Topological invariants of generic flows on compact surfaces and topological flow data analysis

Fluid phenomena is one of important topics in not only science but also the real world. However, the lack of common language among researches was a barrier to proceed interdisciplinary researches. Even it is still hard to describe the slices of flows in $3$-dimensional manifolds (e.g. blood currents) in common words. Therefore we have developed a new classification theory for flows on surfaces by making use of topology. To convert real/numerical input data of $2D$ incompressible flows to words, we have constructed topological classification of such flows and implemented a software using a persistent homology and computer science techniques. Integrable Hamiltonian systems on spheres with finitely many punctures were classified by a complete invariant and the invariant was implemented as a software using a persistent homology. In particular, several real data are described and analyzed without ambiguity by our methods. Moreover, all generic transitions are listed up. In this talk, we introduce the generalization of the finite complete invariant for a generic flows on spheres with punctures. Note that the set of generic flows contains almost all Integrable Hamiltonian systems and all Morse–Smale systems. If time allows, we explain how complete invariants are useful to analyze fluid phenomena.