Lie group, $Γ$ is a discrete subgroup of $G$ and $n=Lie(N)$. We prove a conjecture of Patterson relating the singularities of Selberg zeta functions with the $Γ$-cohomology of…

We study Selberg zeta functions $Z(s,σ)$ associated to locally homogeneous vector bundles over the unit-sphere bundle of a complete odd-dimensional hyperbolic manifold of finite volume. We assume a certain condition on the fundamental group of the manifold. A priori, the Selberg zeta functions are defined only for s in…

Survey on twisted dynamical zeta functions and Fried's conjecture.

For singular metrics, there is no Quillen metric formalism on cohomology determinant. In this paper, we develop an admissible theory, with which the arithmetic Deligne-Riemann-Roch isometry can be established for singular metrics. As an application, we first study Weil-Petersson metrics and Takhtajan-Zograf metrics on …

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmüller space. We then use this formula to determine the asymptotic behavior as $\text{Re} (s) \to \infty$ of the second variation. As a consequence, for $m \in \mathbb{N}$, we obtain the complete expansio…

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.

In this paper we establish a strong multiplicity one type property for the length-holonomy spectrum for the three dimensional compact hyperbolic spaces. We use the analytic properties of Selberg-Gangolli-Wakayama zeta functions associated to compact hyperbolic spaces.

Optimal joint separation condition for radar and communications channels in dual-blind deconvolution.

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X…

In this article we prove a generalization of Selberg's lemma on the existence of torsion free, finite index subgroups of arithmetic groups. Some of the geometric applications are the resolution a conjecture of Nimershiem and answers to questions of Long-Reid and the author.

The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.

A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a …

Let $\overline{M}$ be a compact Riemann surface and let $g^{TM}$ be a metric over $\overline{M} \setminus D_M$, where $D_M \subset \overline{M}$ is a finite set of points. We suppose that $g^{TM}$ is equal to the Poincaré metric over a punctured disks around the points of $D_M$. The metric $g^{TM}$ endows the twisted c…