Introduction To Combinatorial Morse Theory
Morse theory is an important tool for investigating the topology of smooth manifolds. In this talk we introduce a discrete analog of Morse theory developed by Robin Forman which allows combinatorial analysis and direct algorithms. We present the notion of discrete gradient vector ﬁelds, whose critical elements describe the topology of the structure, and establish connections with other concepts such as spanning tree, matching, evasiveness, and the tree-cotree decomposition. At the end we present applications to data analysis, visualization, and graph theory. No background in classical Morse theory will be assumed.