Time-Varying Data as Sheaves: an Invitation to Narratives

2026-09-08

AI Summary

本文提出了一种名为叙事的抽象框架，用于处理时间变化的数据，并通过三个案例展示了如何解决信息丢失、数据分解及多智能体系统建模等问题。

Abstract

Modern science and engineering increasingly rely on time-varying data, yet the mathematical tools used to model temporal phenomena are often developed within separate disciplines, obscuring common principles and limiting the transfer of ideas across fields. This chapter presents the theory of narratives, an abstract framework for time-varying objects of any mathematical kind that supports both theoretical investigations and applications. To illustrate this perspective, the chapter develops three vignettes, each illustrating a different research direction. The first addresses a general concern: What information loss can occur when switching between different representations of temporal data? The second concerns structural and algorithmic approaches: How can we systematically decompose time-varying data into simple pieces and obtain invariants describing its structural complexity? The third is an application to control theory: How can we model multi-agent systems with switching communication topologies? More important than any individual vignette, the central message of this invitation is that a suitable abstract perspective can organize and guide research across remarkably diverse mathematical and scientific domains.

Problem

• time-varying data
• narratives
• information loss
• data decomposition
• multi-agent systems

Innovation

• narratives
• time-varying data
• information loss
• structural complexity
• multi-agent systems