Formality of the constructible derived category for spheres: A combinatorial and a geometric approach

关于$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间的形式化性质

On the Formality of Configuration Spaces of $\mathbb{R}^{n'} \times \mathbb{C}^{n}$

本文针对$\boldsymbol{\reals^{n'} \times \boldsymbol{\reals^{n}}}$的构形空间，通过定义可构造de Rham-Dolbeault上同调理论，分类其形式化性质并给出对应CDGA模型，还将其应用于拓扑-全纯场论的局部算子代数研究。

A combinatorial-topological shape category for polygraphs

We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-be…

A derived isometry theorem for constructible sheaves on $\mathbb{R}$

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in this derived setting, thus expressing the convolution distance of sheaves as a matching distance between combinatorial objects associated to them that we call graded barcodes. This allows to consider sheaf-theoretical constructions as combinatorial, stable topological descriptors of data, and generalizes the situation of persistence with one parameter. To ac…

On locally constructible spheres and balls

Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from …

The stratified homotopy type of the reductive Borel-Serre compactification

We identify the exit path $\infty$-category of the reductive Borel-Serre compactification as the nerve of a $1$-category defined purely in terms of rational parabolic subgroups and their unipotent radicals. As an immediate consequence, we identify the fundamental group of the reductive Borel-Serre compactification, recovering a result of Ji-Murty-Saper-Scherk, and we obtain a combinatorial incarnation of constructible complexes of sheaves on the reductive Borel-Serre compactification as elements in a derived functor category.

图配置空间、合同腺与形式性

Graphical configuration spaces, Contractads and Formality

本文研究图配置空间的合同腺结构，证明其在特定条件下形式性，并探讨了循环图的共形式性障碍及二维合同腺的细胞结构。