Combinatorial Vector Fields for Dynamical Systems and Shape Similarity

我的总体长期目标是在分析和处理多维数据的背景下开发计算拓扑学。 在过去的十年里，由Forman提出的组合向量场的概念，也被称为离散向量场(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开发组合矢量场：在动力系统中和在形状识别研究中。 DVF在动力系统中的应用将基于对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。这一修正允许研究周期轨迹和渐近轨迹，从而获得动力学行为的全部幽灵。我们的Lorenz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。其主要目标是在连续向量场动力学和组合向量场动力学之间建立一一对应关系，特别是在Conley指数和Morse分解的水平上。我们希望我们的第一份出版物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合作开始的研究的继续。我们的匹配算法首次尝试将DVF方法应用于具有多维值的函数，目的是加速多过滤持久同调的计算，即按格而不是按线性阶过滤的子空间同调系统。这种结构将被用来计算精细化的形状相似性度量，目前这带来了计算挑战。该项目涉及计算机编程以及适当地将莫尔斯理论扩展到多维函数的理论发展。 最后，我计划与Mischaikow、Mrozek和Wanner合作，于2004年出版由Springer出版的计算同源一书的新版。施普林格的编辑多次就第二版与我们接洽，但我们推迟了项目，因为自2004年以来，该领域取得了很大发展，这实际上意味着要写一本新书。正如第一版的情况一样，我们预计这本书的写作过程将推动当前研究的进步，并促使许多学生研究项目。

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.