The paper studies Hodge Laplacians from point clouds, proving spectral convergence and harmonic form consistency.
problem Analyzing Riemannian submanifolds from point cloud data.
method Constructing deformed Hodge Laplacians and empirical operators from point clouds, proving convergence properties.
result Empirical spectral cluster contains the k-th Betti number and converges to harmonic k-forms. Proves L2 Frölicher inequality on complex manifolds.
problem Calculating L2 Betti and Hodge numbers. method Uses spectral projectors of (Dh)2 to build an injection. result New proof of classical Frölicher inequality.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
Paper introduces a new multilinear functional for spectral triples and computes its properties.
problem Computing properties of spectral triples and their associated Hodge operators.
method Introduces a new multilinear functional for spectral triples and computes its properties using noncommutative residue and perturbed de-Rham Hodge operators.
result Recover two forms, torsion of the linear connection, and four forms by the noncommutative residue and perturbed de-Rham Hodge Dirac triple.
Promotes spectral functionals to noncommutative fields and proves a theorem.
problem Spectral functionals on noncommutative fields and manifolds with boundary.
method Carries out promotion to spectral functionals, associates with noncommutative residue, and proves theorem.
result Proves Dabrowski-Sitarz-Zalecki type theorem for statistical de Rham Hodge operators on manifolds with boundary.
The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.
problem Bounding bandwidth and focal radius for manifolds with positive isotropic curvature.
method Using spectral properties of a twisted de Rham-Hodge operator.
result Upper bounds on bandwidth and focal radius are derived for hypersurfaces in PIC manifolds.
Study examines metrics and functionals for Hodge-Dirac operator on manifolds.
problem Examining metrics and functionals for Hodge-Dirac operator on manifolds.
method Analyzing the metric and Einstein bilinear functionals of differential forms for Hodge-Dirac operator d+δ. result The functionals reproduce those for the canonical Dirac operator on a spin manifold up to a numerical factor.
Paper introduces magnetic Hodge Laplacian for differential forms.
problem No specific problem stated; general spectral analysis of differential forms.
method Introduced magnetic Hodge Laplacian, discussed spectral results.
result Similarities and differences with magnetic Laplacian on functions.
Study on deformations of (p,q)-forms and spectral sequence degenerations.
problem Understanding deformations of (p,q)-forms under complex structure changes. method Analyzing Frölicher spectral sequence conditions for (p,q)-form deformations. result Unobstructed deformations of (p,q)-forms under specific spectral sequence conditions. For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of…
These are the notes for the talk "Hodge numbers of a hypothetical complex structure on S6" given by the author at the MAM1 "(Non)-existence of complex structures on S6" held in Marburg in March 2017. They are based on [A. Gray, A property of a hypothetical complex structure on the six sphere, Boll. Un. Mat. Ital.…
Sharp lower bound for Hodge Laplacian on Kähler hyperbolic manifolds.
problem Finding a sharp lower bound for the spectrum of the Hodge Laplacian.
method Explicitly expressed in terms of the supremum norm of the 1-form.
result Explicit spectral lower bounds for bounded symmetric domains.
We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of (∙,0)- and (0,∙)-Hodge numbers on a compact complex manifold, and obtain the equality for the number…
This paper extends Dabrowski-Sitarz-Zalecki theorems to manifolds with boundary.
problem Generalizing theorems to manifolds with boundary.
method Extending results of Dabrowski etc. to 4D oriented Riemannian manifolds with boundary.
result Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary.
Study spectral and index properties of Hodge-Dirac operator on compact manifolds.
problem Investigate spectral and index-theoretic properties of Hodge-Dirac operator on compact Riemannian manifolds.
method Establish bisectoriality and H∞ functional calculus without curvature assumptions. result Prove compact Banach spectral triple and recover classical topological invariants as Lp-indices. Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.
problem Simplicity of Hodge Laplacian and curl operator eigenvalues along metric families.
method Generalized Teytel's method to compute meagre codimension of metrics with specific eigenvalue multiplicities.
result Simplicity of Hodge Laplacian and curl operator is not a meagre codimension 2 property.
