从拓扑学角度出发的Helly型问题

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tildeβ_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tildeβ_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These …

重叠-Helly定理

Overlap-Helly theorems

本文提出Helly定理的推广，与Bárány-Gromov重叠定理紧密相关，主要结果蕴含拓扑彩色Helly定理和拓扑中心点定理，并进一步证明稠密复形的重叠定理。

拓扑绘图与凸几何经典定理的相遇

Topological Drawings meet Classical Theorems from Convex Geometry

本文研究凸几何经典定理在拓扑绘图中的应用，提出了纯组合性质的 Kirchberger 定理推广，引入广义 signotopes 结构，并展示了具有任意大 Helly 数的拓扑绘图及 Carathéodory 定理的新证明。

Sheaf-Theoretic Stratification Learning from Geometric and Topological Perspectives

In this paper, we investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given i…

从树状图和梯度采样角度分析循环社区

Cycles Communities from the Perspective of Dendrograms and Gradient Sampling

本文提出通过树状图和梯度采样方法构建循环社区，结合层次聚类和拓扑优化框架，实现对不同拓扑对象中循环特征的识别与比较。

A Geometric Perspective on Sparse Filtrations

We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any convex metric. We also give an algorithm for finding the simplices in such a filtration and prove that the vertex removal can be implemented as a sequence of elementary edge collapses.