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.
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.
Discretizes Hodge-Dirac operators on a torus.
problem Capturing geometric aspects of continuum Hodge theory in discrete settings.
method Discrete exterior calculus framework, Hodge-Dirac and Laplace operators.
result Proves discrete Hodge decomposition theorem on combinatorial torus.
We prove R-bisectoriality and boundedness of the H∞-functional calculus in Lp for all 1<p<∞ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on k-forms.
Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
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. 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.
In order to facilitate the comparison of Riemannian homogeneous spaces of compact Lie groups with noncommutative geometries ("quantizations") that approximate them, we develop here the basic facts concerning equivariant vector bundles and Dirac operators over them in a way that uses only global constructions and argume…
We define local Hardy spaces of differential forms hDp(∧T∗M) for all p∈[1,∞] that are adapted to a class of first order differential operators D on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge--Dirac operator on $…
We establish new Calderón reproducing formulas for self-adjoint operators D that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with D through holomorphic functional calculus whilst the synthesising function interacts with D through funct…
The paper constructs instanton complexes on stratified pseudomanifolds.
problem Analyzing functions with non-isolated critical points on singular spaces.
method Constructing Witten instanton complexes and Hilbert complexes.
result Proves Morse inequalities for stratified pseudomanifolds.
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
GJMS operators connect geometry, analysis, and physics.
problem None explicitly stated; focus on operators and their impact.
method Construction of conformally invariant differential operators.
result GJMS operators have significant impact in geometry, analysis, and physics.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
The paper proves new theorems about specific types of operator perturbations.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Mixtures of neural operators reduce active complexity in operator learning.
problem Reduction of active complexity in operator learning models.
method Constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction.
result Every scalar uniformly continuous nonlinear operator can be approximated by a MoNO whose active expert has smaller depth, width, and rank scaling.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
Researchers create a family of conformally covariant operators.
problem Developing a comprehensive set of conformally covariant operators.
method Constructing a family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space.
result Symmetrization of ambient operators is formally self-adjoint.
The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.
problem Constructing an elliptic operator analogous to the ∂∂ operator in complex or Kähler manifolds.
method Local analysis estimates and demonstration using the Atiyah-Hitchin-Singer operator.
result Every d-exact (1,1)-form is globally tmd-exact for compact taming symplectic 4-manifolds.
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.
The paper classifies and proves properties of symmetry breaking operators for specific groups.
problem Classifying and understanding symmetry breaking operators for de Sitter and Lorentz groups.
method Constructing and classifying differential symmetry breaking operators, proving localness, and showing sporadic nature.
result All symmetry breaking operators are differential and sporadic, not obtainable by residue formulas.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Researchers describe local properties of Haantjes operators.
problem Understanding Haantjes operators with vanishing torsion.
method Complete local description of gl-regular Haantjes operators.
result Complete local description of gl-regular Haantjes operators.
We use the spectra of Dirac type operators on the sphere Sn to produce sharp L2 inequalities on the sphere. These operators include the Dirac operator on Sn, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers o…
Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
GNPs learn operators on non-Euclidean geometries using neural networks.
problem Learning operators on complex geometries like manifolds.
method Geometric Neural Operators (GNPs) that incorporate geometric properties.
result GNPs can estimate metrics, solve PDEs, and learn LB operators on manifolds.
Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
Introduces a new operator generating higher Koszul brackets on differential forms.
problem Developing a new operator for higher Koszul brackets on differential forms.
method Introducing a formal ℏ-differential operator Δ generating higher Koszul brackets on differential forms. result Established properties of the introduced BV type operator and its inclusion in a one-parameter family.