Improved remainder estimate for eigenvalue asymptotics on manifolds with group actions.
problem Asymptotic distribution of eigenvalues of invariant elliptic operators.
method Refined stationary phase approximation and singular critical sets analysis.
result Asymptotic multiplicity formula for families of irreducible representations.
Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
problem Finding generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
method Weyl asymptotic law for G-equivariant volume spectrum, generic density result. result Generic density of equivariant min-max hypersurfaces in Riemannian manifolds.
Study shows Riemannian manifold volume spectrum follows a Weyl law.
problem Understanding volume spectrum of Riemannian manifolds.
method Proved a Weyl law for volume spectrum.
result Volume spectrum of Riemannian manifolds follows a Weyl law.
Proves Weyl's law for metric spaces with Ricci curvature.
problem Proving Weyl's law for metric measure spaces with bounded Ricci curvature.
method Analyzes RCD∗(K,N) spaces to prove asymptotic eigenvalue formula. result Establishes Weyl's law for Dirichlet eigenvalues in metric measure spaces.
Derives Weyl law for volume spectrum using parametric inequalities.
problem Deriving the Weyl law for the volume spectrum in compact Riemannian manifolds.
method Proves parametric generalizations of isoperimetric and coarea inequalities to derive the Weyl law.
result Derives the Weyl law for 1-cycles in 3-manifolds.
Proves the Weyl law for 1-cycles in manifolds.
problem Proving the Weyl law for the volume spectrum of 1-cycles in n-dimensional manifolds.
method Using parametric versions of the coarea inequality and isoperimetric inequality, along with a localized approximation method.
result Proves the Weyl law for 1-cycles in manifolds.
The paper confirms a Weyl law for the p-Laplacian on closed Riemannian manifolds.
problem The variational spectrum of the p-Laplacian on closed Riemannian manifolds.
method Based on ideas of Gromov and Liokumovich, Marques, Neves.
result A Weyl law holds for the variational spectrum of the p-Laplacian.
Improved estimate for remainder term in Weyl law for congruence subgroups of Chevalley groups.
problem Improving the remainder term estimate in the Weyl law for congruence subgroups of Chevalley groups.
method Sharpened the Weyl law remainder term estimate for simply connected Chevalley groups.
result Power saving estimate for the remainder term.
The paper studies heat kernel behavior in RCD spaces and initiates Weyl's law study.
problem Understanding heat kernel behavior in RCD spaces.
method Proved pointwise convergence of heat kernels for mGH-convergent sequences of RCD spaces.
result Initiated study of Weyl's law in RCD spaces.
Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.
problem Proving Weyl laws for Schrödinger operators on noncompact manifolds.
method Heat kernel asymptotics, Karamata-Hardy-Littlewood Tauberian theorem, and semiclassical analysis.
result Established both classical and semiclassical Weyl laws for Schrödinger operators on noncompact manifolds.
New criterion for Weyl law on Riemannian manifolds without standard assumptions.
problem Establishing Weyl law for Schrödinger operators on complete Riemannian manifolds.
method Identifying a geometric-analytic invariant cδ(λ) that balances manifold geometry, potential growth, and oscillation scale. result Weyl asymptotic holds if cδ(λ) approaches 0 as λ goes to infinity. New method connects neural networks to diagrammatic algebra.
problem Constructing permutation equivariant neural networks.
method Schur-Weyl duality between symmetric group and partition algebra.
result Simple diagrammatic method for calculating weight matrices.
The study improves bounds on the number of closed geodesics and logarithmic improvements in the Weyl law.
problem Estimating the number of closed geodesics and improving logarithmic bounds in the Weyl law.
method Study of non-degeneracy properties of nearly closed orbits for predominant sets of metrics.
result Logarithmic improvements in the Weyl law and exponential bounds on the number of closed geodesics.
Introduces Weyl bundle gerbe and shows it's not D-stably isomorphic.
problem Identifying and distinguishing bundle gerbes.
method Introduces cup product bundle gerbe, defines Weyl bundle gerbe, and uses holonomy to show non-isomorphism.
result Weyl bundle gerbe and basic bundle gerbe are not D-stably isomorphic.
