Conley-Morse Chain Maps

Conley index;Morse theory;Data analysis

Moeller, Todd Keith ; Mathematics

University:Georgia Institute of Technology

Department:Mathematics

关键词: Conley index;    Morse theory;    Data analysis;   

Others  :  https://smartech.gatech.edu/bitstream/1853/7221/1/moeller_todd_k_200508_phd.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

We introduce a new class of Conley-Morse chain maps for the purpose ofcomparing the qualitative structure offlows across multiplescales. Conley index theory generalizes classical Morse theory as a tool for studying the dynamics of flows.The qualitative structure of a flow, given a Morse decomposition, can be stored algebraically as a set of homology groups (Conley indices) and a boundary map between the indices (a connection matrix).We show that as long as the qualitative structures of two flows agree on some, perhaps coarse, level we can construct a chain map between the corresponding chain complexes that preserves the relations between the (coarsened) Morse sets.We present elementary examples to motivate applications to data analysis.