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Explore the foundations and recent advancements in combinatorial dynamics through this 26-minute lecture on Conley-Morse-Forman theory for generalized combinatorial multivector fields. Delve into the origins of combinatorial approaches to dynamics, starting with Forman's concept of combinatorial vector fields and their applications in data science. Examine the extension of this theory to combinatorial multivector fields, as proposed by M. Mrozek, and discover how these concepts are further generalized to overcome previous limitations. Learn about the shift from Lefschetz complexes to finite topological spaces and the redefinition of dynamics associated with combinatorial multivector fields. Investigate key concepts such as isolated invariant sets, the Conley index, attractors, repellers, and Morse decompositions. Gain insights into the additivity property of the Conley index and Morse inequalities. Conclude by exploring the persistence of Morse decomposition and the analysis of parameter spaces in this cutting-edge field of applied algebraic topology.
