Generalized chessboard complexes and discrete Morse theory

凸多面体上的Morse-Smale复形

Morse-Smale complexes on convex polyhedra

受地貌学应用驱动，本文将Morse-Smale理论拓展至定义三维凸多面体的径向距离函数，分析了其与两类相关Morse-Smale复形的异同，并设计实现了相应的显式计算算法。

The Morse theory of Čech and Delaunay complexes

Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Čech, Delaunay-Čech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.

图的幅度同调与 Asao-Izumihara 复形上的离散 Morse 理论

Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes

本文研究 Asao-Izumihara 复形的同伦型与图幅度同调可对角化性的关系，证明 pawful 图的该复形同伦等价于球面的一点并，并推广 pawful 图概念，找到新的直径为 2 的非 pawful 可对角化图。

字长、莫尔斯理论与 Vietoris-Rips 复形

Word length, Morse theory, and Vietoris-Rips complexes

本文提出离散莫尔斯理论方法证明 Vietoris-Rips 复形的高连通性或可收缩性，将其应用于直角 Artin 群的复形可收缩性证明，还给出群有限呈现的充分条件。

Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes

We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric gr…

The geometry of genericity in mapping class groups and Teichmüller spaces via CAT(0) cube complexes

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures a notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We prove that their sublinearly Morse boundaries are visibility spaces and admit continuous equivariant injections into the boundary of the…