Combines noncommutative geometry and spectral theory for new Weyl laws.
problem Developing new Weyl laws for noncommutative manifolds.
method Functional analysis, spectral theory, and Tauberian conditions.
result Generalizes and simplifies recent results on Weyl laws and integration formulas.
Study magnetic Dirac operator on manifolds, proving Weyl law and eta bound.
problem Analyzing the semi-classical Dirac operator with magnetic potentials.
method Almost analytic continuations, Birkhoff normal form, local index theory.
result Sharp local Weyl law and bound on eta invariant.
Paper improves Weyl Law for product manifolds.
problem Improving the error term in the Weyl Law for product manifolds.
method Using Duistermaat and Guillemin's result, reducing the problem to lattice point distribution in Euclidean space.
result Quantified bound for Cartesian product of spheres, showing o(λd−1) error. Study local Weyl law on hyperbolic surfaces, identifying geodesic loops.
problem Understanding the variance of a local Weyl law on hyperbolic surfaces.
method Explicit integration of test functions, stationary phase arguments, and geometric analysis of geodesic loops.
result Identifies length-minimizing geodesic loops and sequences, proving they are simple.
The Weyl Law is proven for phase transitions, with applications to minimal hypersurfaces and separating interfaces.
problem Proving the Weyl Law for phase transition spectrum and understanding the density of limit interfaces.
method Techniques of Liokumovich-Marques-Neves and recent work of Chodosh-Mantoulidis.
result Density of separating limit interfaces and existence of infinitely many minimal hypersurfaces for generic metrics.
New Weyl's laws discovered for compact spaces with Ricci curvature bounds.
problem Understanding growth rates of eigenvalues in compact spaces with Ricci curvature constraints.
method Developed new properties of α-Grushin halfplanes and analyzed singular sets of null capacities. result Established Weyl's laws with power growth and logarithmic corrections for compact spaces.
Study spectral properties of sub-Riemannian Laplacians, proving quantum ergodicity and heat kernel asymptotics.
problem Spectral properties of sub-Riemannian Laplacians.
method Quantum ergodicity results, small-time asymptotics of sub-Riemannian heat kernels, Weyl law.
result Weyl law and spectral concentration on Lie brackets of length r-1.
Researchers study eigenvalues on singular Riemannian manifolds, showing how curvature affects Weyl's law.
problem Analyzing eigenvalues of Laplace-Beltrami operator on singular Riemannian manifolds with unbounded geometrical invariants.
method Developed a new quantitative estimate for the remainder of the heat trace and Weyl's function on Riemannian manifolds.
result Constructed singular Riemannian metrics with prescribed non-classical Weyl's law for various slowly varying functions.
The study provides a formula for the volume of leaf spaces of certain foliations.
problem Calculating the volume of leaf spaces for singular Riemannian foliations.
method Proved a version of Weyl's Law for the basic spectrum of closed singular Riemannian foliations.
result Explicit formula for the volume of leaf spaces in terms of basic polynomials.
Researchers found a Weyl law for Liouville quantum gravity eigenvalues.
problem Understanding the spectral geometry of Liouville quantum gravity.
method Obtained a Weyl law for eigenvalues of Liouville Brownian motion.
result The n-th eigenvalue grows linearly with n, with a constant determined by the Liouville area and a specific cγ. We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus Tθ2 equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed L…
The Planck mass can be derived from gravitational potential behavior in compactifications.
problem Deriving the Planck mass from gravitational potential behavior in compactifications.
method Physical considerations and Weyl law application to gravitational potential behavior.
result The Planck mass can be reconstructed from the asymptotics of the masses of spin 2 Kaluza--Klein modes.
The paper shows equidistribution of geodesics and nets on manifolds.
problem Equidistribution of geodesics and nets on manifolds.
method Generic metrics, Weyl Law for volume spectrum, and equidistribution of geodesic nets.
result Equidistribution of geodesic nets on 3-manifolds.
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…
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the D…
The paper extends Weyl's law to CROSSes, showing sharpness and polynomial improvement.
problem Understanding the error term in Weyl's law for different types of manifolds.
method Analyzing the Laplacian eigenvalues on Compact Rank One Symmetric Spaces (CROSSes).
result For CROSSes, the error term in Weyl's law is sharp, and for products of CROSSes, it can be polynomially improved.
