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.
The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.
problem Classifying real hypersurfaces with a particular Jacobi operator.
method Introducing and classifying real hypersurfaces with a quadratic Killing structure Jacobi operator.
result A classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator.
Researchers create compatibility complexes for Einstein metrics.
problem Constructing solutions to the conformal-to-Einstein operator.
method Using a method that leverages symmetries and geometric properties.
result At most one independent solution exists under genericity assumptions.
Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation…
This is the second part in a series of two papers. The k-Dirac complex is a complex of differential operators which are natural to a particular ∣2∣-graded parabolic geometry. In this paper we will consider the k-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove …
Koopman operator theory simplifies complex systems analysis.
problem Analyzing nonlinear dynamical systems and complex networks.
method Estimating Koopman operator from data to reveal system properties.
result Koopman operators provide insights into system characteristics.
Formula for satellite operators using knot Floer homology.
problem Computing knot Floer homology for satellite knots.
method Using Heegaard Floer Dehn surgery formulas and formal knot Floer complexes.
result Formulas to compute knot Floer homology for satellite knots.
The Plebański complex is a differential operator that squares to the Laplacian and is composed of two Dirac operators.
problem The Plebański complex studies the linearization of equations for hyper-Kähler manifolds.
method Defined and studied properties of the Plebański complex, showing it fits into the elliptic complex framework.
result The Plebański complex is an elliptic differential operator that squares to the Laplacian and is composed of two Dirac operators.
Paper proves non-existence of certain hypersurfaces in complex quadric.
problem Non-existence of Hopf real hypersurfaces with parallel normal Jacobi operator.
method Introducing C-parallel and Reeb parallel normal Jacobi operators, proving non-existence theorems. result Non-existence of Hopf real hypersurfaces with C-parallel normal Jacobi operator. We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacob…
The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.
problem Characterizing Hopf real hypersurfaces with commuting Jacobi operators.
method Investigating the commuting property between normal and structure Jacobi operators.
result A remarkable classification of Hopf real hypersurfaces in the complex quadric with commuting Jacobi operators.
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
Study the boundary operator property on simplicial complexes, proving essential properties for Hodge theory.
problem Characterize the boundary operator property ∂∂=0 on simplicial complexes. method Characterization in ℓ2 terms of recurrence of links, defining relative cohomology, and proving harmonic eigenforms. result Essential properties for Hodge theory, including weak decomposition and existence of harmonic eigenforms.
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established C1,1-regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue. result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
Abstract: Study Kähler identities on almost complex manifolds.
problem Generalize Kähler identities to almost complex manifolds.
method Study via Clifford bundle and Dirac operator, then translate results to exterior bundle.
result Obtain generalization of Kähler identities for compact almost complex manifolds.
The study shows conditions for Kähler manifolds to have rational cohomology of complex projective space.
problem Conditions for Kähler manifolds to have rational cohomology of complex projective space.
method Analyzing the Calabi curvature operator and its positivity conditions.
result Compact Kähler manifolds with specific curvature conditions have rational cohomology of complex projective space.
The paper constructs a complex for the Dirac operator in 4 dimensions.
problem Constructing a complex for the Dirac operator in 4 dimensions.
method Using the Penrose transform, the paper constructs a relative BGG complex and its direct image.
result An explicit construction of a complex starting with the Dirac operator in any number of variables.
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.
Let (X,h) be a compact and irreducible Hermitian complex space of complex dimension v>1. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…
New operators help focus on specific areas in complex math problems.
problem Concentration in complex mathematical structures.
method Construct conjugate-linear perturbations of twisted spinc Dirac operators using the conjugate-linear Hodge star operator.
result These perturbations satisfy the concentration principle.
We prove that there does not exist any real hypersurface in complex Grassmannians of rank two with semi-parallel structure Jacobi operator. With this result, the nonexistence of real hypersurface in complex Grassmannians of rank two with recurrent structure Jacobi operator is proved.
PCA-Net combines PCA and neural networks for operator approximation, with new bounds on complexity.
problem Developing approximation theory for PCA-Net architecture.
method Combines PCA and neural networks, derives universal approximation results and lower bounds on complexity.
result PCA-Net can overcome the curse of parametric complexity for specific 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.
