Combinatorial Vector Fields for Dynamical Systems and Shape Similarity

我的总体长期目标是在多维数据的分析和处理的背景下开发计算拓扑。 在过去的十年中，由Forman提出的组合向量场，又称离散向量场（DVF）的概念成为计算数学、图像学和可视化领域中连续问题离散化的一个非常成功的工具。我目前的研究计划是集中在发展组合向量场沿着两个轴的应用：在动力系统和形状识别的研究。 DVF在动力系统中的应用将基于对Forman的V-路径概念的修改，以及我与Mrozek和Wanner在2014年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性和渐近轨迹，从而获得一个完整的幽灵的动力学行为。我们的Lorenz吸引子的组合模型表明，DVF也提供了一个很好的框架来描述混沌动力学。主要目标是建立一个一对一的连续和组合向量场动力学之间的对应关系，特别是在康利指数和莫尔斯分解的水平。我们希望我们的第一篇文章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合作开展的研究的继续。我们的匹配算法是第一次尝试适应的DVF方法的功能与多维值的目标是加快计算的多过滤持久的同源性，即，子空间的同源系统过滤的格，而不是由一个线性顺序。这种结构将反过来用于计算一个细化的形状相似性度量，目前带来的计算挑战。这个项目涉及计算机编程以及理论发展的适当扩展的莫尔斯理论的多维功能。 最后，我计划与Mischaikow、Mrozek和Wanner合作，编写Springer于2004年出版的《计算同源性》一书的新版本。斯普林格的编辑们曾多次就第二版与我们接洽，但我们推迟了这个项目，因为自2004年以来，该领域发展得太快了，这实际上意味着要写一本新书。正如第一版的情况一样，我们希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.