Selberg's Lemma fails for certain curved manifolds.
problem Applying Selberg's Lemma to negatively curved Hadamard manifolds.
method Proving the failure of Selberg's Lemma for specific groups of manifolds.
result Selberg's Lemma does not hold for discrete isometry groups of negatively curved Hadamard manifolds.
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.
Derives Selberg trace formula on Riemann surfaces and generalizes to other spaces.
problem Deriving and generalizing the Selberg trace formula.
method Supersymmetric localization principle and path integral derivation.
result Derives Selberg trace formula on arbitrary compact Riemann surfaces and generic compact locally symmetric spaces.
Sub-Riemannian Selberg trace formulae for compact quotients of SL(2, R)
problem Computing zeta-regularized determinants of sub-Laplacians
method Using Fourier decomposition and Selberg trace formulae
result Compact determinant formula expressed in terms of base hyperbolic surface and relative Selberg product
Study on Selberg's modified metric in symmetric spaces.
problem Properties of modified metric in symmetric spaces.
method Analysis of SL(n,R)/SO(n,R) with Selberg's premetric. result Generalizations of hyperbolic space properties.
Study asymptotics of Selberg zeta function on spin moduli space.
problem Asymptotic behavior of Selberg zeta function for degenerating metrics.
method Analyzes logarithmic derivative of Selberg zeta function for spin Dirac operator on compact surfaces.
result Proves asymptotic expansion up to order t4logt. Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.
problem Understanding Brownian loops on hyperbolic surfaces and their relation to Selberg zeta function.
method Computed mass of loops and related to Selberg zeta function for geometrically finite surfaces.
result Relate total loop mass to Selberg zeta function, providing probabilistic interpretations of determinants.
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.
problem Analytic continuation of zeta functions for non-unitary twists.
method Analytic continuation for compact locally-symmetric spaces with non-unitary twists.
result Zeta functions admit analytic continuation as meromorphic functions.
Study shows strong multiplicity one property for 3D hyperbolic spaces.
problem Understanding the spectrum of length-holonomy in 3D hyperbolic spaces.
method Analyzing Selberg-Gangolli-Wakayama zeta functions.
result Established a strong multiplicity one type property for length-holonomy spectrum.
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.
problem Determining the maximum eigenvalue multiplicity for closed hyperbolic surfaces.
method Applying the twisted Selberg trace formula to induced representations of triangle groups.
result Found counterexamples with higher eigenvalue multiplicities than previously conjectured.
Upper bounds on geodesics in hyperbolic manifolds.
problem Bounding the number of shortest closed geodesics in hyperbolic manifolds.
method Using the Selberg trace formula, we derive upper bounds on the number of shortest and primitive geodesics.
result Generalized Parlier's theorem to higher dimensions and uniform bounds for manifolds with bounded geometry.
Survey on twisted dynamical zeta functions and Fried's conjecture.
problem Analyzing twisted dynamical zeta functions and their relation to Fried's conjecture.
method Review of existing literature and mini-course presentation.
result Discussion and validation of 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.
problem Eigenvalue rigidity of random hyperbolic surface covers.
method Selberg trace formula and polynomial method.
result Distribution of eigenvalues converges to hyperbolic plane's spectral measure.
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 Re(s)→∞ of the second variation. As a consequence, for m∈N, we obtain the complete expansio…
The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.
problem Counting conjugacy classes of loxodromic elements in Anosov subgroups.
method Interpreting Jordan projections as periods of a flow and proving exponential mixing.
result Proves a counting theorem with a power saving error term for conjugacy classes of loxodromic elements.
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.
New examples of hyperbolic 3-manifolds where Seiberg-Witten equations fail.
problem Finding hyperbolic 3-manifolds without irreducible Seiberg-Witten solutions.
method Combining hyperbolic geometry, upper bound for eigenvalues, Selberg trace formula, and precise numerical bounds.
result First examples of hyperbolic 3-manifolds where Seiberg-Witten equations do not admit irreducible solutions.
The paper calculates the full asymptotics of analytic torsions for compact orbifolds.
problem Analytic torsions of compact locally symmetric orbifolds.
method Using Selberg's trace formula and geometric localization, the paper evaluates the heat trace and orbital integrals.
result Explicit formula for the asymptotic Ray-Singer analytic torsion of compact orbifolds.
Optimal joint separation condition for radar and communications channels in dual-blind deconvolution.
problem Recovering information from overlaid radar and communications signals with unknown channels.
method Extremal functions from Beurling-Selberg interpolation theory for joint separation, nuclear norm minimization for matrix retrieval, and MUSIC for parameter estimation.
result Guaranteed well-conditioned Vandermonde matrix for MUSIC, validating theoretical findings.
