淹没函数的拓扑等价性及其在平面上的叶状结构的拓扑等价性：线性类情形

Two dimensional neighborhoods of elliptic curves: formal classification and foliations

We classify two dimensional neighborhoods of an elliptic curve C with torsion normal bundle, up to formal equivalence. The proof makes use of the existence of a pair (indeed a pencil) of formal foliations having C as a common leaf, and the fact that neighborhoods are completely determined by the holonomy of such a pair. We also discussanalytic equivalence and show, for each formal model, that the corresponding moduli space is infinite dimensional.

Total rigidity of generic quadratic vector fields

We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a…

Foliations and Conjugacy:Anosov Structures in the plane

In a non-compact setting, the notion of hyperbolicity, and the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by W. White and P. Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct…

深度线性网络的熵公式

An entropy formula for the Deep Linear Network

本文研究深度线性网络的黎曼几何，以建立学习过程的热力学描述。通过群作用分析过参数化，并利用参数空间到可观测空间的黎曼子流形，定义并计算玻尔兹曼熵。主要技术步骤是利用雅可比矩阵理论显式构造平衡流形的切空间正交基。

奇异德西特环面相对于其光状双分层的刚性

Rigidity of singular de-Sitter tori with respect to their lightlike bi-foliation

本文研究了具有锥形奇点的洛伦兹面的刚性，证明了德西特环面由其光状双分层的拓扑等价类确定。

Integrable embeddings and foliations

A k-submanifold L of an open n-manifold M is called weakly integrable (WI) [resp. strongly integrable (SI)] if there exists a submersion Φ:M\to R^{n-k} such that L\subset Φ^{-1}(0) [resp. L= Φ^{-1}(0)]. In this work we study the following problem, first stated in a particular case by Costa et al. (Invent. Math. 1988): which submanifolds L of an open manifold M are WI or SI? For general M, we explicitly solve the case k=n-1 and provide necessary and sufficient conditions for submanifolds to be WI and SI in higher codimension. As particular cases we recover the theorem of Bouma and Hector (Indag…

紧叶状结构和阿诺索夫流的刘维尔结构的拓扑不变性

Topological invariance of Liouville structures for taut foliations and Anosov flows

研究闭定向三维流形上紧叶状结构相关的刘维尔结构的拓扑不变性，通过特定方法证明其为拓扑不变量，还涉及阿诺索夫流弱叶状结构及轨道等价情况，证明结合平滑方案和沃格尔结果，附录用于构造新例子助于分类。

On a family of unitary representations of mapping class groups

For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group Mod($S$) based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of Mod($S$) on the Teichmüller space of $S$. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.

非交换螺线管的群胚、等价 bibundle 与双模

Groupoids, equivalence bibundles and bimodules for noncommutative solenoids

本文利用群胚作用与等价 bibundle，刻画在 $GL_2(\mathbb{Z}[1/p])$ 线性分式变换下同一轨道参数对应的非交换螺线管之间的 Morita 等价，并显式构造其 imprimitivity 双模。