We consider a family of compact, oriented and connected Riemannian manifolds shrinking to a metric graph and describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian. We apply our results to produce manifolds with spectral gaps of arbitrarily large size in the spectrum of the Hodge Laplacian.
Researchers prove no unexpected relations between complex manifold numbers.
problem Proving no unexpected universal linear relations between Hodge, Betti, and Chern numbers of compact complex manifolds.
method Developed a framework to tackle more general questions involving all cohomological invariants, solved specific construction problems.
result Obtained full answers to general questions about universal relations and bimeromorphic invariants in low dimensions.
Let A be a dg algebra over F_2 and let M be a dg A-bimodule. We show that under certain technical hypotheses on A, a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product of M with itself and converges to the Hochschild homology of M. We apply th…
Study shows eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds is related to isoperimetric ratio.
problem Eigenvalue of Hodge Laplacian on coexact 1-forms in hyperbolic 3-manifolds.
method Using isoperimetric ratio relating geodesic length and stable commutator length, with comparison constants polynomial in volume and injectivity radius.
result Estimates show spectral gap of 1-form Laplacian vanishing exponentially fast in volume for certain hyperbolic 3-manifolds.
Proves Hodge structures on modular functors, providing formulas for Hodge numbers.
problem Existence and uniqueness of Hodge structures on modular functors.
method Non-Abelian Hodge correspondence and Ocneanu rigidity.
result Explicit formulas for Hodge numbers in modular functors of level 2 times an odd number.
We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .
The Hodge spectra can distinguish orbifolds from manifolds, especially in low dimensions.
problem Distinguishing orbifolds from manifolds using Hodge spectra.
method Computing heat invariants of Hodge Laplacians on p-forms. result The Hodge spectra, particularly the 0- and 1-spectra, can distinguish orbifolds from manifolds in low dimensions. For a noncollapsed Gromov-Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which in…
New spectral sequence for K-manifolds, computing cohomology and harmonic forms.
problem Computing cohomology and harmonic forms of K-manifolds. method Introducing a new spectral sequence and using it to generalize theorems from K-contact geometry. result Computed cohomology ring and harmonic forms of S-manifolds. BIG Laplacians bridge combinatorial and Hodge Laplacians for discrete data.
problem Comparing combinatorial and Hodge Laplacians for discrete data.
method Introducing Boundary-Induced Graph (BIG) Laplacians using DEC.
result BIG Laplacian eigenvalues converge to Hodge Laplacian for simple shapes.
The paper introduces a new class of manifolds based on the Hodge decomposition and spectral sequences.
problem Characterizing and understanding new classes of compact complex manifolds.
method Introducing a new class of page-r-∂∂ˉ-manifolds and using spectral sequences. result Characterized and provided examples of the new class of manifolds.
Analytic lattice cohomology defined for isolated singularities, linking to Heegaard Floer cohomology.
problem Defining and analyzing analytic lattice cohomology for isolated singularities.
method Using a good resolution of the singularity, proving independence of resolution choice, and relating to Hodge spectral numbers.
result Independence of analytic lattice cohomology from the choice of resolution and connection to Hodge spectral numbers.
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and tho…
The Hodge spectra help distinguish orbifolds from manifolds with singularities.
problem Distinguishing orbifolds from manifolds based on their singular sets.
method Computing heat invariants of Hodge Laplacians on p-forms. result The Hodge spectra of 0- and 1-forms distinguish orbifolds from manifolds with singularities. The paper studies spectral convergence of Hodge-Kodaira Laplacians on degenerating Hermitian metrics.
problem Spectral convergence of Hodge-Kodaira Laplacians on degenerating Hermitian metrics.
method Proves spectral convergence theorems for Hodge-Kodaira Laplacians Δ∂,m,0,s under general assumptions. result Eigenvalues, heat operators, and heat kernels converge to those of a self-adjoint operator Δ∂,m,0,abs. Study on Hodge theory for almost complex manifolds.
problem Determining Hodge numbers for almost complex manifolds.
method Review and analysis of recent developments in Hodge theory for almost complex manifolds.
result Hodge numbers are almost complex, almost Kähler, or birational invariants in dimension four.