Study spectral analysis on lens spaces, proving isospectral lens spaces with prime order fundamental groups.
problem Spectral analysis of the Kohn Laplacian on lens spaces.
method Analog of Weyl's law and isospectral lens spaces with prime order fundamental groups.
result Two 3D lens spaces with prime order fundamental groups are isospectral with respect to the Kohn Laplacian if and only if they are CR isometric.
Under suitable invertibility hypothesis, the spectrum of the Dirac operator on certain open spin Riemannian manifolds is discrete, and obeys a growth law depending qualitatively on the (in)finiteness of the volume.
Defines half-volume spectrum for manifolds and proves Weyl law holds.
problem Understanding volume distribution in manifolds.
method Introduces half-volume spectrum and uses Weyl law and Allen-Cahn min-max theory.
result Weyl law holds for half-volume spectrum and half-volume constant achieved by specific surfaces.
Generalizes Quillen's theorem to equivariant settings.
problem Classifying commutative formal group laws in equivariant bordism.
method Combines computations and detailed investigations of equivariant Lazard rings.
result Identifies the Z/2-equivariant unitary bordism ring with the Z/2-equivariant Lazard ring. Harish-Chandra's volume formula shows that the volume of a flag manifold G/T, where the measure is induced by an invariant inner product on the Lie algebra of G, is determined up to a scalar by the algebraic properties of G. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing…
Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
Proves a lower bound for Θ function on manifolds without conjugate points.
problem Bounding the Θ function on manifolds without conjugate points.
method Proves a uniform lower bound for the Θ function using elementary techniques.
result The Bérard remainder in the Weyl law is valid for manifolds without conjugate points.
The study reveals scaling limits of spectral projectors on Riemannian manifolds.
problem Understanding spectral function scaling limits on Riemannian manifolds.
method New off-diagonal estimates and Weyl law application.
result Scaling limit of spectral projector is a normalized Bessel function.
Study on sub-Riemannian geometry in 4D, focusing on abnormal geodesics.
problem Understanding sub-Riemannian geometry and its spectrum.
method Wave trace expansion, Weyl laws, propagation of singularities, quantum ergodicity.
result First appearance of abnormal geodesics in sub-Riemannian spectral geometry.
We will provide a lower bound for the equivariant Lusternik-Schnirelmann category of an arbitrary proper action in terms of the stratification by orbit types, and an upper bound for proper polar actions in terms of the equivariant Lusternik-Schnirelmann category of its generalized Weyl group. As an application we repro…
We prove an off-diagonal expansion for a Toeplitz operator with an indicator function.
problem Asymptotics of Toeplitz operators with indicator function
method Off-diagonal expansion
result We extend two results to the non-compact setting.
We consider the T-equivariant cohomology of Bott-Samelson desingularisations of Schubert varieties in the flag manifold of a connected semi-simple complex algebraic group of adjoint type with maximal torus T. We construct a combinatorially pure (in the sense of T. Braden and R. Macpherson) sheaf on the Bruhat graph of …
Develops quantum character theory for complex reductive groups.
problem Quantum analogue of conjugation equivariant D-modules. method Schur-Weyl functor and double affine Hecke algebra.
result Computes endomorphism algebras of quantum Hotta-Kashiwara modules.
We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…
Study spectral distribution of twisted Laplacian on high genus hyperbolic surfaces.
problem Estimating spectral distribution of twisted Laplacian on hyperbolic surfaces.
method Estimate spectral distribution by supremum norm of harmonic form; show small supremum norm for high genus surfaces; prove uniform Weyl law.
result Prove uniform Weyl law for real parts of spectrum on high genus hyperbolic surfaces.
New cell structure on O(3)/O(1)3 derived from injectivity radius computation.
problem Constructing equivariant cell structures on flag manifolds.
method Injectivity radius computation and Dirichlet-Voronoi domains.
result New S3-equivariant cell structure on O(3)/O(1)3. Study nonlocal minimal surfaces for minimal surfaces in 3D manifolds.
problem Existence and regularity of minimal surfaces in 3D manifolds.
method Min-max methods, fractional perimeters, uniform estimates.
result Uniform estimates for min-max s-minimal surfaces in 3-manifolds, convergence to smooth minimal surfaces.