From Discrete Morse Theory to Combinatorial Topological Dynamics

BARMAK, JONATHAN ARIEL; THOMAS WANNER

Paris

Conferencia; GETCO 2022 ? 11th International Conference on Geometric and Topological Methods in Computer Science; 2022

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 tutorial, 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.The tutorial consists of two parts:The first lecture provides a review of Forman’s discrete Morse theory, and describes some of its applications. We also provide an introduction to the theory of finite topological spaces, its connections to posets, simplicial complexes, and their homotopy properties.The second lecture introduces the concept of multivector fields on finite topological spaces and presents the main ingredients of Conley’s theory, such as isolated invariant sets, their Conley index, and Morse decompositions. We also touch upon the relation between this combinatorial theory and its classical counterpart. Time permitting, we address applications of the theory to establishing recurrence in classical dynamics, and describe advanced topics such as connection matrices.