Georgia Institute of Technology College of Sciences

Morse theory establishes a celebrated link between classical gradient dynamics and the topology of the
underlying phase space. It provided the motivation for two independent developments. On the one hand, Conley's
theory of isolated invariant sets and Morse decompositions, which is a generalization of Morse theory, is able
to encode the global dynamics of general dynamical systems using topological information. On the other hand,
Forman's discrete Morse theory on simplicial complexes, which is a combinatorial version of the classical
theory, and has found numerous applications in mathematics, computer science, and applied sciences.
In this talk, we introduce recent work on combinatorial topological dynamics, which combines both of the
above theories and leads as a special case to a dynamical Conley theory for Forman vector fields, and more
general, for multivectors. This theory has been developed using the general framework of finite topological
spaces, which contain simplicial complexes as a special case.