Fiber-Wise Linear Differential Operators

Algebraic fiber spaces and curvature of higher direct images

In this article we are interested in the differential geometric properties of certain higher direct images of exterior powers of the sheaf of relative differentials twisted with a line bundle. We obtain explicit curvature formulas, especially in case where the said line bundle satisfies a natural curvature assumption. Several applications are obtained, including a proof of a result by Viehweg-Zuo in the context of a canonically polarized family of maximal variation.

The Geometry of Fourth Order Differential Operator

In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem is to determine the sufficient and necessary conditions on two fourth order differential operators such that there exists a fiber-preserving transformation mapping one to the other according to gauge equivalence.

对称群不保持纤维的常微分方程

ODEs whose symmetry groups are not fiber-preserving

本文利用李代数分类刻画了对称群不保持纤维的标量与系统常微分方程的全部例外情形，并揭示了活动标架法在欠定系统上的局限性。

Cartan等价方法在五阶微分算子上的应用

Cartan equivalence method on fifth-order differential operator

本文运用Cartan等价方法，解决了直线上五阶微分算子在纤维保持变换下的直接等价问题及规范等价条件问题。

Triole 代数中的微分演算

Differential Calculus in Triole Algebras

本文扩展 diolic 微分演算形式体系，提出 triole 代数以推广至含纤维度量的向量丛，并研究相关微分演算函子，建立保持向量值纤维度量结构丛上微积分的概念框架。

A new construction of strict deformation quantization for Lagrangian fiber bundles

We give a new construction of strict deformation quantization of symplectic manifolds equipped with a proper Lagrangian fiber bundle structure, whose representation spaces are the quantum Hilbert spaces obtained by geometric quantization. The construction can be regarded as a "lattice approximation of the correspondence between differential operators and principal symbols". We analyze the corresponding formal deformation quantization. We also investigate into relations between our construction and Berezin-Toeplitz deformation quantization.

Invariants of symbols of the linear differential operators

In this paper we classify the symbols of the linear differential operators of order $k$, which act from the module $C^\infty(ξ)$ to the module $C^\infty(ξ^t)$, where $ξ\colon E(ξ)\to M$ is vector bundle over the smooth manifold $M$, bundle $ξ^t$ is either $ξ^*$ with fiber $E^*:=\mathrm{Hom}(E,\mathbb{C})$ or $ξ^\flat$ with fiber $E^\flat:=\mathrm{Hom}(E, Λ^n T^*)$ and $C^\infty(ξ)$, $C^\infty(ξ^t)$ are the modules of their smooth sections. To find invariants of the symbols we associate with every non-degenerated symbol the tuple of linear operators acting on space $E$ and reduce our problem to…

纤维束网络：一种几何机器学习范式

Fiber Bundle Networks: A Geometric Machine Learning Paradigm

纤维束网络通过几何优化提升机器学习的可解释性与效率，结合黎曼度量与变分原型优化实现分类任务的清晰几何解释。

Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold

Let $M$ be a smooth manifold, $\cal S$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, $Vect(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on $\cal S$. This cohomology space is closely related to the $Vect(M)$-modules, ${\cal D}_λ(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $λ$.

Noether identities of a generic differential operator. The Koszul-Tate complex

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 condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency condition restarts all the Noether identities characterizing the degeneracy of an original differential operator.

Spectral theory of von Neumann algebra valued differential operators over non-compact manifolds

We provide criteria for self-adjointness and τ-Fredhomness of first and second order differential operators acting on sections of infinite dimensional bundles, whose fibers are modules of finite type over a von Neumann algebra A endowed with a trace τ. We extend the Callias-type index to operators acting on sections of such bundles and show that this index is stable under compact perturbations.

关于余切丛中投影哈密顿轨道交集的研究

On the intersections of projected Hamiltonian orbits in cotangent bundles

本文研究了余切丛中哈密顿轨道投影后的交集行为，证明了在特定条件下交点可被扰动消除，并应用于Reeb流和Finsler流的研究。