Defines fiber-wise linear differential operators on vector bundles.
problem No specific problem stated; focuses on definition and equivalence.
method Definition and equivalence of fiber-wise linear differential operators to derivations of line bundles.
result Equivalence of fiber-wise linear differential operators to derivations of line bundles.
This paper studies how key tensor properties are inherited in subtensors of tensor train decompositions.
problem Theoretical development of property inheritance for subtensors in tensor train decompositions.
method Theoretical analysis of incoherence and condition number preservation, and tensor train rank preservation through fiber-wise sampling.
result Key tensor properties (incoherence and condition number) can be well preserved to subtensors formed via fiber-wise sampling.
In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsi…
We examine the relationships between the differential invariants of objects and of their images under a surjective map. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably…
New method for tensor completion from specific mode observations.
problem Recovering multiway data tensors from partial observations.
method Tensor train decomposition for fiber-wise observations.
result Deterministic recovery guarantees for specific observation patterns.
In this paper we investigate fiber-wise linear complex Banach sub-Poisson structures defined canonically by the structure of a W*-algebra M. In particular we show that these structures are arranged in the short exact sequence of complex Banach sub-Poisson VB-groupoids with the groupoid of partially invertible elements …
We prove an analogue of Thurston's h-principle for 2-dimensional foliations on manifolds of dimension bigger or equal to 4, in the presence of a fiber-wise non-degenerate 2-form. This helps us understand the flexibility of rank 2 regular Poisson structures on open manifolds with dimension bigger or equal to 4…
In this paper we define K-theoretic secondary invariants attached to a Lie groupoid G. The K-theory of Cr∗(Gad0) (where Gad0 is the adiabatic deformation G restricted to the interval [0,1)) is the receptacle for K-theoretic secondary invariants. We give a Lie groupoid version of construction given b…
Given a Kähler fiber space p:X→Y whose generic fiber is of general type, we prove that the fiberwise singular Kähler-Einstein metric induces a semipositively curved metric on the relative canonical bundle KX/Y of p. We also propose a conjectural generalization of this result for relative twisted Kähler-Eins…
Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant derivative compatible with a given fiber-wise metric. Imposing further conditions resembling standard Levi-Civita connect…
Visible Lagrangians in Hitchin systems are studied for pillowcase covers.
problem Visible Lagrangians in Hitchin systems intersecting non-trivially.
method Computation of Fourier-Mukai transforms and study of mirror dual branes.
result Mirror dual branes are closely related to Hausel's toy model.
Study of intersections in Hamiltonian orbits on cotangent bundles.
problem Understanding intersections of projected Hamiltonian orbits in cotangent bundles.
method Generic submersive level set analysis, multi-jet transversality theorem.
result Projected Hamiltonian orbits have discrete intersections, which can be perturbed away under certain conditions.
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 …
Equivalence of second order differential operators in vector bundles studied.
problem Equivalence problem for second order linear differential operators in vector bundles.
method Description of rational invariants of symbols, finding connections associated with differential operators.
result Solving problems of local and global equivalency of differential operators.
The paper examines scalar fourth-order linear differential operators and their invariants.
problem Equivalence problem of scalar fourth-order linear differential operators.
method Investigation of differential invariants.
result Application of differential invariants to the equivalence problem.
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.
Examines how first-order differential operators can be equivalently transformed.
problem Equivalence of first-order linear differential operators.
method Discussion of equivalency transformations.
result Explains how first-order differential operators can be transformed equivalently.
Study natural invariants for differential operators, simplifying their equivalence problem.
problem Equivalence problem of nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to simplify differential operator equivalence.
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.
Study natural invariants for third order nonlinear operators on 2D manifolds.
problem Equivalence problem of third order nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to equivalence problem.
Study differential operators over maps and their applications in supermanifolds.
problem Understanding differential operators over smooth maps and their applications.
method Recall and study differential operators, formal ℏ-differential operators, pullbacks by thick morphisms, and quantization of symplectic micromorphisms. result Developed constructions and examples of differential operators over maps.
