同伦距离与类群空间

数字图像中的$m-$同伦距离

$m-$homotopic Distances in Digital Images

本文定义了数字图像中的$m-$同伦距离及其高阶版本，探讨了其与$m-$范畴和$m-$复杂度的关联，并考察了相关概念的同伦不变性。

Lipschitz映射间的离散同伦距离

Discrete homotopic distance between Lipschitz maps

本文提出两个Lipschitz映射间的离散同伦距离$D_r$，通过指定步长$r$有效忽略较小孔洞以实现分类，并证明其推广了离散LS范畴与拓扑复杂度且在离散同伦下不变。

参数化同伦距离

Parametrized homotopic distance

本文提出参数化同伦距离，建立其与纤维截面范畴的关系及上下界，并推广拓扑复杂度与 LS 范畴相关定理。

公理化的截面类别、拓扑复杂度与同调距离

Axiomatic Sectional category, Topological complexity, and Homotopic distance

本文通过三个公理构建了截面类别、拓扑复杂度和同调距离的公理化框架，并引入了统一这些概念的提升类别概念。

等变同伦距离

Equivariant homotopic distance

本文引入等变同伦距离D_G，统一了等变Lusternik-Schnirelmann范畴与拓扑复杂度，并建立其性质、界及在Hopf G-空间和等变纤维化中的应用。

分布截面范畴的新界限及其在分布同伦距离中的应用

New Bounds on Distributional Sectional Category and Applications to Distributional Homotopic Distance

本文针对分布截面范畴（dsecat）建立新界限，证明Jauhari猜想并得到不同系数下的下界，应用于分布同伦距离，推导相关等价表述与新不等式。

Homotopy invariants in small categories

Tanaka introduced a notion of Lusternik Schnirelmann category, denoted $\mathrm{ccat}\, \mathcal{C}$, of a small category $\mathcal{C}$. Among other properties, he proved an analog of Varadarajan's theorem for fibrations, relating the LS-categories of the total space, the base and the fiber. In this paper we recall the notion of homotopic distance $\mathrm{D}(F,G)$ between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, later introduced by us, which has $\mathrm{ccat} \mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},\bullet)$ as a particular case. We consider another particular case, the…

重温相对截面范畴

Relative sectional category revisited

本文探讨了相对截面范畴的扩展及其性质，揭示了其如何统一近期文献中多种同伦数值不变量，如映射拓扑复杂度、相对拓扑复杂度和连续映射的同伦距离。

函数-里普斯复形在持久同伦论中的应用：稳定性与持久拉茨谢夫定理

Function-Rips complexes in persistent homotopy theory: Stability and persistent Latschev theorems

本文研究了函数-里普斯复形在持久同伦论中的稳定性及持久拉茨谢夫定理，通过结合固定尺度的里普斯复形与底层空间的滤波数据，推广了拉茨谢夫和哈斯曼的经典结果到具有滤波函数的空间中。

关于连续距离的计算（同调）方法

A Computational (Co)homological Approach to Contiguity Distance

本文提出了一种基于同调的计算方法，用于计算连续映射之间的距离，并通过具体计算验证了该方法的有效性。

The Persistent Homotopy Type Distance

We introduce the persistent homotopy type distance dHT to compare real valued functions defined on possibly different homotopy equivalent topological spaces. The underlying idea in the definition of dHT is to measure the minimal shift that is necessary to apply to one of the two functions in order that the sublevel sets of the two functions become homotopically equivalent. This distance is interesting in connection with persistent homology. Indeed, our main result states that dHT still provides an upper bound for the bottleneck distance between the persistence diagrams of the intervening funct…

An infinite-dimensional phenomenon in finite-dimensional metric topology

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function…

等变相对截面范畴及诱导不变量

Equivariant Relative Sectional Category and Induced Invariants

本文针对G-映射引入等变相对截面范畴，建立其同伦性质，将其应用于定义映射拓扑复杂度等变类似物等，证明相关不变量扩展了经典等变概念。

Persistence-like distance on Tamarkin's category and symplectic displacement energy

We introduce a persistence-like pseudo-distance on Tamarkin's category and prove that the distance between an object and its Hamiltonian deformation is at most the Hofer norm of the Hamiltonian function. Using the distance, we show a quantitative version of Tamarkin's non-displaceability theorem, which gives a lower bound of the displacement energy of compact subsets of cotangent bundles.

局部紧群的几何不变量：同调视角

Geometric invariants of locally compact groups: the homological perspective

本文针对局部紧Hausdorff群构建Σ-理论的同调版本，建立其与同伦版本的关联，给出CPₘ型与同调局部紧Σᵐ的判定准则，并分析短正合列下的Σ-理论性质。

函数空间的持续同调

Persistent homology of function spaces

本文利用持续同调研究两个流形间映射空间上Lipschitz常数作为高度函数的拓扑结构，探讨同伦路径及高维循环是否必须经过高Lipschitz常数的映射，并给出初步结果。