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

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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.