Abstract:
There is canonical partition of set of critical values of smooth
function into pairs "birth-death" and a separate set representing basis
in homology, as was shown in the speaker's paper in 1994. This
partition arises from bringing Morse complex, defined by gradient
trajectories of the function, to so called "canonical form" by a linear
transform respecting the filtration given by order of the critical
values. These "canonical forms" are combinatorial invariants of
filtered complexes. Starting from the beginning of 2000s these
invariants became widely used in applied mathematics under the name of
"Persistence diagrams" and "Persistence Bar-codes". Currently there are
over 400 scientists working on applications of these invariants in
different domains ranging from biology and medicine to artificial neural
nets. The talk is devoted to these invariants and their applications in
mathematics and data analysis.