Computational topololgy and its appliations (topologie computationnelle et ses applications)

我目前的研究兴趣集中在数学的新领域称为计算拓扑。 在这一领域的研究目标是开发一种组合和算法的方法来抽象拓扑结构，目的是将它们应用于科学和工程。 我目前研究的拓扑结构是三次复形、它们的同调以及三次集上连续映射的同调。激发我研究的应用程序分为两类。一个是在非线性分析和动力系统领域，例如计算机辅助动力学证明。这类应用中涉及的主要拓扑工具是同调、不动点指数和Conley指数。另一类应用涉及成像科学领域的数值数据的定性分析和各种工程问题，如材料微观结构中缺陷的自动搜索。在这些应用中涉及的主要拓扑工具是立方和细胞复形，相对同调，经典和离散的莫尔斯理论。

My current research interest is focused on the new domain of mathematics called Computational Topology.  The objective of research in this domain is to develop a combinatorial and algorithmic approach to abstract structures of topology with the aim of applying them to sciences and engineering.  The topological structures I am presently working with are cubical complexes, their homology, and homology of continuous maps on cubical sets. The applications that motivate my research fall into two categories. One is in the area of nonlinear analysis and dynamical systems, for example computer-assisted proofs in dynamics. The main topological tools involved in this category of applications are homology, fixed point index, and Conley index. The other category of applications is related to qualitative analysis of numerical data in the area of imaging science and in a variety of engineering problems such as automated search for defects in the microstructure of materials. The main topological tools involved in those applications are cubical and cellular complexes, relative homology, classical and discrete Morse theory.