We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type ZΓ,Σo(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…
The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.
problem Understanding the relationship between geodesic flows and higher-dimensional Reidemeister torsion.
method Using the integral expression of the Ruelle zeta function and the Selberg zeta function.
result The absolute value at zero of the Ruelle zeta function equals the higher-dimensional Reidemeister torsion.
Explains the Schwarz lemma in lecture notes.
problem None explicitly stated; focuses on explanation.
method Expository notes on the Schwarz lemma.
result Explains the Schwarz lemma.
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 …
Author provides an alternate proof of the free ribbon lemma.
problem Proving that every free sphere-link in the 4-sphere is a ribbon sphere-link.
method An alternate proof of the free ribbon lemma.
result Provides an alternate proof of the free ribbon lemma.
Study on Dirac operator spectrum on hyperbolic surfaces with shrinking geodesics.
problem Spectrum of spin Dirac operator on hyperbolic surfaces with pinched geodesics.
method Trace formula for Dirac operator, Huber's theorem, small-time heat trace asymptotic expansion.
result Convergence of Selberg zeta function for degenerating hyperbolic surfaces.
The paper extends Schwarz's lemma to RC-positivity and complex manifolds.
problem Comparing metrics with RC-positivity in complex manifolds.
method Establishing Schwarz lemmas for RC-positivity and applying them to complex manifolds.
result New diameter and volume comparison theorems.
New CR-structures lemma simplifies CR-manifold deformation proof.
problem Deformation unobstructedness of CR-manifolds.
method New Tian-Todorov lemma applied to CR-manifolds.
result Reproved deformation unobstructedness of CR-manifolds.
Survey on strong closing lemmas in Hamiltonian dynamics.
problem Understanding dynamics in Hamiltonian systems.
method Use spectral invariants in symplectic geometry.
result Proofs of strong closing lemmas in various dimensions.
Study several weak forms of a lemma on compact complex manifolds.
problem Understanding weak forms of a lemma on compact complex manifolds.
method Complete unified study of weak forms of the $\ddb-$Lemma.
result Unified understanding of various weak forms of the $\ddb-$Lemma.
Unified Schwarz lemma in Kähler and Hermitian geometry.
problem Various forms of the Schwarz lemma in Kähler and Hermitian geometry.
method Introducing new curvatures to refine and elucidate the real bisectional curvature.
result Unified Chern-Lu, Aubin-Yau, and Chen-Cheng-Look Schwarz lemmas.
Paper generalizes Schwarz Lemma for VT harmonic maps with conditions.
problem Generalizing Schwarz Lemma for a specific type of harmonic maps.
method Conditions on eigenvalues and Ricci curvature are used to prove the lemma.
result Schwarz Lemma for VT harmonic maps proved with distance and volume decreasing properties.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.
New Schwarz Lemma for Bergman metrics in bounded domains.
problem Finding bounds for Bergman metrics in bounded domains.
method Using Cauchy-Schwarz inequality from probability theory.
result Established a new Schwarz Lemma for Bergman metrics.
Tucker and Ky Fan's lemma are combinatorial analogs of the Borsuk-Ulam theorem (BUT). In 1996, Yu. A. Shashkin proved a version of Fan's lemma, which is a combinatorial analog of the odd mapping theorem (OMT). We consider generalizations of these lemmas for BUT-manifolds, i.e. for manifolds that satisfy BUT. Proofs rel…
Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.
problem Comparing geometric quantities of circle packings with different boundary values.
method Combinatorial Calabi flows and maximum principle.
result Discrete Schwarz-Pick lemma proven for generalized circle packings.
The paper improves Zakalyukin's lemma for frontals and applies it to surface singularities.
problem Improving the conditions under which wave front germs imply map germs.
method Generalization of Zakalyukin's lemma for frontals and applications to surface singularities.
result The paper provides a more general version of Zakalyukin's lemma for map germs.
Meridian lemma extended to fully alternating links in thickened surfaces.
problem Extending Menasco's meridian lemma to fully alternating links in thickened surfaces.
method Developed a new meridian lemma for fully alternating links in thickened orientable surfaces of positive genus.
result The meridian lemma holds for fully alternating links in thickened surfaces.
Proves a quantitative closing lemma for negatively curved manifolds.
problem Closing lemma for negatively curved manifolds.
method Quantitative closing lemma proof.
result Study of partner and pseudo-partner orbits for self-crossing closed geodesics.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Extends Margulis Lemma to RCD(K,N) spaces.
problem Applying Margulis Lemma to new geometric structures.
method Improved Regularity Estimates for Regular Langrangian Flows.
result Margulis Lemma extended to RCD(K,N) spaces.