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…
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
The study examines differential singularities in 3D Nijenhuis operators.
problem Characterizing differential singularities in three-dimensional Nijenhuis operators.
method Examined cases of proportional differentials of invariants and fold-type singularities.
result New examples of Nijenhuis operators with specified singularities constructed.
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.
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.
We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.
We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of authomorphisms.
In this paper, we study eigenvalues of the closed eigenvalue problem of the differential operator L, which is introduced by Colding and Minicozzi in [4], on an n-dimensional compact self-shrinker in Rn+p. Estimates for eigenvalues of the differential operator L are obtained. Our estimates for eigenvalues…
Paper introduces magnetic Steklov operator on differential forms and its properties.
problem Analyzing the boundary value problem of magnetic Steklov operator.
method Introduced magnetic Steklov operator and proved its well-posedness. Also, computed spectral properties.
result An analogue of Diamagnetic Inequality does not always hold for magnetic Steklov operators.
The paper extends Lie bracket to noncommutative geometry using differential operators.
problem Generalizing Lie bracket to noncommutative geometry.
method Antisymmetrizing compositions of vector fields and treating symbols of differential operators.
result Provided necessary and sufficient conditions for jet modules to represent differential operators.
Given a generic Lagrangian system, its Euler-Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condi…
Lecture notes introduce differential geometry using sheaves and differential operators.
problem Exploring differential geometry concepts.
method Using sheaves, differential operators, and horizontal subbundles.
result Presented an approach to fundamental differential geometry structures.
The paper studies differential operator invariants and equivalence under Lie pseudogroups.
problem Understanding invariants and equivalence of differential operators under Lie pseudogroups.
method Analysis of invariants, use of n-invariants, and application of local symplectomorphisms as an example.
result Normal forms and solutions to equivalence problems for differential operators.
The paper defines and analyzes abla-Sobolev spaces and operators on manifolds.
problem Defining and analyzing Sobolev spaces and differential operators on manifolds.
method Coordinate-free approach using connections, proving properties of abla-Sobolev spaces and operators. result Equivalent definitions of abla-Sobolev spaces and operators under certain conditions. We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result i…
We present Bernstein-Sato identities for scalar-, spinor- and differential form-valued distribution kernels on Euclidean space associated to conformal symmetry breaking operators. The associated Bernstein-Sato operators lead to partially new formulae for conformal symmetry breaking differential operators on functions, …
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.
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. ICON learns differential equation operators from examples, revealing probabilistic inference.
problem Learning operators for differential equations from limited examples.
method Probabilistic operator learning using ICON architectures trained on diverse datasets.
result ICON implicitly performs Bayesian inference on solution operators.
In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…
Paper solves equivalence problems for fifth-order differential operators using Cartan's method.
problem Equivalence problem for fifth-order differential operators under fiber-preserving transformations.
method Cartan's method of equivalence applied to solve two versions of the equivalence problem.
result Sufficient and necessary conditions for fiber-preserving transformations between fifth-order differential operators.
This paper deals with sheaves of differential operators on noncommutative algebras. The sheaves are defined by quotienting a the tensor algebra of vector fields (suitably deformed by a covariant derivative) to ensure zero curvature. As an example we can obtain enveloping algebra like relations for Hopf algebras with di…
Study linear differential operators on special manifolds.
problem Analyzing elliptic differential operators on specific types of manifolds.
method Examining a linear elliptic differential operator of the form Δ + V - λ on quasi-asymptotically conical manifolds.
result Established an isomorphism theorem for these operators.
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
ICON learns differential equation operators from prompts, reducing retraining and improving few-shot learning.
problem Training neural networks to solve differential equations without retraining for new problems.
method In-Context Operator Networks (ICON) that learns operators from prompted data and applies them to new problems.
result ICON can generalize to new operators beyond the training distribution and requires only a few demos.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
We study the duality between M-theory on compact holonomy G2-manifolds and the heterotic string on Calabi-Yau three-folds. The duality is studied for K3-fibered G2-manifolds, called twisted connected sums, which lend themselves to an application of fiber-wise M-theory/Heterotic Duality. For a large class of such G2-man…