三角化奇异空间上的通用拓扑统计
Universal topological statistics on triangulated singular spaces
AI总结：
该研究针对一类可三角化空间证明了随机持续同调的普适性定理，得到与空间及概率密度无关的期望持续比测度普适极限，还将方法推广至尺度不变泛函，适用于多种非欧空间。
AI中文摘要：
我们证明了一类可三角化空间上随机持续同调的普适性定理。更准确地说，设 $M \subset \mathbb{R}^D$ 是满足几何质量条件的紧 $C^2$ 可三角化空间，$f: M \to \mathbb{R}$ 是概率密度，则由强度为 $nf$ 的泊松点过程的Čech或Vietoris-Rips复形计算得到的期望持续比测度，存在与 $(M, f)$ 无关的普适极限。由于光滑流形、代数簇、半代数集及Whitney分层空间均为可三角化空间，该定理适用于大量非欧空间。除持续同调外，证明还涵盖一类尺度不变泛函，其依赖几何转移方法，通过Freudenthal-Kuhn三角剖分的逐次逼近将欧氏空间的构造适配到三角化空间，并控制奇异层间的干扰。
英文摘要：
We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M \subset \mathbb{R}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f: M \to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the Čech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M, f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.
