两层分层空间中的邻域

分层空间和点的构形空间的谱序列

A spectral sequence for stratified spaces and configuration spaces of points

构造了一个与分层空间相关联的谱序列，利用闭层的紧支上同调群和层偏序集的约化上同调群计算开层的紧支上同调群.几种常见的谱序列是作为特例出现的。该结构是层理论的，适用于拓扑空间和代数簇的平展上同调。作为应用，我们证明了点配置空间的一个非常一般的表示稳定性定理。

切向条件下的分层莫尔斯理论

Stratified Morse Theory with Tangential Conditions

我们的目标是发展一个切向条件下的分层莫尔斯理论。利用梯度向量场上的控制条件和径向假设，在抽象分层空间上定义了一个连续的分层光滑Morse函数。对于Morse函数的临界点，可以证明局部不稳定集是流形，而局部独立集本身是抽象分层空间。我们还给出了梯度动力学在临界点附近的规范形。对于满足一般Morse-Smale条件的分层Morse对，我们可以构造Morse-Witten复形，并证明其同调等价于空间的奇异同调.

分层同伦论中的有限性与有限支配

Finiteness and finite domination in stratified homotopy theory

本文利用层与同伦链刻画带保守函子的 ∞-范畴紧性和有限性，证明锥形及锥形光滑分层空间出口路径范畴的相应有限性定理，并以 Quinn 的 C^0 反例说明拓扑情形失效。

分层辫群：单值性

Stratified braid groups: monodromy

本文通过将对数导数关联的无限面积平移曲面引入新方法，确定了首一无平方因子复多项式空间分层在辫群中的单值性，并揭示其由水平叶状结构诱导的圆盘框架所支配。

Equivariant control data and neighborhood deformation retractions

In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group $G$ which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a $G$-stratified space carries a system of $G$-equivariant control data. As an application, we show that if $A \subset X$ is a closed $G$-stratified subspace which is a union of strata of $X$, then the inclusion $i : A \hookrightarrow X$ is a $G$-equivariant cofibration. In particular, this theorem applies whenever $X$ is a $G$-invariant analytic subspace of an analytic $G$-manifold $M$ and …

实多项式的分层空间与遍历流的轨迹空间

The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

本文是研究Morse理论、梯度流、光滑紧致流形的复杂性的系列文章的第三篇。利用\cite{K2}中的局部解析模型，对于$(n+1)$ -流形$X$上的\emph{traversally generic flows}，我们对它们关于边界$\d X$的切向模式的泛组合学进行了详细而有些繁琐的研究.这个组合是由一个通用偏序集$Ω^\bullet_{'\langle n]}$，它只依赖于$X$的尺寸。它与不超过$2(n+1)$的单变量次数实多项式的实因子的组合模式密切相关。这种模式是另一个自然偏序集$Ω_{\langle 2n+2]}$的元素，它描述了实根在实多项式变形下合并、分裂、出现和消失的方式。实次数$d$多项式$\mathcal P^d$的空间是分层的，使得其纯层是由偏序集$Ω_{\langle d]}$的元素标记的细胞。$\mathcal P^d$中的这种细胞结构本身就很有趣（参见定理\ref {th4.1}和定理\ref {th4.2}）。此外，它还有助于理解遍历泛域$v$的轨迹空间$\mathcal T(v)$的\emph{localized}结构，这是定理\ref {th5.2}和定理\ref {th5.3}的主要内容.

惠特尼范畴与缠结假说

Whitney categories and the Tangle Hypothesis

我们提出了一个新的概念，“n-范畴与对偶”，我们称之为惠特尼n-范畴。有两个动机。第一，贝兹和多兰的缠结假设（几乎）重言式时，解释为一个关于惠特尼范畴的声明。第二个是我们可以从每个光滑分层空间X中函式地构造“基本Whitney n-范畴”。这些是通过考虑光滑映射到X的同伦理论得到的，X是所有层的横截。这使得具体的另一个想法贝兹和多兰的这是一个合适的版本同伦理论分层空间应允许一个概括之间的关系空间和groupoid一个分层空间和类别之间的联系。

Witten's perturbation on strata with general adapted metrics

Let $M$ be a stratum of a compact stratified space $A$. It is equipped with a general adapted metric $g$, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, $g$ has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then $g$ is called good. We consider the maximum/minimun ideal boundary condition, $d_{\text{\rm max/min}}$, of the compactly supported de~Rham complex on $M$, in the sense of Brüning-Lesch. Let $H^*_{\text{\rm max/min}}(M)$ and $Δ_{\text{\rm max/min}…

Continuity argument revisited: geometry of root clustering via symmetric products

We study the spaces of polynomials stratified into the sets of polynomial with fixed number of roots inside certain semialgebraic region $Ω$, on its border, and at the complement to its closure. Presented approach is a generalisation, unification and development of several classical approaches to stability problems in control theory: root clustering ($D$-stability) developed by R.E. Kalman, B.R. Barmish, S. Gutman et al., $D$-decomposition(Yu.I. Neimark, B.T. Polyak, E.N. Gryazina) and universal parameter space method(A. Fam, J. Meditch, J.Ackermann). Our approach is based on the interpretatio…