The paper develops a new Laplacian for manifold learning from data.
problem Learning manifold structures from point cloud data.
method Constructs deformed Hodge Laplacians and proves their spectral convergence.
result Empirical operators converge to the classical Hodge Laplacian.
Study Hodge-de Rham numbers for almost complex 4-manifolds, extending properties from complex surfaces.
problem Understanding Hodge-de Rham numbers for almost complex 4-manifolds.
method Introduced and studied Hodge-de Rham numbers, extending properties from complex surfaces.
result All Hodge-de Rham numbers for compact almost complex 4-manifolds are determined by the cohomology, except for one (the irregularity).
For any positive integer m and any dimension n, we show that any n-dimensional Hodge diamond with values in Z/mZ is attained by the Hodge numbers of an n-dimensional smooth complex projective variety. As a corollary, there are no polynomial relations among the Hodge numbers of n-dimensional smooth complex projective va…
Introduces a new Hodge theory using vector fields on manifolds.
problem Developing a new Hodge theory for manifolds with vector fields.
method Defines a vector field induced Hodge L2-inner product, codifferential, and Laplacian. result Established de Rham-Hodge theory for closed and boundary manifolds.
For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; if k=2m is even, then we have to impose that h^{m,m} is bigger than some quadratic bound in m. Combining these results for different weights, we so…
Study contact manifold heat kernels under Riemannian metrics blow-up.
problem Analyze spectral invariants on contact manifolds.
method Examine Hodge Laplacian heat kernel behavior under Riemannian metrics.
result Contact versions of eta-invariant and analytic torsion are topological.
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
We study solutions for the Hodge laplace equation Δu=ω on p forms with Lr estimates for r>1. Our main hypothesis is that Δ has a spectral gap in L2. We use this to get non classical Lr Hodge decomposition theorems. An interesting feature is …
TopoNTK kernel captures higher-order interactions in simplicial complexes.
problem Graph neural networks miss higher-order interactions in relational systems.
method Introduces TopoNTK, an infinite-width kernel for simplicial message passing.
result TopoNTK captures topology invisible to graph kernels, improving expressivity and interpretability.
The paper introduces a trilinear functional to recover torsion in spectral triples.
problem Recovering torsion in noncommutative spectral triples.
method Introduces a trilinear functional for spectral triples and demonstrates its application to recover torsion.
result The trilinear functional recovers the torsion of the linear connection in canonical spectral triples.
New Sasaki structures identified by Hodge numbers in odd dimensions.
problem Identifying Sasaki structures with distinct Hodge numbers.
method Producing examples of manifolds with pairs of Sasaki structures having different basic Hodge numbers.
result Examples of manifolds with pairs of Sasaki structures having different basic Hodge numbers in odd dimensions.
Gaussian processes classify graphs using vertex and edge features.
problem Graph classification in machine learning.
method Transform graph features into spectral Euclidean features, apply Hodge decomposition.
result Gaussian processes can classify graphs using vertex and edge features.
The paper proves wave operator existence and completeness for Hodge Laplacians.
problem Proving the existence and completeness of wave operators for Hodge Laplacians.
method Integral criterion, probabilistic Bismut-type formulae, heat semigroup, local curvature bounds.
result Absolutely continuous spectra of Hodge Laplacians coincide under quasi-isometry.
Novel algorithm learns sparse signal representations over topological spaces.
problem Sparse representation of signals over combinatorial topological spaces.
method Leveraging Hodge theory, the paper embeds topology into a dictionary structure via concatenated sub-dictionaries, each as a polynomial of Hodge Laplacians, and optimizes the dictionary coefficients and sparse signal representation via iterative alternating algorithms.
result Efficiently learned sparse representations and underlying relational structure of topological signals.
Paper shows examples of almost Kähler manifolds satisfying Hard Lefschetz but not Betti-Hodge equality.
problem Understanding the Hard Lefschetz condition in almost Kähler manifolds.
method Examples and counterexamples of compact almost Kähler manifolds.
result The Hard Lefschetz condition does not imply the equality between Betti and Hodge numbers in almost Kähler manifolds.