Combinatorial Vector Fields for Dynamical Systems and Shape Similarity

我的总体长期目标是在多维数据分析和处理的背景下发展计算拓扑。******在过去的十年中，Forman提出的组合向量场的概念，也称为离散向量场（简称DVF），成为计算数学、成像和可视化中连续问题离散化的一个非常成功的工具。我目前的研究计划集中在沿着两个应用轴发展组合向量场：在动力系统和形状识别的研究中。****** DVF在动力系统中的应用将基于Forman的v路径概念的修改，以及我在2014年与Mrozek和Wanner的联合工作中引入的多值映射动力学的使用。这种修改允许研究周期和渐近轨迹，从而获得动力学行为的完整幽灵。我们对洛伦兹吸引子的组合模型表明，dvf也为描述混沌动力学提供了一个很好的框架。主要目标是建立连续和组合矢量场动力学之间的一一对应关系，特别是在Conley指数和Morse分解水平上。我们希望我们的第一篇论文是一个非常有前途的研究方向的起点。****** DVF在形状识别中的应用是我最近与Allili和Landi合作开展的研究的延续。本文的匹配算法是将DVF方法应用于具有多维值的函数的首次尝试，目的是加快多滤波持久同调的计算速度，即由格而不是线性阶滤波的子空间同调系统。这种结构将被用于计算一种精确的形状相似性度量，目前这给计算带来了挑战。该项目涉及计算机编程以及莫尔斯理论对多维函数的适当扩展的理论发展。******最后，我计划与Mischaikow、Mrozek和Wanner合作，编写施普林格于2004年出版的《计算同源学》（Computational Homology）一书的新版本。施普林格的编辑曾多次就第二版的事与我们接洽，但我们推迟了这个项目，因为自2004年以来，这个领域发展得如此之快，这实际上意味着要写一本新书。正如第一版的情况一样，我们期望这本书的写作过程将推动当前研究的进展，并促进许多学生的研究项目

My general long-term goal has been to develop Computational Topology in the context of analysis and processing of multidimensional data.******In the past decade, the concept of combinatorial vector field, also called discrete vector field (abbreviation DVF) introduced by Forman became a highly successful tool for discretization of continuous problems in Computational Mathematics, Imaging and Visualization. My current research proposal is focused on developing combinatorial vector fields along two axes of applications: in dynamical systems and in the study of shape recognition.******The applications of DVF to dynamical systems will be based on a modification of Forman's concept of V-paths and the use of a multivalued mapping dynamics introduced in my joint 2014 work with Mrozek and Wanner. This modification permits studying both periodic and asymptotic trajectories, thus giving access to a full specter of dynamical behavior. Our combinatorial model for the Lorenz attractor shows that DVFs also provide a good framework for describing chaotic dynamics. The primary goal is to establish a one-to-one correspondence between continuous and combinatorial vector field dynamics, in particular on the level of Conley index and Morse decomposition. We expect that our first publication is a starting point for a very promising research direction. ******The applications of DVF to shape recognition are a continuation of my recently initiated research in collaboration with Allili and Landi. Our matching algorithm is the first tentative of adapting the DVF methods to functions with multidimensional values with the goal of speeding up the computation of multi-filtered persistent homology, that is, the subspace homology system filtered by a lattice rather than by a linear order. This structure will in turn be used for computing a refined shape similarity measure, currently bringing computational challenges. This project involves computer programming as well as a theoretical development of an appropriate extension of the Morse theory to multidimensional functions. ******Finally, I plan to work, jointly with Mischaikow, Mrozek, and Wanner, on a new edition of the Computational Homology book published by Springer in 2004. We were approached by Springer editors about the second edition several times but we postponed the project because the field developed so much since 2004 that this will mean writing practically a new book. As it was the case with the first edition, we expect that the process of writing the book will motivate the progress in current research and prompt for many student research projects.**