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)

Study on Selberg's modified metric in symmetric spaces.

Study asymptotics of Selberg zeta function on spin moduli space.

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

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 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…

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…

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…

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…

Three counterexamples show higher eigenvalue multiplicities than conjectured.

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.

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…

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

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.

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

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.

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.

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…

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 …

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

Uniform spectral gap for convex cocompact hyperbolic surfaces and expanders.

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…

Given a sequence of regular finite coverings of complete Riemannian manifolds, we consider the covering solenoid associated with the sequence. We study the leaf-wise Laplacian on the covering solenoid. The main result is that the spectrum of the Laplacian on the covering solenoid equals the closure of the union of the …

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 …

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by countting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspe…

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, we will compare the order of the Alexander invariant at t=1 and one of Ruelle-Selberg L-function at s=0. Our result may be considered as a geometric analog of the Iwasawa main conjecture in the algebraic …

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on ${\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $\exp (C |s|^δ)$ where $δ$ is the dimension of the limit set of the group. This bound is more precise than the optimal…

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 give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian $n$-manifolds. For some non-compact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we us…

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 define geometric zeta functions for locally symmetric spaces as generalizations of the zeta functions of Ruelle and Selberg. As a special value at zero we obtain the Reidemeister torsion of the manifold. For hermitian spaces these zeta functions have as special value the quotient of the holomorphic torsion of Ray an…

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)…

This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of the laplacian on differential forms. The method of proof is representation theoritic, we hope the different chapters may as well serve …

New expanders found using origami surfaces with spectral gap.

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-…

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cus…

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