Аннотация:
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.
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