Let J be a unitary almost complex structure on a Riemannian manifold (M,g). If x is a unit tangent vector, let P be the associated complex line spanned by x and by Jx. We show that if (M,g) is Hermitian or if (M,g) is nearly Kaehler, then either the complex Jacobi operator (JC(P)y=R(y,x)x+R(y,Jx)Jx) or the complex curv…
New Kähler manifolds found with nonpositive curvature operators.
problem Rigidity of curvature operators in Kähler manifolds.
method Proved existence of Kähler manifolds with specific curvature properties.
result Kähler manifolds can have nonpositive curvature operators, unlike quaternionic and Cayley hyperbolic manifolds.
Near-optimal rates for multi-task learning with shared representations.
problem Approximation and statistical complexity of learning multiple operators.
method Multiple Neural Operators (MNO) architecture and comparison with DeepONet.
result Near-optimal upper and lower bounds for approximation and generalization.
Study classifies real hypersurfaces in complex projective spaces based on Lie derivatives and structure Jacobi operator properties.
problem Classifying real hypersurfaces based on Lie derivatives and structure Jacobi operator properties.
method Defined a tensor field RξT(k) from structure Jacobi operator Rξ and Lie derivative, and studied its symmetry and skew-symmetry. result Obtained classifications of real hypersurfaces for which RξT(k) is either symmetric or skew symmetric. A notion of up and down Grover walks on simplicial complexes are proposed and their properties are investigated. These are abstract Szegedy walks, which is a special kind of unitary operators on a Hilbert space. The operators introduced in the present paper are usual Grover walks on graphs defined by using combinatoria…
New methods solve MI problems with locally Lipschitz operators, improving solution efficiency.
problem Solving monotone inclusions with locally Lipschitz continuous operators.
method Primal-dual extrapolation methods using backtracking line search.
result Improved operation complexity for solving MI problems.
We prove several approximation theorems of the complex Monge-Ampere operator on a compact Kahler manifold. As an application we give a new proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex Monge-Ampere operator in the class of w-plurisubharmonic functions with vanishing…
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…
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
In this note, we prove that the $\pd$- and $\barpd$-operators introduced by Gualtieri for a generalized complex structure coincide with the $\bdees$- and $\bdel$-operators introduced by Alekseev-Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
Study bounds Rademacher complexity of Fourier neural operators.
problem Bounding Rademacher complexity for Fourier neural operators.
method Investigated using specific group norms and capacity.
result Inferred that group norms determine model information.
We show that the exterior derivative operator on a symplectic manifold has a natural decomposition into two linear differential operators, analogous to the Dolbeault operators in complex geometry. These operators map primitive forms into primitive forms and therefore lead directly to the construction of primitive cohom…
We use mathematical induction to prove that the horizontal composition in the class of coherently diagonal complexes is indeed a binary operation. That is to say, the embedding of two coherently diagonal complexes in an alternating planar diagram produces a coherently diagonal complex.
Proves the Hodge conjecture for complex projective manifolds.
problem Proving the Hodge conjecture for complex projective manifolds.
method Utilizing the Dirac-Dolbeault operator and Nash-Moser generalized inverse function theorem.
result Existence of complex submanifolds whose fundamental classes span rational Hodge classes.
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
Estimates for complex Hessian equations on Hermitian manifolds.
problem Establishing estimates for solutions to complex Hessian equations.
method Using concavity inequality for complex sum-of-Hessian operators.
result Second-order estimates for admissible solutions on Hermitian manifolds.
The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassm…
We show that elliptic complexes of (pseudo)differential operators on smooth compact manifolds with boundary can always be complemented to a Fredholm problem by boundary conditions involving global pseudodifferential projections on the boundary (similarly as the spectral boundary conditions of Atiyah, Patodi and Singer …
Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.
problem Analyzing complex structures with perturbed differential operators.
method Perturbing the standard differential operator to a first-order operator Dη and computing Bochner-Kodaira-Nakano-type formulae. result Obtained vanishing results for certain harmonic spaces and Dolbeault cohomology.
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.
First we introduce the notion of parallel structure Jacobi operator for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2 . Next we give a complete classification of real hypersurfaces in Qm=SOm+2/SOmSO2 with parallel structure Jacobi operator.
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, RξφT=TRξφ, where T=A or T=S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…
In this paper, we introduce new notions of semi-parallel shape operators and structure Jacobi operators in complex two-plane Grassmannians G2(Cm+2). By using such a semi-parallel condition, we give a complete classification of Hopf hypersurfaces in G2(Cm+2).